The Acoustic Representation of Bending Waves M.C. Brink

M.Sc. Thesis

Supervisor:

Dr.ir. D. de Vries

Laboratory of Acoustic Imaging and Sound Control Delft University of Technology

i

Delft, November 2002

 Copyright 2002 the Laboratory of Acoustic Imaging and Sound Control All rights reserved. No parts of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission of the Laboratory of Acoustic Imaging and Sound Control, Delft University of Technology.

ii

Graduation Committee:

Prof.dr.ir. D. Gisolf Laboratory of Acoustic Imaging and Sound Control Department of Imaging Science and Technology Delft University of Technology Dr.ir. D. de Vries Laboratory of Acoustic Imaging and Sound Control Department of Imaging Science and Technology Delft University of Technology Prof.dr.ir. A.J. Berkhout Department of Technology, Strategy and Entrepreneurship Faculty of Technology, Policy and Management Delft University of Technology Dr.ir. M.M. Boone Laboratory of Acoustic Imaging and Sound Control Department of Imaging Science and Technology Delft University of Technology Prof.dr.ir. J.J.M. Braat Optics Research Group Department of Imaging Science and Technology Delft University of Technology Prof.ir. J.J.M. Cauberg Section Building Engineering Faculty of Civil Engineering and Geosciences Delft University of Technology

iii

Abstract When an object radiates sound, it is caused by vibrations at the surface that is in contact with the air. The enclosed room the object is in will be filled with the radiated sound. Should a sudden pulse be emitted in all directions, the wavefront will be spherical and increasing in size until it reaches the floor, ceiling and walls. Depending on the materials of the surfaces the wavefront will be reflected to a certain degree and in time an increasingly complicated wavefield appears. When we measure the sound pressure with microphones along a surface in the room, we can actually see what acoustically is happening. During wavefront incidence on a surface, vibrations in the surface will appear, which are usually bending (flexural) waves, waves that have their displacement perpendicular to the surface. Also in the surfaces of the sound-radiating object, for example a piano, the bending waves usually cause the radiated sound. By pressing a key on the piano, vibrations will be introduced at certain positions on the plates and the bending waves that appear spread out, just like other waves, over the medium. Depending on the construction, these waves will reflect at the boundaries as well and again a complex wavefield appears. To learn more about this process, best is to first investigate a simple model of reality: a rectangular homogenous plate. When this model is fully understood, we can move forward to more complex structures, like multiple plates, other shapes, combination of materials, etc. In this thesis, the theory of bending waves in a plate will be presented along the theory of acoustic waves in a hall. The ideal situation of investigating bending waves would be to use the present techniques of room acoustics for our bending wave research. However, a disadvantage is, that, compared with acoustic waves in air, bending waves are dispersive. This means that inside the material each frequency travels with a different velocity and consequently a pulse will change into a wide wavelet. Images of bending waves with reflections are not very useful for further investigation. This thesis, however, will discuss a technique to transform the recorded bending waves into pseudo-acoustic waves. The so-called Dispersion Transform gives us the opportunity to see what happens with a plate when a source applies a transversal force on the surface. By mathematical analysing this transform, we learn about her positive and negative properties and will use her for simulations and measurements of plates.

v

Samenvatting Wanneer een object geluid afstraalt, is dit ten gevolge van trillingen aan het oppervlak dat in contact staat met de lucht. De ruimte waarin het object zich bevindt zal gevuld worden met het uitgestraalde geluid. Indien een plotselinge puls in alle richtingen wordt uitgezonden, is het golffront bolvormig en zal groter worden totdat het vloer, plafond en muren bereikt. Afhankelijk van de materialen van de oppervlakten zal het golffront in bepaalde mate gereflecteerd worden en met de tijd ontstaat een steeds gecompliceerder golfveld. Wanneer we tegelijkertijd met microfoons langs een oppervlak in de ruimte de geluidsdruk meten, kunnen we zien wat er akoestisch gebeurt. Bij het invallen van een geluidsgolf op een wand zullen er trillingen in de wand ontstaan, over het algemeen de transversale buiggolven, die hun uitwijking loodrecht op het oppervlak hebben. Ook in de oppervlakten van het object van geluidsafstraling, bijvoorbeeld die van een piano, zijn het de buiggolven die voor het geluid zorgen. Door op een pianotoets te drukken, zal op bepaalde plaatsen trillingen op de platen worden geïntroduceerd waarna de ontstane buiggolven zich, net als alle golven, over het medium verspreiden. Afhankelijk van de constructie zullen ook deze golven reflecteren aan de randen en wederom ontstaat een complex golfveld. Om meer over dit proces te weten te komen, kunnen we het beste als eerste een eenvoudig model van de werkelijkheid onderzoeken: een rechthoekige uniforme plaat. Als dit model eenmaal volledig begrepen is, kan de stap gemaakt worden naar meer complexere structuren, zoals meerdere gekoppelde platen, niet rechthoekige platen, combinaties van materialen etc. In dit afstudeerverslag zal de theorie van de buiggolven in een plaat naast die van die geluidsgolven in een zaal worden gepresenteerd. Ideaal in het onderzoek van buiggolven zou zijn dat we de bestaande technieken van de zaalakoestiek kunnen gebruiken voor buiggolfonderzoek. Nadeel is echter dat, in tegenstelling tot geluidsgolven, de buiggolven dispersie vertonen. Dit wil zeggen dat in het materiaal elke frequentie een andere snelheid heeft en hierdoor zal een puls veranderen in een breed signaal. Afbeeldingen van buiggolven met reflecties lijken hierdoor onbruikbaar te zijn voor verdere analyse. Dit verslag zal echter een techniek behandelen om de gemeten buiggolven te transformeren in pseudo-akoestische golven. Deze Dispersie Transformatie geeft ons de mogelijkheid om te kunnen zien wat er in zo'n plaat gebeurt wanneer een bron er een transversale kracht op uitoefent. Door een wiskundige analyse van deze transformatie leren we haar positieve en negatieve eigenschappen kennen en tenslotte passen we haar toe op simulaties en metingen van platen.

vii

Contents Abstract

v

Samenvatting

vii

Contents

ix

List of Variables

xi

1. Introduction

1

2. Acoustic wave fields in enclosures 2.1 Introduction 2.2 The three-dimensional wave field theory in air 2.2.1 Wave propagation and the wave equation 2.2.2 The solution of the wave equation in infinite media 2.2.3 Pulse propagation 2.2.4 Array recording 2.3 Reflections in enclosures 2.3.1 Single reflection: mirror image source 2.3.2 Multiple reflections: mirror image sources 2.3.3 The diffuse field 2.4 Array recordings of a concert hall 2.4.1 Array recordings and the hyperbolae 2.4.2 Normal Move-Out (NMO) and separation

3 3 3 3 4 5 6 7 7 9 10 12 12 14

3. Bending wave fields 3.1 Introduction 3.2 Bending wave field theories for infinite media 3.2.1 One-dimensional bending wave equation 3.2.2 Two-dimensional bending wave equation 3.2.3 Solution of the one-dimensional bending wave equation 3.2.4 Solution of the two-dimensional bending wave equation 3.2.5 Dispersion and phase/group velocity 3.2.6 Pulse propagation 3.3 Bending wave field theories for finite media 3.3.1 Reflections 3.3.2 Damping

17 17 17 17 20 21 22 22 24 25 25 26

4. Dispersion Transform 4.1 Introduction 4.2 Two-dimensional acoustic and bending waves

29 29 29

ix

4.3 Static Dispersion Transform 4.3.1 Single pulse dispersion deconvolution 4.3.2 Multiple pulse deconvolution 4.4 The convolution process 4.4.1 Continuous time static convolution 4.4.2 Discrete static convolution 4.5 Dynamic Dispersion Transform 4.5.1 Continuous dynamic convolution 4.5.2 Discrete dynamic convolution 4.5.3 Single pulse deconvolution 4.5.4 Multiple pulse deconvolution 4.6 Scaling and spectral changes 4.6.1 Scaling in the frequency-domain 4.6.2 Dispersion constant A 4.6.3 Spectral optimisation 4.6.4 Dispersion transform and noise 4.7 Array recording simulations 4.7.1 Free-field simulations 4.7.2 Plate simulations

30 30 35 37 37 38 39 39 40 40 42 43 43 44 46 49 51 51 53

5. Measurements 5.1 Introduction 5.2 Steel plate 5.2.1 Set-up and measurement results 5.2.2 Dispersion Transform 5.2.3 Acoustic image processing 5.2.4 Damping 5.3 Aluminium and Glare® plates 5.3.1 Measurement set-up 5.3.2 Measurement results of the aluminium plate 5.3.3 Measurement results of the Glare® plate

55 55 55 55 57 58 60 61 61 62 63

6. Conclusions and recommendations

65

Appendix

67

Bibliography

69

x

List of Variables A A A,B,C,D B ca cb cB cg d E f f' f(t), F(ω) F F h I K m', m'' M p(t), P(ω) pR(t), PR(ω) q(t), Q(ω) r ri rn R sA0(t), SA0(ω) Sa(ω), SA(ω) sA(ri;t), SA(ri,ω) Sb0(ω), SB0(ω) Sb(ω), SB(ω) sB(ri;t), SB(ri,ω) t tn T Ts T V Vz w (x,y,z) xn

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

total absorption dispersion constant [m/s0.5] amplitude factors bending stiffness propagation velocity of acoustic waves in air (here: ca = 343 m/s) one-dimensional bending wave phase velocity [m/s] two-dimensional bending wave phase velocity [m/s] group speed [m/s] distance between array elements [m] Young’s modulus of elasticity frequency [Hz] pseudo-acoustic frequency [Hz] force source [N] dispersion filter dispersion transform matrix height/thickness of the bar/plate [m] moment of inertia [m4] compression modulus [Pa] mass per usint length, unit surface [kg/m, kg/m2] bending moment [Nm] acoustic pressure [Pa] acoustic pressure at receiver position [Pa] vertical displacement [m] distance from source to receiver [m] distance from mirror image source to receiver [m] distance from source to nth array element [m] radius [m] acoustic wave source function in air two- and three-dimensional acoustic far-field source solutions in air three-dimensional acoustic mirror image source bending wave source function one- and two-dimensional far-field bending wave source solutions two-dimensional bending wave mirror image source (travel)time [s] traveltime from source to nth array element [s] general reverberation time [s] specific reverberation time (for a certain frequency band) [s] length of discrete time recording [s] volume of enclosure [m3] particle velocity of perpendicular to a plate [m/s] width of the bar/plate [m] spatial coordinates [m] offset position of nth array element [m]

xi

(xs ,zs)

α γ η µ ρ τ Φ ω

= = = = = = = = =

position of the source [m] absorption factor at reflection boundary scaling ratio [s-0.5] damping factor Poisson constant density [kg/m3] time variable [s] phase part of spectrum angular frequency [rad/s]

xii

CHAPTER 1 Introduction The use of arrays of receivers in seismics has been successfully introduced in the investigation of room acoustics. An impulse response recording by a microphone returns valuable acoustic information about the enclosure it has been operating in. When measuring with an array of microphones, or multiple measurements with a microphone along a line in the enclosure, we obtain more then just the single microphone acoustic information at the points. Plotting the data in a time-offset graph (multi-trace representation) reveals spatial correlation: the direct and reflected wavefronts, passing the array from all directions, become visible. Analysing these wavefronts gives us information about the boundaries with which they coincided and insight into the influence of these boundaries on the total wavefield. A new field in which array technology could be useful is the investigation of bending (also called flexural) waves in structures. These vibrations, that travel through plates and walls, have an important influence on the radiated sound field. Understanding how a wavefront propagates through a construction would give us insight in how this sound field is built up. For example, more efficient anti-noise or anti-vibrations systems could be possible, when we know exactly how these vibrations in plates behave. Within the Carrouso project, the Laboratory of Acoustic Imaging and Sound Control is responsible for the realistic reproduction of recorded sound waves (Wave Field Synthesis) using Dynamic Mode Loudspeakers (DML's). These flat panel loudspeakers are made out of light sandwich plates and attached exciters introduce the bending waves. The propagation of these waves through the panels is what determines the final sound field, but how the mechanism works and how it can be controlled is not yet fully understood. Compared with acoustic waves in air or seismic waves in the earth, bending waves have a big disadvantage: they are highly dispersive, meaning that each frequency component travels with its own velocity. Introducing a pulse at the source position will not simply result in a pulse with time delay at the receiver position, but in a sweep signal. Taken into account the fact that a wavefront reflects (most of) its energy at the boundaries of the plate, the receiver array will probably record multiple sweeps that overlap instead of clear wavefronts and further analysis of the reflection is impossible. Already in 1998 J. Martens stated in his M.Sc. thesis that removing the dispersion from these bending wave recordings is possible, resulting in a plot of the measurement as if it were executed in a non-dispersive medium. Being able to remove the dispersion gives us the opportunity to apply a range of seismic and acoustic processing techniques on a bending wave array recording of a plate. However, the process of removing dispersion, the Dispersion Transform, was not yet completely understood and a publication about the subject remained unfinished. This M.Sc. thesis is about dispersion transform too. Bending waves in plates will be transformed into pseudo-acoustic waves in air. A thorough mathematical investigation, found

1

The acoustical analysis of bending waves.

Maarten Brink

in Chapter 4, is necessary to understand the mechanism and simulations help us to test it. But before arriving at the dispersion transform, we have to discuss the behaviour of bending waves in general (Chapter 3) and the behaviour of acoustic waves in air, in which they are transformed to (Chapter 2). Also some room acoustics processing is introduced (Chapter 2), which are methods that can only be applied on bending wave recordings after dispersion removal. Finally some measurement results are shown (Chapter 5) to see to what extend the dispersion transform is functional in real life situations. Parallel to this thesis the work on the mentioned publication is continued, or better say restarted, and at this moment almost finished. It is mentioned in the bibliography of this thesis as a pre-submitted publication to the Journal of the Acoustic Society of America. The research presented in this M.Sc. thesis has been executed at the Laboratory of Acoustic Imaging and Sound Control, part of the new department of Imaging Science and Technology at the faculty of Applied Sciences, Delft University of Technology. I would like to thank my mentor Diemer de Vries for supervising my graduation and proofreading this thesis and my parents for the support during my entire study.

2

CHAPTER 2 Acoustic wave fields in enclosures 2.1

Introduction

The acoustics of a room can be very well described in subjective terms as warmth, spaciousness or brilliance, and, depending on the purpose of the room and the type of sound that is audible, it can be more or less quantified as good or bad, but discussion is possible. Describing the acoustics in an objective manner, means of impulse response, reverberation time and by absorption, gives us the possibility to calculate, predict and understand the behaviour of sound in enclosures, which can lead to control and improvement. In this chapter, an overview is given of the principles of acoustics in 3D enclosures. The general behaviour of waves in air without boundaries will be explained to understand how a sound wave travels through the medium. When adding boundaries to the system we will see a complex interference of direct and reflected waves. With recording and processing techniques developed for seismic exploration we will be able to see how the sound field is built up and how wave fronts spread out over the room. In addition, a separation technique will be explained how to filter out a single wavefront for further analysis.

2.2

The three-dimensional wave field theory in air

2.2.1

Wave propagation and the wave equation

The behaviour of sound in air can be described by the wave equation for the acoustic pressure in a homogeneous, absorption-free 3D fluid, derived from the laws of Newton and Hooke. The relation between pressure p, space (x,y,z) and time t for the source-free situation is, according to [1],

∇ 2 p ( x, y , z , t ) =

ρ ∂2 K ∂t 2

p ( x, y , z , t ) ,

(2.1)

where ρ is the density, K is the compression modulus of the medium and the Laplace operator ∇2 is defined as

 ∂2 ∂2 ∂2  ∇2 =  2 + 2 + 2  .  ∂x ∂y ∂z 

(2.2)

Introducing a monopole source signal at position (xs,ys,zs) formula (2.1) changes to

∇ 2 p ( x, y , z ) −

ρ ∂2 K ∂t 2

p( x, y, z , t ) = − sA0 (t )δ( x − xs )δ( y − ys )δ( z − zs ) ,

3

(2.3)

The acoustical analysis of bending waves.

Maarten Brink

where δ(x)δ(y)δ(z) is the spatial delta pulse, also called the three-dimensional Dirac delta function and sA0(t) is the source signal, for example the mass flux per unit time. The Fourier transform of the space-time domain (x,y,z,t) situation of formula (2.3) gives us for the pressure P in the space-frequency domain (x,y,z, ω)

(∇ -k ) P( x, y, z,ω ) = −S 2

2

a

A0

(ω )δ( x − xs )δ( y − ys )δ( z − zs ) ,

(2.4)

where ω is the angular frequency and ka is the wavenumber in air, defined as

ka ≡

ω

(2.5)

ca

with ca the propagation velocity of an acoustic wave in air, considered to be a constant 343 m/s in this thesis. For notational convenience, we use the angular frequency variable ω in most formulas instead of the normal frequency variable f. The plots in this thesis, however, will show frequency f on the axes, using the following relation:

f ≡ 2.2.2

1 ω. 2π

(2.6)

The solution of the wave equation in infinite media

The free-field solution of formula (2.3) is defined as the pressure p as a function of time for a certain receiver position (x,y,z) in infinite media, given by

sA0 (t − p ( x, y , z , t ) = p ( r , t ) =

r ) ca

(2.7)

r

with for r the distance between monopole source and receiver, defined as

r = ( x − xs ) 2 ( y − ys ) 2 ( z − zs ) 2 .

(2.8)

In words, this means that the source signal sA0 at (xs,ys,zs) undergoes a time-delay of r/ca and an amplitude decrease by 1/r (attenuation of 6 dB per distance doubling) before arriving at the detector receiver position (x,y,z). Fourier transforming equation (2.7) to the space-frequency domain (or solving equation (2.4)) gives us for the pressure P:

P ( x, y, z , ω ) = ∫ p( x, y, z , t )e − jωt dt = SA0 (ω )

e − j ka r . r

(2.9)

When we compare this relation with equation (2.7) we can see that the 1/r factor is still present, while the time delay is transformed into a linear phase delay of ka⋅r, conform the rules of Fourier transform.

4

Chapter 2: Acoustic wave fields in enclosures 2.2.3

Pulse propagation

Figure 2.1 shows the set-up of a simulated experiment: a source and two receivers at distances of r1 = 2 m and r2 = 3 m from each other. At t = 0 the source generates a zero-phase pulse with a frequency band of 0 – 5 kHz. The pulse propagates through the air with a spherical wavefront obeying equation (2.7) and arrives at the receiver positions with certain time delays and amplitude decays (see figure 2.2). receiver 1

r1 source

Figure 2.1:

*

o

r2

*receiver 2

Schematic representation of the source and receiver in the free field. In this simulation the distances between source and receivers are: r1 = 2 m and r2 = 3 m and the source sends out a zero-phase pulse at t = 0.

(a)

Time-signal and amplitude and phase spectrum of receiver 1.

(b)

Time-signal and amplitude and phase spectrum of receiver 2.

Figure 2.2:

Simulated signals from the set-up shown in figure 1.1. Notice in (b) that the amplitude has decayed with a factor 2/3 compared with (a) due to distance increase. Also a time delay of (r2 – r1) / ca = 1/343 = 2.9 ms can be noticed between (a) and (b). The two phase plots show a linear phase delay of – k⋅r = – 2π f r/ca = – 0.018 r f.

5

The acoustical analysis of bending waves. 2.2.4

Maarten Brink

Array recording

We can execute the same experiment as in figure 2.2, but now for an array of receivers. The schematic set-up of such a recording is illustrated in figure 2.3 and the result of this multitrace impulse response is shown as a wiggle plot representation in figure 2.4.

receiver array

z

y=0 x

*

zs

d

(0,0,0) offset x

source

. o o o o o o o o o o o o o o o o o o .

Figure 2.4: Multi-trace impulse the set-up of figure 2.3. The wavefront approaches the represented by the dashed lines, formula (2.11).

Figure 2.3: Schematic representation of the source and receiver array in the free field. The shortest distance r from source to the array is 3 m and the distance d between the receivers is 0,25 m.

response of hyperbolic asymptotes defined by

We consider the centre of the array to be positioned at (x,y,z) = (0,0,0). The hyperbolic shaped wavefront should obey the following travel time–offset relation for the elements of the array:

tn ( xn ) =

rn 1 = ca ca

2

zs + ( xn − xs ) 2 ,

(2.10)

where rn is the distance from the source to the nth element of the array, zs is the z-coordinate of the source, xn is the position on the nth array element and xs the source offset in the array direction, which in this case is equal to 0. Formula (2.10) shows that with increasing offset x the distance zs can be neglected. The hyperbola will approach the following asymptotes (offset x can be either positive or negative):

t ( x) =

6

| x| . ca

(2.11)

Chapter 2: Acoustic wave fields in enclosures

2.3

Reflections in enclosures

The situation described in the previous section is called the ‘free-field’ situation, meaning an infinitely large room. The pulse from the source of figure 2.1 will keep expanding in a spherical shape with the source point as centre and boundaries are never met. Complex situations arise when changing to the real life situation where boundaries reflect the acoustic waves inside an enclosure. Even the simplified situation of a rectangular room with rigid walls gives us a complex pattern of reflections at the receiver point when the same experiment as in figures 2.1 and 2.2 is executed. It goes beyond this thesis to discuss the reflection principles in detail. Extensive explanations can be found in [2]. For the purpose of this and the following chapters, an explanation of mirror image sources and ray paths will be sufficient. 2.3.1

Single reflection: mirror image source

When zooming in at the spherical wavefront locally as shown in figure 2.5 we see that the wavefront can be considered a plane wave approximately in the zoomed area, meaning that the propagation direction is constant within this small area.

source s0

o

Figure 2.5:

Schematic representation in one plane of a spherical wavefront. Zooming in at a small part of the wavefront (on the right of the illustration) shows that locally the wave can be considered a plane wave. The arrows indicate the direction of the wave locally and for a plane wave, all wave elements have the same direction.

According to [1] the angle of reflection of a plane wave at a flat rigid surface is equal to the angle of incidence and the factor of energy that is absorbed at the reflector we call α (thus the factor of energy that is reflected is equal to 1-α). Looking at a larger scale (figure 2.6), we see that the wavefront keeps expanding spherically after reflection on a wall. The reflected wavefront has the shape of a wavefront from a virtual source S' behind the wall, the mirror image source, at an equal distance from the wall as source S. S' and S excite the same pulse at the same time (see also the explanation in [3]). Further on in this thesis we will not show the complete spheres anymore that emit the sources, but only the paths from source (or image source) to receiver, called the ray paths.

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The acoustical analysis of bending waves.

Maarten Brink α

R * r

r0 o SA0

Figure 2.6:

o S'A0

Illustration of source S, mirror image source S’ and receiver point R, r0 is the distance source SA0 – receiver R (the direct wave) and r is the distance mirror image-source S’A0 to receiver R. The arrows, also called ray-paths, indicate the shortest route between source and receiver, without and with reflection.

In [2] is explained that the pressure from the image source, prefl, at the receiver point R can be written as 1

prefl = (1 − α ) 2

1 r sA0 (t − ) , r ca

(2.12)

with for r the distance from image source to receiver and the power ½ is due to energy being proportional with p2 instead of p. Combining equation (2.7) and (2.12) we get for the total pressure at receiver point R:

ptotal = pdirect + prefl =

1 r 1 1 r sA0 (t − 0 ) + (1 − α ) 2 sA0 (t − ) . r0 ca r ca

(2.13)

Fourier transforming this result gives us for the pressure in the frequency domain

Ptotal (ω ) = ∫ ptotal e-jω t dt = ∫ ( pdirect + prefl )e-jω t dt = Pdirect (ω ) + Prefl (ω ) .

(2.14)

Simulating what the measurement would be at the receiver position for the situation that r0 = 2 m and r = 3 m and presuming that no absorption takes places at the wall (α = 0), we get the result shown in figure 2.7. Figure 2.7 (a) is a plot of the pressure recording in time, with a pulse at t = 2/343 = 5.8 ms and t = 3/343 = 8.7 ms (compare with figure 2.2). Both formula (2.14) and figure 2.7 (b) and (c) show that the amplitude and phase of Ptotal are not simply the sum of the amplitudes and phases of Pdirect and Prefl, but have a more complicated shape, called a comb filter [4]. The distance ∆f between the lobes in (b) is equal to 1/∆t, where ∆t is the time between the arrival of the direct and the reflected pulse. In this case, ∆t equals 2.9 ms and therefore ∆f will be 343 Hz.

8

Chapter 2: Acoustic wave fields in enclosures

Figure 2.7:

2.3.2

Simulation of a measurement at receiver position R (see figure 2.4) for a direct wave (r0 = 2 m) and a reflection (r = 3 m). The time-plot (left) shows the time measurement as described in formula (2.13), the amplitude spectrum (centre) shows a comb filter and the phase spectrum (right) is almost linear.

Multiple reflections: mirror image sources

Applying the principles of subsection 2.3.1 to a room with walls on each side instead of the single wall reflection (floor and ceiling are ignored), we will see a complicated pattern of reflections, also called specular reflections [5]. The room will have multiple mirror image sources, as seen in figure 2.8. Instead of showing the ray paths of all reflective waves inside the room (which would be a complex and incomprehensible representation), we will show the ray-paths from the mirror image sources to the receiver. Following one ray-path from image source to receiver the arrow will cross as many boundaries as the real wave would reflect at the walls travelling from source to receiver. *

*

*

*

*

*

*

o x *

= receiver = original source = mirror image source

o

Figure 2.8:

*

*

*

*

*

*

x

*

*

*

*

*

*

*

Schematic representation of a room with source and receiver and its mirror image sources. The arrows indicate the ray paths for some of the image sources. Floor and ceiling are ignored.

Looking at the special case that there are no losses or phase changes at the enclosure boundaries, we can assign the source time signal sA0(t) and spectrum SA0(ω) to every mirror image source, such that

9

The acoustical analysis of bending waves.

Maarten Brink

 r  sA  t − i  ca  sA (ri ; t ) =  ri

(2.15)

and

S (ω )e S A (ri ; ω ) = A0 ri

−j

ω cA

ri

,

(2.16)

with the index i representing the ith image source. Note the difference between sA0(t) and sA(ri;t): the first one is the source signal and the second one the wave field expression. The pressure signal at the receiver position, PR, now equals the sum of all mirror image sources:

pR (t ) = ∑ sA ( ri ; t ) ,

(2.17)

i

and in the Fourier domain this equals:

PR (ω ) = ∑ SA (ri ; ω ) .

(2.18)

i

At some time after the source has emitted a pulse (and the image sources theirs), there will be so many pulses arriving at the receiver position around the same time that the single reflections will be indistinguishable: waves will be constantly arriving and come from every direction. This will be called a diffuse field and will be discussed in the following subsection. 2.3.3

The diffuse field

The diffuse field, extensively discussed in [2], is considered to be a homogeneous and isotropic field and statistically the sound pressure is constant all over the room. After the source has generated a pulse and the distinguishable early reflections have passed, the sound field in the room will become increasingly diffuse. Due to absorption at the boundaries, the sounds pressure level Lp will decay and it can be shown that this decay is linear. The reverberation time T, the time needed for the sound pressure level to decrease 60 dB, can be obtained by plotting this sounds pressure level and determine the time needed for Lp to decay 60 dB. According to [2] we can use the following formula to plot the sounds pressure level decay of a recorded impulse response h(t):

∞  2 Lp (t ) = 10 log  ∫ h (τ ) dτ  . t 

(2.19)

Reverberation time T depends highly on the absorption of the room; it can also be predicted by the following approximation:

10

Chapter 2: Acoustic wave fields in enclosures

T=

1V 6A

(2.20)

where V is the volume of the room and A is the total absorption:

A = αS

(2.21)

where α is the average absorption coefficient at the boundaries and S the total boundary surface. In real situations, the absorption α is frequency dependent and reverberation times are given in octaves. After bandpass filtering the impulse response recording h(t) of the room, we can plot the sound pressure level using formula (2.19) and determine the specific reverberation time Ts for that frequency band. The general reverberation time of the enclosure is the average of the 60 dB decays of the 500 Hz and 1000 Hz octave bands. As an example of reverberation time and multiple reflections, an impulse response measurement of the Amsterdam Concertgebouw is shown in figure 2.9 (a). Obviously, the amplitude is zero until the direct wave arrives. After the direct wave has passed, reflections from all directions arrive, coming from chairs, floor, ceiling, walls etc. Taken the average reverberation time T of the octave bands of 500 Hz and 1000 Hz (figure 2.9 (c) and (d)) we get for the general reverberation time at the recorded position a value of about 2.3 s.

Figure 2.9:

Impulse response recorded in the Amsterdam Concertgebouw and derived decay plots. (a) shows the recorded impulse response, which clearly indicates the direct wave and the decay in amplitude of the defuse field. (b) – (g) show the sound level decay curves for the octave bands around the frequencies given in the graphs.

11

The acoustical analysis of bending waves.

2.4

Maarten Brink

Array recordings of a concert hall

The acoustic measurement where a pulse is generated and the ‘behaviour’ of the enclosure is measured yields an impulse response. The ideal pulse is infinitely short and has infinite power and can therefore never be realized in real situations. Methods are developed to approximate impulse responses by introducing sweeps or noise containing a wide spectrum of frequencies into a room and deconvolving the measured response with the source signal. 2.4.1

Array recordings and the hyperbolae

When impulse response measurements or simulations are executed for an array of receivers, instead of a single receiver, interesting patterns will appear when plotting the impulse responses next to each other as multi-trace impulse response (see also subsection 2.4.4). The time-space relationship along a line in the enclosure reveals much of the temporal and spatial structure of the wavefield. Figure 2.10 shows the Amsterdam Concertgebouw plan with the position of the microphone array and of the omni-directional source. Instead of doing a measurement at a single position, as in figure 2.9, recordings have been made over the complete width of the hall [6]. Figure 2.11 shows the result.

array

| 0m

source o

offset x array *

source o

Figure 2.11: Impulse response data-set measured along the array of figure 2.10. The direct wave and first reflections on the right and left walls can be very well distinguished.

Figure 2.10: Plan of the Amsterdam Concertgebouw with microphone array and omni-directional sound source. the shortest distance from source to array equals 11,3 m.

12

Knowing the hall geometry from figure 2.10 and the responses from figure 2.11, we can identify reflected wavefronts and their reflectors. Further processing of the dataset can give us insight in acoustic properties like diffusivity and their relation to the position in the enclosure. When isolating a certain wavefront, we can, for example, investigate the changes that have taken place after a reflection due to choice of material or structure of a wall. A simple but effective way to show the hyperbolic shape that the wavefront has in the array recording, explained in subsection 2.4.4, is by mirroring the recorded impulse responses along the offset boundaries. After adding the necessary empty recordings between the datasets to correct for the fact that measurements have not been taken place over the full width of the hall but with a certain distance of the side walls, the dataset should be mirrored multiple times along the side boundaries. The result, seen in figure 2.12, is a hyperbola that contains parts of the direct wavefront and the wavefronts that are reflected at the side walls only. Note that reflections from the front or back wall will become hyperbolae too, but reflections from objects in the hall, non-specular reflections, can be seen as sources that emit a wave later then t = 0 and will have a more curved hyperbolic shape.

Figure 2.12: The same dataset as figure 2.09 but then mirrored along the offset axis and empty traces are added at the mirror positions to compensate for not measuring completely up to the sidewalls of the hall. The central wavefront (A) can be followed downwards and seems to have a hyperbolic shape, as is expected when recording a spherical wavefront along a linear axis. The hyperbola approaches the dashed asymptotes that indicate ca = offset/travel time = 343 m/s.

13

The acoustical analysis of bending waves.

Maarten Brink

A possible way to get a clearer view of the hyperbolic wavefront A in the representation of figure 2.10 is using the Radon Transform. In this thesis however, we will use a different one as an example: the Normal Move-Out procedure. 2.4.2

Normal Move-Out (NMO) and separation

Using formula (2.10) together with the measured source-receiver distance, zs = 11.3 m, we can correct for the hyperbolic shape using the Normal Move-Out operation [5]. Every array element recording will be shifted upwards in the plot (back in time), depending on the offset position of the element. This way the hyperbolic wavefront, shown in figure 2.12, will be plotted as a horizontal line (see figure 2.13). To filter out the horizontal wavefront in figure 2.11, we can apply a filter in the wavenumberfrequency domain filtering out the wavefronts that are not parallel with the offset axis (with in a certain angle). On the filter result, we can apply an Inverse Normal Move-Out (INMO) operation to show us the data in the normal travel time-offset axis again. As can be seen in figure 2.12 all wavefronts are more or less filtered out except the one that belongs to the hyperbola of the direct wave. Now a more secure examination of this wavefront is possible.

Figure 2.13: Data of figure 2.12 after the Normal Move-Out correction, applied for the direct wave plus side reflections in the same hyperbola.

14

Chapter 2: Acoustic wave fields in enclosures

Figure 2.14: Plot of the data of figure 2.12 after executing the Normal Move-Out, a spatial filter and the Inverse Normal Move-out.

15

CHAPTER 3 Bending wave fields 3.1

Introduction

In the previous chapter, we have seen how acoustic wavefronts in air, which can be seen as small pressure variations in space, behave in a three-dimensional enclosure. On a molecular scale, we see a mechanism of particles vibrating around an equilibrium transferring energy to next particles in line. This wave phenomenon is of course not restricted to the medium air or to fluids but also applies to solids. In fact, in solids more mechanisms of transferring energy are possible than the pressure variation in air; besides the longitudinal (compression) wave motion also transversal motions as shear and bending wave phenomena can take place. Information about the first two wave types can be found in [1] and [7]; the bending wave principle will be explained in this chapter. Bending waves, also called flexural waves, are surface waves that appear in thin media (thickness is small compared to the wavelength), for example plates and bars. A measurable property of a bending wave is the local displacement, perpendicular to the surface, depending on position and time. First, the general laws for infinite one-dimensional and twodimensional media will be discussed. After that, we will have a closer look to finite media: a bar and a plate. Bending waves are not 'acoustics' when we define acoustics as dealing with audible waves. However, a bending wave field can easily generate acoustic waves in the air that surrounds the material; another reason to discuss and try to understand them.

3.2

Bending wave field theories for infinite media

3.2.1

One-dimensional bending wave equation

Theoretically a one-dimensional object is a line with infinite thickness, but, investigating wave propagation, we also consider bars or beams with thickness and height much smaller then its length and smaller then the wavelengths travelling through it to be one-dimensional. In addition, plane waves in air (subsection 2.3.1) can be seen as one-dimensional waves in a three-dimensional medium. Looking at figure 3.1, we see the cross-section of a bar through which bending waves are travelling. Figure 3.2 shows a small element of the bar with length dx and the local forces. The dotted line represents the cross-section of the neutral fiber plane of the bar with the paper: this plane is not compressed or stretched during bending. The element is small enough to neglect the curvature (see also figure 2.3) and a the forces in the z-direction that act on the element have a resulting force Fz,res.

17

The acoustical analysis of bending waves.

Figure 3.1:

Maarten Brink

Illustration of bending waves in a bar. The bending wave amplitudes have been highly exaggerated.

Fz(x,t) z h

neutral fiber

Fz(x+dx,t)

dx Figure 3.2:

x

Illustration of the cross section of a bar element with the (bending) forces acting on it.

Applying Newton’s second law of motion on the bar gives us the following relation:

Fz , res = Fz ( x, t ) − Fz ( x + dx, t ) = −

∂ ∂2 Fz (x)dx = m ' dx 2 q( x, t ) , ∂x ∂t

(3.1)

where q is the vertical displacement in the z-direction and m′ is the mass per unit length of the bar,

m ' = ρ wh ,

(3.2)

with for ρ the density, h the height and w the width of the bar. Due to bending of the material, compression and stretch in the x-direction takes place between the neutral fiber and the surfaces (see figure 3.3). The bending moment M that has to be applied to the bar element to make this deformation happen, is in [2] formulated as

∂2q M = −B 2 , ∂x

(3.3)

with for B the bending stiffness, in [7] defined as

B = EI ,

(3.4)

where E is Young’s modulus of elasticity and I the moment of inertia of the cross section. This moment of inertia depends on the shape of the cross section. For rectangular cross sections this moment of inertia is

I rect = 121 wh3 ,

18

(3.5)

Chapter 3: Bending wave fields and for cylinders with radius R

I circ = 14 π R 4 .

M

Figure 3.3:

(3.6)

M

x

Illustration of the cross section of the bar element with the bending moment acting on it forcing the upper half of the bar to compress and the lower half to stretch in the xdirection.

Bending moment M and force F have the following relationship:

∂M , ∂x

(3.7)

∂3 q ( x, t ) . ∂x 3

(3.8)

Fz = − and with formula (3.3) this results in:

Fz ( x, t ) = B

Combining formula (3.1) and (3.8) we obtain the one-dimensional bending wave equation, describing the space-time dependency of the transversal displacement q due to bending waves:

 ∂4 ∂2   B 4 + m ' 2  q ( x, t ) = 0 . ∂t   ∂x

(3.9)

In the space-frequency domain this can be written as

 ∂4 4  4 − k b  Q ( x, ω ) = 0 ,  ∂x 

(3.10)

with for the bending wavenumber kb:

kb = ω 4

m' 12 ρ = ω4 . B Eh 2

(3.11)

Note that the bending wavenumber has frequency dependence different from the acoustic wavenumber in air in formula (2.5). This important effect will be discussed in subsection 3.2.5.

19

The acoustical analysis of bending waves. 3.2.2

Maarten Brink

Two-dimensional bending wave equation

Expanding the theory about one-dimensional bending waves to two dimensions requires adding the derivative to the y-direction to formula (3.9), but also the Poisson constant µ will start playing a role. It describes the magnitude of expansion in the direction perpendicular to the direction of compression. As stated in [2] and [7], formula (3.9) will change to the following two-dimensional bending wave equation:

 ∂2  4 B ' m '' ∇ +   q ( x, y , t ) = 0 , ∂t 2  

(3.12)

where m'' is the mass per unit area of the plate, defined as

m '' = ρ h ,

(3.13)

were ∇4 is the square of the Laplace operator for two dimensions (compare with formula (2.2)): 2

 ∂2 ∂2  ∇ = 2 + 2  ,  ∂x ∂y  4

(3.14)

and where B′ is the two-dimensional bending stiffness per unit length, defined as

B' =

EI ' , 1− µ 2

(3.15)

with for µ the Poisson constant and for I′ the cross sectional moment of inertia per unit length defined as

I plate ' = 121 h3 ,

(3.16)

with for h the thickness of the plate. In the space-frequency domain the two-dimensional bending wave equation reads

(∇

4

)

− k B q ( x, y , ω ) = 0 4

(3.17)

with kB being the two-dimensional bending wavenumber (compare with the one-dimensional kb in formula (3.11)), defined as

k B (ω ) = ω 4

m '' 12 ρ (1 − µ 2 ) . = ω4 B' Eh 2

(3.18)

We will make use of subscript B instead of b to show that we are dealing with the twodimensional bending wave situation and not the one-dimensional version.

20

Chapter 3: Bending wave fields 3.2.3

Solution of the one-dimensional bending wave equation

Returning to the one-dimensional situation of subsection 3.2.1, we introduce a force signal f(t) at position x = xs on the bar, positive when directed upwards just like displacement q(t). Formula (3.9) will be extended as follows:

 ∂4 ∂2  + B m ' q ( x, t ) = f (t )δ ( x − xs ) ,  4 2  ∂ ∂ x t  

(3.19)

where δ(x) is the one-dimensional Dirac delta function. In the space-frequency domain formula (3.19) will lead to

 ∂4 F (ω ) 4 δ ( x − xs ) .  4 − k b  Q ( x, ω ) = B  ∂x 

(3.20)

Unlike formula (2.3), the wave equation in the space-time domain for air, we cannot directly solve the bending wave equations of formula (3.19). The displacement q at position x is not simply the source signal with a time delay. Formula (3.20), however, can be rewritten and split into two equations as follows: 2  ∂2 2  ∂ 2  2 − kb   2 + kb  Q( x, ω ) = S b0 (ω )δ ( x − xs ) ,  ∂x   ∂x 

(3.21)

with Sb0(ω) the source signal, defined as F(ω), the Fourier transform of the force signal, divided by the bending stiffness B:

S b0 (ω ) =

F (ω ) . B

(3.22)

The solution of formula (3.21) for an infinite bar at x ≠ xs gives us the displacement at a position x along the bar due to a force applied at xs and is written as follows:

Q( x, ω ) = A(ω )e − jkb ( x − xs ) + B (ω )e + jkb ( x − xs ) + C (ω )e− kb ( x − xs ) + D(ω )e+ kb ( x − xs ) . (3.23) The first two terms are travelling waves in the +x and –x direction, respectively. The last two are evanescent waves, which appear close to edges or, in this case, the source and have an exponential decay in the offset position. In [8] the following can be found about the source terms A(ω), B(ω), C(ω) and D(ω) in formula (3.23):

A(ω ) = B (ω ) = jC (ω ) = jD(ω ) , A(ω ) = − j

S (ω ) F (ω ) = − j b0 3 . 3 4Bkb 4k b

21

(3.24) (3.25)

The acoustical analysis of bending waves.

Maarten Brink

Reformulating the distance to the source as r = ( x − xs ) 2 and only considering the far-field non-evanescent part, where kbr >> 1, we can write for the displacement Q as a function of place and frequency

Q(r , ω ) = − j

S b0 (ω ) − jkb r e = S b (ω )e − jkb r . 3 4k b

(3.26)

Different from the situation in air of Chapter 2 is that we have two kinds of source signals: in this thesis Sb0 is called the source signal, defined by formula (3.22), and Sb is called the source function, the far-field solution of the bending wave equation. 3.2.4

Solution of the two-dimensional bending wave equation

Working similar to subsection 3.2.3 we first introduce a source force signal f(t) to formula (3.12) and its space-frequency version formula (3.17):

 ∂2  4 ∇ + B ' m ''   q ( x, y, t ) = f (t )δ ( x − xs )δ ( y − ys ) , ∂t 2  

(∇

4

)

− k B Q ( x, y , ω ) = 4

F (ω ) δ ( x − xs )δ ( y − ys ) . B'

(3.27)

(3.28)

Splitting similar to formula (3.21) gives us

(∇

2

− kB

2

)( ∇

2

)

+ kB Q( x, y, ω ) = SB0 (ω )δ ( x − xs )δ ( y − ys ) . 2

(3.29)

The far-field solution for this equation is, according to [9],

Q(r ; ω ) = S B0 (ω )

1 32 jπ kB

5

e − jkBr e − jkB r , = S B (ω ) r r

(3.30)

with for the distance to the source r = ( x − xs )( y − ys ) . The 1/ r decay is identical to the decay of two-dimensional acoustic waves in air. For the displacement q in the place – time domain we now get the following result:

 e − jk B r  q (r ; t ) = FT −1  S B (ω ) , r  

(3.31)

where FT −1 {...} is the inverse Fourier transform. 3.2.5

Dispersion and phase/group velocity

Looking closer at the two-dimensional bending wave, we see that the bending wavenumber depends on the square root of the frequency, instead of the linear frequency dependence of

22

Chapter 3: Bending wave fields the acoustic wavenumber in air. With the definition of wavenumber, given in formula (2.5), we conclude that the bending wave phase velocity cB is frequency dependent:

cB (ω ) ≡

ω = k B (ω )

ω

ω

12 ρ (1 − µ 2 ) 4

=

4

Eh 2 ω=A ω. 12 ρ (1 − µ 2 )

(3.32)

Eh 2

This non-linear dependence of the velocity cB on the (angular) frequency ω is called dispersion. The dispersion constant A is material dependent. Figure 3.4 shows us this bending wave phase velocity plotted against the frequency f for three different media: a glass window with thickness h = 4 mm, a steel plate with h = 30 mm and a concrete wall with h = 150 mm. The horizontal line represents the velocity in air, being constant with the frequency.

Figure 3.4: Bending wave (phase) velocity of three materials plotted against the frequency. Note the square root dependence. The dispersion constants are: Aglass = 2.53 ms -0.5, Asteel = 6.76 ms -0.5 and Aconcrete = 11.4 ms -0.5.

A dispersion curve as shown in figure 3.4 can also tell us something about the speed at which energy is transported by a wave: the group speed cg. It is defined as

 dk  cg ≡    dω 

−1

(3.33)

and is constant for acoustic waves in air: cg = ca. The bending wave group speed, however, has the following relation: −1   ∂k   1 = cg,B (ω ) ≡   2 ω  ∂ω  

−1

12 ρ (1 − µ 2 )  4  = 2 A ω = 2c (ω ) . B 2  Eh 

23

(3.34)

The acoustical analysis of bending waves. 3.2.6

Maarten Brink

Pulse propagation

With the information from subsection 3.2.5, figure 3.4 and formula (3.31) we can simulate how a pulse would propagate through, for example, a glass window. Analogue to the simulation in air from subsection 2.2.3 a pulse with frequency band 0 – 5 kHz is introduced and the displacement q as a function of time is simulated for several distances. The set-up is shown in figure 3.5 and results can be found in figure 3.6. Note that we are still discussing infinite plates: no reflections or standing waves take place.

source

receivers

o

*

*

*

*

*

r Figure 3.5:

Schematic illustration of the source and receivers on a glass window with 3 mm thickness, dispersion constant A = 2.5 ms-0.5 and infinite size. Receivers are placed every 1 m and the source sends out a zero-phase pulse at t = 0.

source signal

r=1m

0

−5

0

0

5 10 15 time (ms)

20

−5

0

0

r=3m

5 10 15 time (ms)

20

Figure 3.6:

5 10 time (ms)

0

15

20

−5

5 10 15 time (ms)

20

r=5m

0

0

−5

r=4m

0

−5

r=2m

0

0

5 10 time (ms)

15

20

−5

0

5 10 time (ms)

15

20

Simulated signals from the set-up in figure 3.5. d at r = 0 m, that propagates through a glass window (A = 2.5 ms-0.5). Due to dispersion the higher frequencies in the pulse travel faster: the simulated time measurements at r > 0 m show a sweep.

24

Chapter 3: Bending wave fields Due to dispersion, the pulse will change into a sweep. In figure 3.7 the source signal and the signal for r = 2 m are shown, together with the accompanying amplitude and phase spectra (compare with figure 2.2). The amplitude and phase factor

1 32 jπ kB5

between source signal

and source function (see formula (3.30) and subsection 3.2.4) is not taken into account. Figure 3.7 (b) does shows the 1/ r decay compared with (a) and the non-linear phase spectrum that is equal to –kB(ω)r.

(a)

Source signal with its amplitude and phase spectrum.

(b)

Simulated signal at r = 2 m with its amplitude and phase spectrum.

Figure 3.7:

Simulated signals from the set-up of figure 3.5. (a) shows the source signal and (b) the signal at receiver position r = 2 m from the source. Notice in (b) the 1/√r = 1/√2 amplitude decay compared with the source signal and the non-linearity in the phase due to the dispersion: Φ(ω) = -kB(ω)r.

3.3

Bending wave field theories for finite media

3.3.1

Reflections

Just like acoustic waves, bending waves in finite bars or plates reflect at the boundaries as well. The many types of boundary conditions will not be discussed in this thesis. A detailed outline of the boundaries and the plate eigenfunctions can be found in [10]. In the special case of free boundaries with no absorption, similar to the formulas in subsection 2.3.2, we can describe the displacement signal q(t) at the receiver point in a plate

25

The acoustical analysis of bending waves.

Maarten Brink

as the sum of source functions sB(t) with certain delays and decays, or as the sum of all mirror image sources sB(ri;t):

 ri  sB  t −  cB (ω )   q(t ) = ∑ = ∑ sB ( ri ; t ) . ri i i

(3.35)

In the frequency domain this leads to −j

S (ω )e Q(ω ) = ∑ B ri i

ω

cB (ω )

ri

= ∑ SB (ri ; ω ) .

(3.36)

i

Unfortunately, due to the changing into a sweep of a single pulse we can not investigate reflections in plates as we do in acoustic enclosures (Chapter 1). When we execute the same reflection simulation as in subsection 2.3.1 with a direct wave at ri = 2 m and a reflection at ri = 3 m, but now for a bending wave in glass, we can clearly see why. In figure 3.8 (a) (compare with figure 2.5), it is seen that the two pulses partly coincide, and it is not really clear how many pulses the wavelet contains. From the shape of the phase-spectrum however, we can see that we are dealing with bending waves.

Figure 3.8:

3.3.2

Simulation of a direct bending wave (ri = 2 m) and a reflection (ri = 3 m) forming a signal q(t) in (a) with its amplitude (b) and phase spectrum (c). The same set-up in air would give the result shown in figure 2.5 and in (a) the two pulses are not distinguishable anymore.

Damping

In Chapter 2, the damping of acoustic waves in air due to boundary effects and absorption was discussed; the internal damping due to the medium, air, was neglected. In plates we can distinguish three different mechanisms of damping: internal damping, damping due to sound radiation and damping due to boundary effects. Internal damping The one and two-dimensional bending wave equations discussed make use of the bending stiffness B, which was presumed to be a real value. However, in the ‘real world’ Young's modulus of elasticity E, and therefore B is complex:

E ' = E (1 + jη ) ,

26

(3.37)

Chapter 3: Bending wave fields where η is the damping factor [8]. It can be shown that due to this extra factor in the wave equation the bending wave number will change as follows:

kB ' = kB ⋅ (1 − 14 jη ) .

(3.38)

For the two-dimensional solution of the wave equation, formula (3.30), this leads to an extra decay factor:

e − jkB r − 14η kB r e Q(r , ω ) = S B (ω ) . r

(3.39)

Notice that due to the variable kB in the decay exponent, the decay is not constant but depends on frequency. Damping due to sound radiation The stiff boundary plate – air causes the air close to the plate to vibrate when the plate vibrates. The displacement Q(x,y, ω) and the vertical component of the particle velocity in air have the following relationship:

Vz (x,y,ω )=

∂ Q(x,y,ω )=jω Q(x,y,ω ) . ∂t

(3.40)

Although being a very important acoustic aspect of bending waves, it goes beyond this thesis to discuss sound radiation of plates. More about this subject can be found in [8]. Damping due to boundary effects Similar to the damping due to boundary effects in subsection 2.3.1, a bending wave in a bar or plate can loose a lot of energy during reflection at a boundary. Although presumed rigid, most boundaries absorb a part of the vibrational energy of the object. Bending (or other type of) waves appear in the construction that surrounds the bar or plate. Compared to the previous two damping mechanisms, damping by boundary effects is usually the bigger part. According to [3] and [7] we can use the reduction term in formula (3.39) not only for internal damping but also for damping in general. In that case the damping factor η should not be seen anymore as a material property, but instead as a general constant of damping representing the complete set-up of the plate and all three damping effects. Now the damping factor for a specific bar or plate in a specific situation cannot be predicted properly anymore, nor found in the literature, so should be derived practically. It can be shown from diffuse bending wave theory that the damping constant can be determined using following formula:

η=

2.2 , Tf

(3.41)

where f is the frequency and T is the plate’s 60 dB decay, similar to the reverberation time of a room (see formula (2.20)).

27

CHAPTER 4 Dispersion Transform 4.1

Introduction

Chapter 3 has shown us the dispersive behaviour of bending waves and we have seen how a short zero phase pulse at the source position is transformed into a sweep when arriving at the receiver position. We have also seen that a pulse and even a single reflection already mix into each other after a certain travelling distance, and a fuzzy wavelet remains; the spatial information is lost. The acoustic imaging techniques of which Chapter 2 gives a glance, can only be of use when pulses remain pulses and wavefronts remain wavefronts. This chapter will discuss signal processing techniques to remove the unwanted dispersion from a bending wave signal, returning a signal that reveals the hidden spatial information of the medium.

4.2

Two-dimensional acoustic and bending waves

As described in Chapter 3, formula (3.30), a bending wave travels through an infinite plate (free field) according to the following formula in the frequency domain:

S B (r ; ω ) = S B0 (ω )

1 32 jπ kB

5

1 − jkB r 1 − j kB r , = S B (ω ) e e r r

(4.1)

where SB(ω) is far-field source function (see subsection 3.2.4) and kB(ω) is defined as follows:

kB (ω )=

ω 1 = ω. cB (ω ) A

(4.2)

The multiplication in formula (4.1) of the source function SB(ω) with the amplitude factor

1/ r and non-linear phase-shift e − jkB r in the frequency domain is similar to a convolution of these factors in the time domain. The last factor, the phase-shift, is the one that transforms a small pulse into a sweep along the path. Would the two-dimensional wave be travelling through air (a two-dimensional wave in the three-dimensional medium air can be generated by a line source), the equation (4.1) would, according to [9], be as follows:

 2π  1 − jka r 1 − jka r Sa (r ; ω ) =  S A0 (ω ) e = Sa (ω ) e ,  jk a  r r 

29

(4.3)

The acoustical analysis of bending waves.

Maarten Brink

where SA0(ω) is the Fourier transform of sA0(t), the point source signal introduced in formula (2.3). The source function Sa(ω), is the far-field solution of the integration over an infinite number of point source signals SA0(ω) along the line (see figure 4.1). Index a and A in the source terms tell us whether we are dealing with two- or three-dimensional acoustic wavefronts respectively. The velocity ca and wavenumber ka of the waves in air do not need this specification since they are equal. Formula (4.3) is dispersion-free, as ka is linear to the frequency:

ka (ω ) =

ω ca

,

(4.4)

with ca at a constant 343 m/s.

infinitely small point source element SA0 =

r

receiver

o

infinitly long source line

z

y x

cilindrical wave in 3D air

Figure 4.1:

Schematic illustration of a cylindrical wave in air. Every point source element in the infinite source line sends out a spherical wave and the resulting wave in a plane along the x,y-direction is described by formula (4.3). The wavefront of the line source is cylindrical, so its intersection in the plane is circular.

4.3

Static Dispersion Transform

4.3.1

Single pulse dispersion deconvolution

Looking at formula (4.1) and (4.3) we can easily transpose a recorded bending wave into a two-dimensional acoustic wave in air if we want to cancel/filter out the dispersion in the

30

Chapter 4: Dispersion Transform recording. A multiplication by a filter function F(r, ω) in the frequency domain transforms SB(r,ω) into Sa(r, ω):

Sa (r ; ω ) = F (r; ω ) SB (r;ω ) ,

(4.5)

with for F(ω):

F (r; ω ) =

Sa (r ; ω ) Sa (ω ) e − jka r Sa (ω ) +jkB r − jka r = = e e . SB (r ; ω ) SB (ω ) e − jkB r SB (ω )

(4.6)

The signal Sa(r,ω) will be called the pseudo-acoustic equivalent of the bending wave response SB(r,ω). The bending wave displacement data is transformed into a fictive pressure recording of a cylindrical wave in air: the Dispersion Transform has been applied. Because we do not want to throw away the useful amplitude information of the bending waves, we will only use the phase domain part of the dispersion removal operation:

F (r ; ω ) = e jkB r e − jka r = e jkB r e

−j

ω ca

r

= G (r ; ω )e

−j

ω ca

r

,

(4.7) ω

−j r

with for G(r,ω) the dispersion removal part e b . The linear phase-delay part e ca depends on the acoustic velocity ca and the source-receiver distance r and represents a time shift. jk r

Figure 4.2 illustrates how the operation looks like in the frequency domain and figure 4.3 the time-dependent version of this convolution. Figure 4.2 (a) shows the amplitude and phase spectrum of a simulated bending wave signal SB(r;ω) in glass for r = 3 m. Similar to figure 3.7, the phase spectrum of this input data is non-linear. (b) shows the amplitude and phase spectrum of the function G(r;ω). G is built up as a rectangular window, not to affect the amplitude information of the input data, and has smooth edges to shorten the time-signal version g(t) for more efficient processing of the dispersion removal operation. The phase spectrum of G is inverse of that of the input signal SB and multiplication with it will result in a zero-phase signal (c). Applying the appropriate time-delay to the filter signal G results in a filter signal F(r;ω). (d). Multiplication of SB with filter signal F results in the correct pseudoacoustic pulse (e).

31

The acoustical analysis of bending waves.

Maarten Brink

(a)

Amplitude spectrum (left) and phase spectrum (right) of bending wave signal SB(r,ω)

(b)

id. of zero-phasing filter G(r;ω)

(c)

id. of the multiplication SB(r;ω)G(r;ω)

(d)

id. of the dispersion removal operator F(r;ω)

(e)

id. of the resulting pseudo-acoustic pulse Sa(r;ω)

Figure 4.2:

Frequency domain illustration of the static deconvolution of a bending wave.

32

Chapter 4: Dispersion Transform Looking at F(r;ω) in formula (4.7) we can understand that the time-dependent version f(r;t) is built up from an ‘anti-causal’ dispersive signal g(r;t) and a time delay δ(t − r/ca):

 r f ( r ; t ) = FT −1 {e +jkB r } ∗ FT −1 e − jka r = g ( r ; t ) ∗ δ  t −  ca

{

}

  r  = g  r; t − ca  

 . 

(4.8)

In the time-domain, formula (4.5) now transforms into

 r sa (r ; ta ) t →t = sB ( r ; t ) ∗ f ( r ; t ) = sB ( r ; t ) ∗ g  r ; t − a ca 

 , 

(4.9)

with ta being the pseudo-acoustic traveltime, defined as

ta ≡

r . ca

(4.10)

This procedure is illustrated in figure 4.3. (a) shows the bending wave input function sB(r;t); (b) shows g(r;t), which is approximately the opposite of (a) and is longer due to a flat spectrum as seen in figure 4.2 (d); (c) shows the pseudo-acoustic result of the convolution of (a) and (b); (d) shows the same signal as (b) but delayed in time and (e) shows the pseudoacoustic result of the convolution of (a) and (d). We now have a small pulse at the correct delay position of a measurement in air: ta ≡ r/ca = 3/343 = 8.7 ms. In (f), the same plot as (e) but now plotted against the distance r, we see the pulse appearing at the correct distance r = 3 m.

33

The acoustical analysis of bending waves.

Maarten Brink

(a)

Bending wave sB(r;t) in glass, r = 3 m.

(b)

Zero-phasing signal g(r;t) for glass, r = 3 m.

(c)

Pseudo-acoustic result of the zero-phasing operation sB(r;t) ∗ g(r;t).

(d)

Dispersion removal signal f(r;t) = g(r;t-r/ca).

(e)

Pseudo-acoustic result of the dispersion-removal operation, ta = r/ca = 3/343 = 8.7 ms.

(f)

Same plot as in (e) but then plotted along the distance axis r using the scaling formula (4.10).

Figure 4.3

Illustration of the dispersion-removal operation of equation (4.9), which is also shown in figure 4.2, but now in the time domain.

34

Chapter 4: Dispersion Transform 4.3.2

Multiple pulse deconvolution

In subsection 3.3.1 and figure 3.8 we have already seen that multiple dispersive pulses can not be distinguished from each other, because a pulse transforms into a wider sweep. With the technique of the previous subsection, the static dispersion transform, we can remove the sweep-like shape from a pulse and transform it into a narrow pulse. However, filter-signal f(r;t) in formula (4.9) requires a constant distance r and our multiple reflections have different (mirror image source) distances ri. Again, in figure 4.4 we see the simulation of a signal q(t), which is built up of a direct bending wave pulse at ri = 2 m, and two reflections at ri = 3 m and ri = 5 m. In (b) the result of the convolution between q(t) and f(ri;t) with ri = 2 m can be seen (a peak appears at ta= 2/343 = 5.8 ms). (c) shows the same result for ri = 3 m and (d) for ri = 5 m. Each time one wavelet is transformed into a narrow pulse, and the other two either remain dispersive or transform into opposite dispersive signals. A possible second step would be isolating the narrow pulses and throwing away everything with a dispersive character, which is in this case very well possible: the three pulses have enough spacing between them. Unfortunately, this will not be possible when we deal with the bending wave field of a normal plate, which will be very complex. A more advanced method of dispersion removal will be discussed later in this chapter.

35

The acoustical analysis of bending waves.

Maarten Brink

(a)

Bending wave q(t): a simulation of a direct pulse (ri = 2 m) and two reflections (ri = [3,5] m).

(b)

Result of convolution q(t)*f(ri;t) with ri = 2 m (ta = 2/343 = 5.8 ms).

(c)

Result of convolution q(t)*f(ri;t) with ri = 3 m (ta = 2/343 = 8.7 ms).

(d)

Result of convolution q(t)*f(ri;t) with ri = 5 m(ta = 2/343 = 14.6 ms).

Figure 4.4:

Static dispersion deconvolution of a signal q(t) with three different filter signals f(ri;t). For each mirror source distance ri time the matching wavelet transforms into a small pulse, but the other two remain dispersive.

36

Chapter 4: Dispersion Transform

4.4

The convolution process

4.4.1

Continuous time static convolution

In general, a convolution between two time-dependent signals x(t) and y(t) is described as follows:

z (t ) = x (t ) ∗ y (t ) ≡

+∞

∫ x (τ ) y ( t − τ ) dτ ,

(4.11)

−∞

where * is the convolution operator. Since t and τ are identical time-variables, we can write formula (4.12) as +∞

z (τ ) = x (τ ) ∗ y (τ ) =

∫ x ( t ) y (τ − t ) dt .

(4.12)

−∞

In words, the inner product1 between input signal x(t) and convolution signal y(−t) is calculated for every time-shift τ. The resulting scalar will be the value of z(τ) for that specific time-shift. The shape of signal y will be the same throughout the whole process, it is only reversed in time and shifted along the time-axis by τ. Substituting τ by ta and introducing our single pulse dispersion deconvolution formula (4.9) for a pulse with distance r = ri , we will get:

sa (ri ; ta ) =

+∞

∫

sB ( ri ; t ) f ( ri ; ta − t ) dt =

-∞

+∞

 r  s r ; t g r ; t − t − ( )   dt B i i a ∫ ca   -∞

(4.13)

Again, we prefer rewriting the process into the filter signal g(t) instead of signal f(t), because g(t) and sB(t) are each others opposites (see subsection 4.3.1). But, because the amplitude spectra |SB(ri;ω)| and |G(ri;ω)| are different (see figure 4.2, G(ri;ω) has a flat spectrum) we will use the following notation:

g (ri ; −t ) = sˆB ( ri ; t ) ,

(4.14)

where sˆB (ri;t) is the white-spectrum version of sB(ri;t). Applying the substitution of formula (4.14) to formula (4.13), sa(ri;ta) now becomes

sa (ri ; ta ) =

+∞

∫ s ( r ; t ) sˆ B

i

B

−∞

  ri  ri ; t + − ta  dt . ca  

(4.15)

Note that ri in these formulas is a constant and ta a variable representing the (time)shift. In words, for a certain value of ta, the defined signal sˆB (ri;t) is shifted to the left along the time1

The inner product can be seen as a correlation process. The correlation version of the Dispersion Transform can be found in the Appendix.

37

The acoustical analysis of bending waves.

Maarten Brink

axis by the constant ri/ca and to the right by ta and the inner product is taken. For ta = ri/ca, the sˆB -wavelet will have the same position as the input signal sB and the signals will match: the inner product (or correlation – see the Appendix) returns a high value and a peak will appear for this ta, just like in figure 4.3 (e). 4.4.2

Discrete static convolution

In practical situations we always measure with a computer and have to deal with discrete signals. The discrete version of formula (4.12) is written as follows: M

z ( n+τ ) = x ( n+τ ) ∗ y ( n+τ ) =+t ∑ x ( m+t ) y ( n+τ − m+t ) ,

(4.16)

m=0

where M∆t is the length of discrete signals x and y and N∆τ is the length of z. For notational convenience will omit the use of ∆ in the notation for discrete signals from now on. Our bending wave static deconvolution operation of formula (4.15) now becomes M  r  sa ( ri ;nta ) = ∑ sB ( ri ;mt ) sˆB  ri ; i + mt − nta  . m =0  ca 

(4.17)

This discrete convolution can be formulated as a matrix multiplication. We will use a filter matrix F, filled with the signal sˆB (ri; ri/ca + t) for ri = 3 m, that is shifted over a certain timeoffset ta according to its row position in the matrix (see figure 4.5). A multiplication of F and

G

G

descrete input signal s B (ri;t) results in a descrete pseudo-acoustic signal s a (ri;t):

G G s a (ri ; ta ) = F s B (ri ; t ) .

(4.18)

Again the example of figure 4.3 is used: the single bending wave in glass at a distance of ri = 3 m. Unfortunately the procedure explained in the subsections 4.4.2 and 4.4.3 still only works for single dispersive pulses and we need to know the source−receiver distance ri to generate the correct filter signal. The next section will discuss a solution to this problem, called the Dynamic Dispersion Transform.

38

Chapter 4: Dispersion Transform

F

sa(ri;ta), ri = 3 m

sB(ri;t), ri = 3 m

t

sˆB ( r ; + t − ta ), ri = 3 m ri i ca

ta = ri/ca “= 3/343 = 8.7 ms

sˆB (ri ; crai + t − ta ), ri = 3 m

ta

sˆB ( ri ; cria + t − ta ), ri = 3 m

ta

t

=

Figure 4.5:

Matrix representation of the static dispersion transform operation of formula (4.17). Matrix F is filled with filter signal sˆB ( ri ; cri + t − ta ) with increasing offset ta in the a

vertical direction. For a clearer schematic illustration, not all sˆB signals are drawn in matrix F.

4.5

Dynamic Dispersion Transform

4.5.1

Continuous dynamic convolution

As explained in section 4.4, the static dispersion transform gives us a good result when the distance ri from source to receiver is known. For a bending wave recording sB with unknown distance, formula (4.15) will change into

sa ( ri ; ta ) =

+∞

∫ s ( t ) sˆ B

B

−∞

  ri  ri ; t + − ta  dt ca  

(4.19)

In subsection 4.4.2 we have seen that the matrix F is filled with a generated wavelet sˆB ( ri;t) , its shape determined by ri, and shifted along the time axis by a variable ta. Because we now have to deal with an unknown source-to-receiver distance, F has to be designed in such a way that it can deal with all the sweep-like wavelets pulses that are possible. Formula (4.10) describes the relation between source-to-receiver distance and pseudoacoustic traveltime. We could substitute ri in formula (4.19) by cata . This way the wavelet in matrix F is filled with all possible shapes of the bending wave: it changes from a pulse to a wide sweep with increasing time-shift ta (increasing row number). The time-continuous equation, formula (4.19), changes into:

sa (ta ) =

+∞

∫ s ( t ) sˆ B

−∞

B

  ca ta − t a  dt .  ca ta ; t + ca  

39

(4.20)

The acoustical analysis of bending waves.

Maarten Brink

Or, simplified to the following final result:

sa ( ta ) =

+∞

∫ s ( t ) sˆ ( t ; t ) dt . B

B

(4.21)

a

−∞

In words this means that when we want to remove the dispersion from a bending wave signal sB(t) with known material properties but unknown source−receiver distance, the inner product with all possible wavelets sˆB (ta;t) should be executed. Important aspect is that a peak (the highest outcome of an inner product is the inner product between two signals with the same shape) will appear in the pseudo-acoustic result when both signals match. 4.5.2

Discrete dynamic convolution

For better understanding of the operation, again an illustration of the matrix F is given, but now for the dynamic convolution variant (see figure 4.6). Matrix multiplication formula (4.18) will apply for this dynamic convolution too, but now F is not filled by one wavelet shifted along the time-axis, but with a changing wavelet: M

sa ( nta ) = ∑ sB ( mt ) sˆB ( nta ; mt ) = F sB (m t ) .

(4.22)

m =0

t

F=

Figure 4.6:

t

sˆB (ta ; t ) sˆB (ta ; t ) sˆB (ta ; t ) sˆB (ta ; t )

ta

Schematic illustration of dynamic dispersion transform matrix F. Instead of one signal that is shifted (static), now the signal itself changes with increasing ta. Again not all signals sˆB (ta ; t ) in F are shown.

In the following two subsections we will investigate what will happen to the dispersive signal when multiplied with matrix F. 4.5.3

Single pulse deconvolution

Not being a normal (de)convolution, with the dynamic dispersion transform we step away from a physical wave. Figure 4.7 (a) shows the single bending wave pulse in glass, which we have used before, with its amplitude spectrum. After multiplication of this (discrete) signal with a matrix F, filled for this type of glass plates, we get the pseudo-acoustic result, shown in (b).

40

Chapter 4: Dispersion Transform Just like the static deconvolution in figure 4.3 (e), the resulting pulse is placed at ta = 8,7 ms, the time needed for a pulse to travel 3 m in air, but the shape of the pulse is different. Due to the non-static character of the convolution matrix, we will not obtain the source signal but a modified one. Looking at the two phase-plots of figure 4.7, we see that the goal of a dispersive-free signal (which is indicated by a straight line) has been fulfilled. The phase in (b) shows some strange behaviour for low and high frequencies, but for the bandwidth area where the amplitude has a significant value, it is a straight line.

(a)

Bending wave sB(ri;t) in glass (left), A = 2.53 m/s0.5 and ri = 3 m, with its amplitude (centre) and phase spectrum (right).

(b)

Pseudo-acoustic result (left) after multiplication of the signal sB(ri;t) from (a) with a filter matrix F, designed for glass with dispersion constant A = 2.53 m/s0.5, and amplitude (centre) and phase spectrum (right).

Figure 4.7:

Bending wave pulse (a) and the dispersion-free pseudo-acoustic variant (b) after applying the dispersion transform by multiplication with matrix F shown in figure 4.6. In (b) we can see that the resulting pulse is positioned at the correct time ta = ri/ca = 3/343 = 8,7 ms, but the shape is different then after the static dispersion deconvolution. The amplitude spectrum (right) has changed, although we have not multiplied with an amplitude factor (see subsection 4.3.1). The two phase-plots show that the dispersion curve has disappeared.

Looking at both spectra in figure 4.7, we see the change in amplitude spectrum of the dispersive pulse. For straight-line phased signals in general, a smooth and symmetric amplitude spectrum will have a narrow pulse in the time-domain. Our pseudo-acoustic spectrum is asymmetrical and therefore has a wider pulse (compare figure 4.7 (b) with figure 4.3 (e)). In subsection 4.3.1 we explained that the goal is not to change the spectrum of

41

The acoustical analysis of bending waves.

Maarten Brink

the signal during the dispersion transform operation, because it contains valuable information, but we can conclude that with the use of a dynamic deconvolution, a change in spectrum is inevitable.

(a)

Bending wave data-set q(t): a simulation of a direct pulse (ri = 2 m) and two reflections (ri = [3,5] m) in glass.

(b)

Pseudo-acoustic result of the multiplication p(t) = F.q(t). Three wavelets become visible.

(c)

Idem as (b) but then plotted against a distance-axis using the relation r = taca. As expected, the operation has removed the dispersion from the input signal and placed the three wavelets at their correct distances.

Figure 4.8:

4.5.4

Data-set of direct pulse (ri = 2 m) and two reflections (ri = [3,5] m) in (a), identical to figure 4.4 (a). Multiplication of this (discrete) signal with matrix F of figure 4.6 results in (b) and (c).

Multiple pulse deconvolution

Let us now look at the advantage of the dynamic dispersion transform: the distanceindependence. Figure 4.4 has shown that the static dispersion transform is not able to deconvolve dispersion from more then one pulses at the same time, but the dynamic version is said to do so. In figure 4.8 (a) the same data-set q(t) of three pulses is shown as in figure 4.4. The result of applying the dynamic dispersion transform is shown in (b,c), having used the following formula:

p ( t ) = Fq ( t ) .

42

(4.23)

Chapter 4: Dispersion Transform The dispersion-free pulses for the three different distances ri seem to have the same symmetrical shape. In matrix representation, the operation of removing the dispersion from the data-set of figure 4.8 is illustrated in figure 4.9.

p(t)

q(t)

F t

ta

Figure 4.9:

4.6

t

ta

Matrix representation of the dynamic dispersion transform, using the data-set from figure 4.8 (compare with figure 4.5).

Scaling and spectral changes

The previous section has shown that the dynamic dispersion transform works: the dispersion of the input signal is removed and the original three pulses become visible. the only disadvantage, the spectral change (and thus a change of the shape of the signal) of the data will be investigated in this section. 4.6.1

Scaling in the frequency-domain

In subsection 4.3.1 we have defined the static dispersion transform in the frequency-domain as follows:

Sa ( r; ω ) = F ( r; ω ) SB ( r ;ω ) = SB ( r ;ω ) e

(

j

1 − 1 cB ca

)ω r ,

(4.24)

and, with the definition of the dynamic dispersion transform, we can substitute car by ta again. We now get

Sa (ta ; ω ) = S B (ta ; ω )e

(

j

)ωt = S (t ; ω )e jω ( )t e − jωt B a

ca −1 cB

ca cB

a

a

a

.

(4.25)

To obtain the time-domain signal sa(ta), we have to inverse-Fourier transform formula (4.25):

1 sa ( ta ) ≡ FT {Sa ( ta ; ω )} ≡ 2π −1

+∞

∫

S a ( ta ; ω ) e

−∞

jω ta

1 dω = 2π

+∞

∫

SB (ta ; ω )e

jω

( )t dω . ca cB

a

(4.26)

−∞

When we want sa(ta) to have the same spectrum as the input signal SB, we just have to take the inverse Fourier transform of it. The right part of formula (4.26) looks like the normal inverse Fourier transform of SB, except for the factor ca/cB factor in the exponent. The data-set is not

43

The acoustical analysis of bending waves.

Maarten Brink

transformed using the normal ω, but using a scaled version of the angular frequency: ω' = ω .ca/cB. Or, in other words, sa(ta) is the inverse Fourier transform of the bending wave data-set, scaled to a new angular frequency ω’:

sa ( ta ) ≡

+∞

1 2π

∫S

B

(ta ; ω ')e jω 'ta dω ' ,

(4.27)

−∞

with for ω' using formula (3.32):

ω'=

ca c ω= a ω cB A

(4.28)

ca dω . 2A ω

(4.29)

and for dω':

dω ' =

This means for the relation between dispersive data-set SB and dispersion-free data-set Sa:

S a (ω ) =

2A c  ω SB  a ω  . ca  A 

(4.30)

We can now understand why the phase and amplitude spectrum change, as observed in subsection 4.5.3, while applying the dynamic dispersion transform. The spectrum SB is plotted against a scaled frequency axis (formula (4.28))and multiplied with a factor

2A ca

ω

that depends on the frequency and on the construction material. 4.6.2

Dispersion constant A

Formula (4.30) describes the change of the signal when dispersion is removed. It contains the dispersion factor A in both the scaling and the amplitude factor, a material constant described in formula (3.32). Looking only at the maximum frequency component of the pulse (in this thesis the simulations have a maximum frequency of fmax = 5 kHz) we can predict what the maximum frequency component of the resulting pseudo-acoustic signal will be using the following formula:

f max ' =

1 ca 2π f max . 2π A

(4.31)

For the three construction materials mentioned in figure 3.4, fmax' is given in table 4.1. We now know where the spectrum of the pulse is placed during the Dispersion Transform, but the actual shape depends on the factor

2A ca

ω and, of course, the shape of the input

spectrum too. Calculating, for example, the frequency of the peak of the window (the input spectrum has its peak at the central frequency fc = 2.5 kHz), will be difficult due to this frequency depending factor: a cosine window with its peak in the centre will not have a the peak in its centre after multiplication with a square-root shaped scaling factor.

44

Chapter 4: Dispersion Transform Table 4.1:

The three construction materials with thickness, dispersion constant and maximum frequency component with for the source fmax = 5 kHz.

material

glass steel concrete

thickness h [mm] 4 30 200

dispersion constant A [m/s0.5] 2.53 6.76 11.4

fmax' [kHz]

3.8 1.4 0.85

What the shape will be for the three construction materials is shown in figure 4.10. The two upper figures of (a), (b) and (c) show the bending sB(ri;t) wave with its amplitude spectrum SB(ri;ω) for ri = 3 m. The two lower figures show the result after multiplication of the simulated pulse with the appropriate Dispersion transform matrix F. Maximum frequency components are conform the theory and the values in table 4.1. Visible in the figures is that the higher the dispersion constant A, the lower the dispersion and therefore the smaller the bending wave sweep, but the wider the pseudo-acoustic pulse.

(a)

Bending and pseudo-acoustic wave in glass (A = 2.53 m/s0.5) with amplitude spectra.

45

The acoustical analysis of bending waves.

(b)

Idem of steel (A = 6.76 m/s0.5).

(c)

Idem of concrete (A = 11.4 m/s0.5).

Maarten Brink

Figure 4.10: Direct bending wave sB(ri;t)and its amplitude spectrum |SB(ri ;ω)| and pseudo-acoustic equivalent sa(ta) and its amplitude spectrum |Sa(ω)| for the different construction materials, ri = 3 m. The bending wave has a 3dB bandwidth of 4.9 kHz, the pseudoacoustic versions have a 3dB bandwidth of: glass (a): 1.3 kHz, steel (b): 0.48 kHz and concrete (c): 0.29 kHz.

4.6.3

Spectral optimisation

The previous subsections have shown that the spectral change is inevitable when we want to remove the dispersion, and the scale of spectral change depends on the material property A. But, looking at formula (4.30) we see that the scaling factors also depend on the acoustic velocity in air. If we step away from the goal stated in the beginning of this chapter of

46

Chapter 4: Dispersion Transform transforming the bending wave signal into a pseudo-acoustic one in air (with ca = 343 m/s), we can influence the scaling mechanism by choosing a new velocity ca. , which we will call ca'. For example, when we want the ratio γ,

γ=

ca ' , A

(4.32)

taken from formula (4.30), to be constant and not depending on the material and the thickness of the plate, we have to choose the appropriate ca' accordingly. Table 4.2 shows us the ca'values needed when we want the spectral changes we saw in glass in figure 4.10 for all three materials. The values for the acoustic velocity from table 4.2 have been applied to dispersion transform simulations in the three materials. Pseudo-acoustic results are shown in figure 4.11. Table 4.2:

Choice of acoustic velocities ca' to keep the spectral change ratio γ constant.

material glass steel concrete

A [m/s0.5] 2.53 6.76 11.4

ca' [m/s] 343 916 1546

γ [s-0.5] 136 136 136

(a)

Pseudo-acoustic pulse and amplitude spectrum for glass with ca' = 343 m/s.

(b)

Idem for steel with ca' = 916 m/s.

47

The acoustical analysis of bending waves.

(c)

Maarten Brink

Idem for concrete with ca' = 1546 m/s.

Figure 4.11: Pseudo-acoustic pulses and amplitude spectra for the three construction materials. The value of ca' is chosen such that fmax' does not depend on the material and is equal to the one of a pseudo-acoustic wave transformed from glass to air.

A different but also interesting choice would be to use the acoustic velocities of the materials themselves, instead of the acoustic velocity in air. For thin plates this material dependent nondispersive velocity is called the quasi-longitudinal velocity ca,ℓ and reads, according to [7],

ca,A =

E 12 A2 = . ρ (1 − µ 2 ) h

(4.33)

Table 4.3 shows the dispersion constant A again, the chosen acoustic velocity ca,ℓ and the ratio γ. Due to extreme high velocities compared with, for example, the velocity in air, γ is very small and aliasing should be prevented. Figure 4.12 shows the pseudo-acoustic results, plotted with the same axes to better show the differences between the different materials. Table 4.3:Values of ratio γ using the quasi-longitudinal wavespeed ca,ℓ.

material glass steel concrete

A [m/s0.5] 2.53 6.76 11.4

ca' [m/s] 5543 5277 2251

48

γ [s-0.5]

0.46·10-3 1.3·10-3 5.1·10-3

Chapter 4: Dispersion Transform

(a)

Pseudo-acoustic pulse and amplitude spectrum for glass using the quasi-longitudinal velocity ca,ℓ = 5543 m/s.

(b)

Idem for steel with ca,ℓ = 5277 m/s.

(c)

Idem for concrete with ca,ℓ = 2251 m/s.

Figure 4.12: As figure 4.11, but now using the acoustic velocities from table 4.3.

From the illustration we can conclude that the spectrum of the pseudo-acoustic waves can be modified by changing the acoustic wave speed, according to our desires. However, the only thing we do is scaling the time-axis of the pseudo-acoustic pulse, the shape and width of the pulse is not improving. 4.6.4

Dispersion transform and noise

The amplitude spectra in figure 4.10 already reveal that removing the dispersion from a signal will be good for the signal-to-noise ratio. The phase-operation, executed during processing, changes the sweep into a pulse: the same energy is arriving at the receiver position in a shorter time and the signal/noise ration will increase. Figure 4.13 shows the simulations of the same situation in glass again, but now two different levels of noise are added to the signal. The dispersion transform matrix F is, of course, free of

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The acoustical analysis of bending waves.

Maarten Brink

noise and identical to the matrix used in previous simulations in glass, with the ca equal to 343 m/s.

(a)

Bending wave with noise (signal/noise = 20 dB) with amplitude spectrum and the result after dispersion transform.

(b)

Same as in (a) but now with a signal/noise ratio of 10 dB.

Figure 4.13: Simulation of bending wave recording in glass (A = 2.53 m/s0.5, r = 3 m) and result after dispersion transform with amplitude spectra. Noise has been added to the simulated time-signal and the pseudo-acoustic signal shows that removing the dispersion increases the signal/noise ratio.

50

Chapter 4: Dispersion Transform Figure 4.13 (a) shows a bending wave signal/noise level of 20 dB, and after dispersion transform this level is about 35 dB. In figure 4.13 (b) even more noise is seen, a signal/noise level of 10 dB, resulting into 20 dB after removing the dispersion. Note that due to the spectral changes of the dispersion transform the noise will become pink-like (amplitude decreases with increasing frequency).

4.7

Array recording simulations

Section 2.4 has shown us the big advantage of using an array of detectors above a single one to investigate the acoustic properties of the enclosure. The simulations in this section will show that the combination of Dispersion transform and an array of detectors (or one detector and an array of sources and presuming reciprocity) can reveal the bending wave distribution in a plate. 4.7.1

Free-field simulations

First let us simulate the unbounded bending wave for our three construction materials and the same situation in air. The shortest distance source-array is 3 m and the set-up is sketched in figure 4.14.

Figure 4.14: Set-up sketch of source and array for the unbounded situation. Source-array minimum distance is 3 m.

The responses in our three construction materials after applying a zero-phase displacement pulse source with bandwidth 20 Hz – 5 kHz, are shown in figure 4.15. Also the response as if the medium would be air, is given using a monopole source. As expected, the bending wave velocity in the glass window is, per frequency, much lower than in the concrete plate. In concrete the wavefront arrives sooner at the detector array and the front almost resembles a plane wave. The wavefront in air is, of course, non-dispersive.

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The acoustical analysis of bending waves. air

Maarten Brink

glass

steel

concrete

Figure 4.15: Free-field array recordings of the responses in air and on the three construction materials, using the set-up of figure 4.14. The source emitted a pulse with bandwidth 20 Hz – 5 kHz.

Now let us apply the dynamic dispersion transform to the bending wave recordings of figure 4.15, using the acoustic velocity in air, ca = 343 m/s. Formula (4.23) can be easily expanded to array recordings to transform an entire data-set at once:

P = FQ ,

(4.34)

where Q is a matrix containing all the array recordings (each column represents once receiver recording q(xn,t)) and P is the dispersion-free result. The matrices F are the same as in formula (4.23) and figure 4.9, depending on the dispersion constant A of the material, on the number of array elements n and on the length of recording. Figure 4.16 shows the same recordings of figure 4.15, but now after dispersion transform. The recording in air has not been processed and we see the dispersion-free pseudo-acoustic wavefronts appear at the same position. Due to scaling effects, explained in subsection 4.6.3, and the choice of a constant ca, the bandwidth of the result depends on the material.

52

Chapter 4: Dispersion Transform air

glass

steel

concrete

Figure 4.16: Measurement in air and pseudo-acoustic versions of bending wave recordings of figure 4.15. As seen, applying the Dispersion transform removes the dispersion. Because we used the acoustic velocity of air, ca = 343 m/s, for each matrix F, the wavefronts are placed at the same position in time as the air-recording. Note that with increasing A, the pulse becomes wider, as explained in subsection 4.6.3.

4.7.2

Plate simulations

Adding (rigid) boundaries to the simulation with the use of a mirror image source script, will returns an incomprehensible data-set of with only a direct wave recognisable. Figure 4.17 shows the new plate set-up that is used for the simulations and figure 4.18 the simulation results for the three different plate materials.

2.5 m

1.8 m

source *

oooooooooooooooooooooo detector array

0.5 m

Figure 4.17: Set-up of plate with source and detector array position.

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The acoustical analysis of bending waves. glass

Maarten Brink steel

concrete

Figure 4.18: Simulated array responses of the three construction materials using the set-up of figure 4.17.

Again, we multiply these data-sets Q with exactly the same dispersion transform matrices as in subsection 4.7.1. Results are shown in figure 4.19 and reflections become visible. All three materials show the same spatial information (the three plates had the same size), but, as expected, the simulated glass recording returns the 'sharpest' picture. Since we are dealing with simulations, further processing to improve the images is not necessary. glass

steel

Figure 4.19: Pseudo-acoustic version of the array simulations of figure 4.17.

54

concrete

CHAPTER 5 Measurements 5.1

Introduction

Chapter 4 has shown that the dynamic dispersion transform works: a bending wave field was generated using the mirror image model and material properties and, after multiplying this simulated measurement with a dispersion transform matrix, dispersion free wavefronts became visible. This chapter will introduce real measurements to investigate to what extend the dispersion transform is functional in 'real-life' situations, where boundaries never are ideal, measurement equipment has its limitations and materials aren't homogeneous.

5.2

Steel plate

5.2.1

Set-up and measurement results

From [11] we will use the measurements executed on a steel plate. A large plate is used with free boundaries, placed on a rubber mat. Size and thickness are given in table 5.1. Using formula (3.32) we come to a dispersion constant of A = 6.76 m/s0.5. Table 5.1:

Properties of the steel plate used in the experiment from [11].

Steel plate

length [cm] width [cm] thickness [cm] density [·103 kg/m3] Mod. of Elasticity [GPa] Poisson constant

250 180 3.0 7.85 210 0.2

In figure 5.1 the experimental set-up is shown. Six source positions have been used and the receiver array consisted out of 36 points of measurement with distance 5 cm. The results of the six positions are shown in figure 5.2.

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The acoustical analysis of bending waves.

Maarten Brink 1.8 m

* s1

2.5 m

* s2 * s3

* s5

* s4

* s6

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

|| 5 cm

offset

0.5 m

Figure 5.1:

Steel plate (A = 6.76 m/s0.5) with the array detector and the six source positions s1 − s6.

Figure 5.2:

Results of the six measurements on the steel plate in the set-up of figure 5.1. The shape of the direct wave is visible and first reflections can more or less be recognized.

56

Chapter 5: Data processing 5.2.2

Dispersion Transform

To remove the dispersion from the measurements and obtain pseudo-acoustic versions, we multiply with the same dispersion removal matrix F as in the simulations of Chapter 4. The results are shown in figure 5.3, and we see that, just as in the simulations, the early reflections become visible as clear wavefronts. With the geometry of figure 5.1, we can identify the reflections from the sides of the plates.

Figure 5.3:

Dispersion-free version of figure 5.2. The data-sets are multiplied with Dispersion transform matrix F, designed for steel with thickness 30 mm.

For better comparison the pseudo-acoustic data-set no. 4 and the simulation in figure 4.19 are given in figure 5.4. Note that for the simulation the same plate-size, dispersion constant, source and receiver position, number of receivers and approximately the same bandwidth is used as in the real measurements.

57

The acoustical analysis of bending waves.

Figure 5.4:

Maarten Brink

Pseudo-acoustic data-sets of the simulation (left) and measurement at point 4 of the setup of figure 5.1.

Looking at figure 5.4 (a) and (b) we see that the images are more or less the same, but a closer look reveals that the reflected wavefronts in the simulation appear earlier in time. Apparently the real reflections at the boundaries behave different from the simulation program, which is originally designed from acoustic simulations in air. 5.2.3

Acoustic image processing

In Chapter 2, section 4, we have visualized the hyperbolic shapes by mirroring the data-sets along the offset boundaries. Now that we have a applied the Dispersion transform on a multitrace bending wave impulse response of a plate, we can do the same with the pseudo-acoustic data-set. Figure 5.5 shows the fold-out version of figure 5.4 (b) and two hyperbolic wavefronts become visible. The white-dotted line (this time only shown at one side not to spoil the plot too much) represents the asymptotes of acoustic velocity ca, explained in subsection 2.2.4. On the data-set of figure 5.5 we can apply the Normal Move-Out, spatial filtering and Inverse Normal Move-Out procedure, this time for all possible distances. The result is shown in figure 5.6. We can 'fold up' this data-set again, going back to the normal representation. Figure 5.7 (a) shows the original pseudo-acoustic data-set and figure 5.7 (b) the data-set after fold-up of figure 5.6.

58

Chapter 5: Data processing

Figure 5.5:

Fold-out representation of the data-set of figure 5.4 (b). The white dashed line represents the acoustic velocity asymptote in one offset direction . As expected, the two visible hyperbolae approach this asymptote.

Figure 5.6:

Fold-out representation as in figure 5.5, but after applying Normal Move-Out, spatial filtering and Inverse Normal Move-Out.

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The acoustical analysis of bending waves.

Figure 5.7:

Maarten Brink

Pseudo-acoustic data-sets. (a) is identical to figure 5.3, no. 4, and (b) is the data-set of figure 5.6 after fold-up.

Comparing figure 5.7 (a) and (b), we see that in between the wavefronts the noise has been reduced and the wavefronts are narrower. However, the 'visibility' of the total image and wavefronts hasn't improved much. 5.2.4

Damping

Similar to the measurement of the concert hall from subsection 2.3.3 we can plot the decay curves for different frequencies in a single plate measurement. Formula (2.19) is applied to one of the bending wave impulse responses and the results are shown in figure 5.8. Note that we do not execute these decay and reverberation time calculations on a pseudo-acoustic dataset, but on a real measurement. The frequency shift and the dispersion transform will affect the reverberation time, while we only want to investigate the displacement decay for a certain frequency band. From the plots we can derive a general (60 dB decay) reverberation time T of approximately 11 s and using formula (3.41) this leads to an approximate damping factor of η = 0.20/f. We can also use the specific reverberation time Ts of the six octave bands to derive the damping factor η; results are shown in table 5.2.

60

Chapter 5: Data processing

Figure 5.8:

Raw impulse response and reverberation graphs from a measurement of one of the array elements on the steel plate. The reverberation time, the average of 60 dB decays of the octave bands of 500 and 1000 Hz is approximately 11 s.

Table 5.2:

From the plots derived specific reverberation times Ts and damping factors η for the six measured octave bands with centre frequency fc.

fc [Hz] (octave band) 250 500 1000 2000 4000 8000

5.3

Ts [s] 14 10 12 13 12 11

η [·10-3] 0.6 0.4 0.2 0.08 0.05 0.03

Aluminium and Glare® plates

Similar to the large and heavy steel plate, measurements have been executed with two smaller plates: aluminium and Glare, the last one being a composite aircraft material. 5.3.1

Measurement set-up

For both plates the same set-up has been used, illustrated in figure 5.9. The edges are clamped by round metal bars and the properties of the plates are shown in table 5.3. For

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The acoustical analysis of bending waves.

Maarten Brink 0.5

aluminium this results into a dispersion constant of A = 1.2 m/s . For the Glare plate A is unknown. The set-up of the experiment with the positions of the source and the array is illustrated in figure 5.9. Table 5.3:

Properties of the aluminium and Glare plate used in the experiment. Some properties of the Glare plate are unknown.

Glare® 100 70 0.12 ? ? ?

aluminium 100 70 0.10 71 2.7 0.1

length [cm] width [cm] thickness [cm] Mod. of Elasticity [GPa] density [·103 kg/m3] Poisson constant

70 cm

15 cm

1.0 m

* S

o o o o o o o o o o o o o o o o o o o o o o o o o o o

|| 2.5 cm

Figure 5.9:

5.3.2

offset

10 cm

Measurement set-up of both the aluminium and glare plate.

Measurement results of the aluminium plate

Figure 5.10 (a) shows the measurement result of aluminium and (b) shows the data-set after applying the dynamic dispersion transform with A = 1.2 m/s0.5. A direct wave and a few side reflections become visible, but the quality is low.

62

Chapter 5: Data processing

Figure 5.10: Measurement (a) and pseudo-acoustic version (b) of the aluminium plate. Shortest distance from source to array is 75 cm and the direct wavefront, visible in the pseudoacoustic result (b), arrives at approximately 2 ms.

5.3.3

Measurement results of the Glare® plate

Figure 5.11 shows the measurement data-set of the Glare plate. Because we don't know the dispersion constant A for this plate, we have generated eight different dispersion transform matrices, for A = 0.8 m/s0.5 to 1.5 m/s0.5. The eight pseudo-acoustic results are shown in figure 5.12. A direct wave and a few reflections become visible, but again the quality is low. Knowing the measurements of the set-up and comparing the results with the result for aluminium in figure 5.10 (b), we can estimate the dispersion constant for Glare with thickness h = 1.2 mm to be approximately 1.0 m/s0.5.

Figure 5.11: Measurement result of the Glare plate.

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The acoustical analysis of bending waves.

Maarten Brink

Figure 5.12: Pseudo-acoustic result after multiplication of the data-set of figure 5.11 with matrix F, for 8 different values of dispersion constant A.

64

CHAPTER 6 Conclusions and recommendations In 1998 Martens has showed that it is possible to remove the dispersion from a signal when the properties that cause the dispersion are known [11]. Instead of seeing an incomprehensible pattern of displacements when doing array measurements on a plate, clear wavefronts appear after applying, what he called, the Dispersion Transform on them. Now, identification of reflections from one or more of the boundaries of the plate is possible. Instead of the ‘scaled Fourier transform’ approach in [11], this thesis has tried to explain the dispersion removal procedure by means of a convolution, first introduced by Berkhout et al. in [9]. This approach sees the dispersion as a property of the signal that can be filtered out by convolution with an opposite signal. Subsequently, the obtained zero-phase signal can be shifted to the correct pseudo-acoustic time. When recording the bending wave impulse response of a plate, we will have a signal with multiple reflections. We have seen that for such a dispersive signal a simple convolution does not work anymore. The Dynamic Dispersion Transform is introduced and the input signal is convolved with a signal that changes during the convolution process. This operation is best represented by a matrix multiplication. Important aspect of the dispersion transform is the flexibility of choosing the pseudo-acoustic velocity. So far we have been transforming the dispersive measurements into pseudo-acoustic ones in air. We could also choose the non-dispersive (quasi-)longitudinal velocity of the plate as pseudo-acoustic reference. This way the measured transversal displacements from bending waves are transformed into longitudinal displacements and we remain within the physics and acoustics of the structure material. However, the three materials used all have different and such high longitudinal velocities, that in this thesis is chosen for the uniform acoustic velocity of air. The greater part of the research for this thesis has been simulating bending waves. Single and multiple bending waves and even multi-trace impulse responses of plates have been simulated to test the dispersion transform. As a result we now understand how and to what extend the dispersion transform is useful and what the positive and negative properties are. Next step after the many simulations is doing real measurements. Once a measurement is transformed into pseudo-acoustics, the processing and investigation of the data can start. An inevitable disadvantage of analysing the plate data is the limited size of the object. A zerophase pulse has, depending on the bandwidth, a certain 'width' in the time recording. The wavefronts in the pseudo-acoustic data-set of a plate follow each other so quickly that isolating a certain reflection is much more difficult than in room acoustics. This was especially a problem for the smaller aluminium and glare plates. Changing the pseudoacoustic velocity doesn’t improve things, because the wavefronts and the time between the wavefronts are both scaled at the same time. A solution could be introducing a source with a wider spectrum, but this is not possible when hitting the plate with a hammer as source pulse.

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The acoustical analysis of bending waves.

Maarten Brink

Attaching a heavy exciter will influence the whole vibration behaviour of the plate. A possible solution to this problem could be the light and flat piezo-polymer exciters, that bend when a voltage is introduced. Another practical problem was measuring the displacement. The steel plate measurements have been done with normal accelerometers attached on them, but for the lighter aluminium and glare plates these would influence the vibrations too much. For the last two a non-contact capacitive transducer is used, which was noise sensitive and has a relatively large surface of measurement. Solutions to this can be the modern 'ultra-light' accelerometers or the advanced method of laser interferometry. In the flat loudspeakers (DML's), mentioned in Chapter 1, even lighter plate material is used, especially chosen for its lightness and stiffness. For this material the laser interferometer is advised, because accelerometers will probably have a too big influence and the capacitive transducer only works well with conductive materials. Exciting a pulse with a hammer will not be an option for DML plates, but the panels have one or more exciters attached that can be used as a source. The heavy exciters are a relatively big load on such a plate and to what extent this influences the wavefronts is not discussed in this thesis. The knowledge of the dispersion transform is not only restricted to bending waves in plates. The square-root dependence of the velocity on the frequency, typical for these bending waves, is changeable to any other 'dispersion power' ωn.

66

Appendix Convolution and correlation The convolution operator is closely related to another process, called correlation. We define the cross-correlation function as follows:

z (τ ) = x ( t ) ⊗ y ( t ) ,

(A.1)

with for ⊗ the correlation operator. This is a symbolic notation for

z (τ ) =

+∞

∫ x ( t ) y (τ + t ) dt .

(A.2)

−∞

As we can see is τ a normal time variable, in this process representing a time shift. Using the substitutions t = −t and the knowledge of the convolution from subsection 4.4.1, we come to the following: −∞

+∞

+∞

−∞

z (τ ) = − ∫ x ( −t ) y (τ − t ) dt =

∫ x ( −t ) y (τ − t ) dt = x ( −t ) ∗ y ( t )

→ z (τ ) = x ( t ) ⊗ y ( t ) = x ( −t ) ∗ y ( t ) = y ( t ) ∗ x ( −t ) .

(A.3)

Mirroring the signal x(t) in t = 0 leads to

z (τ ) = y ( t ) ∗ x ( t ) = x ( −t ) ⊗ y ( t ) .

(A.4)

Applying formula (A.4) on the convolution of bending wave sB(r,t) with filter signals f(r,t) and g(r,t) in formula (4.9), we get the following formula for the pseudo-acoustic result sa(ri;t) (see also figure 4.3):

sa (ri ; ta ) = sB ( ri ; t ) ∗ f ( ri ; t ) = f ( ri ; −t ) ⊗ sB ( ri ; t )

 r = sB ( ri ; t ) ∗ g  ri ; t − i ca 

  ri  = g  ri ; −t − ca  

  ⊗ sB ( ri ; t ) 

(A.5)

and instead of using the time-variable τ, we used the pseudo-acoustic traveltime ta:

ta =

67

ri . ca

(A.6)

Using the substitution of formula (4.14) in formula (A.5), sa(ri;t) now becomes

sa (ri ; ta ) =

+∞

   ri ri  ˆ ˆ + − = + s r ; t s r ; t t d t s r ; t ( )     ⊗ sB ( ri ; t ) . B i B i a B i ∫−∞ c c a a    

(A.7)

Finally, the dynamic version of formula (A.7) is:

sa ( ta ) =

+∞

∫ s ( t ) sˆ ( t ; t ) dt = sˆ ( t ; t + t ) ⊗ s ( t ) . B

B

a

B

a

a

B

(A.8)

−∞

From formula (A.8) we can conclude that the dynamic dispersion transform can be seen as a convolution of the measured signal sB (t ) with all ideal bending wave sˆB ( t ) , but, to obtain a peak at the correct pseudo-acoustic traveltime position, the ideal bending wave is shifted to the left in time by ta and becomes sˆB ( t + ta ) .

68

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[1]

A.J. Berkhout, Applied Seismic Wave Theory, Elsevier, Amsterdam, 1987

[2]

M.M. Boone & D. de Vries, Geluidsbeheersing/Sound Control, Delft University of Technology, Delft, 1995

[3]

D.W. van Wulfften Palthe & D. de Vries, Inleiding in de Akoestiek (c36), Technische Hogeschool Delft, Delft, 1976

[4]

H. Kuttruff, Room Acoustics, Applied Science Publishers LTD, London, 1973

[5]

J. Baan & D. de Vries, Array Technology for Discrimination between Specular and Non-specular Reflections in Enclosed Spaces, preprint 4713, 104th Convention Audio Eng. Soc., Amsterdam, 1998

[6]

J.J. Sonke, Variable Acoustics by Wave Field Synthesis, Ph.D. Thesis, J.J. Sonke, Amsterdam, 2000

[7]

L. Cremer & M. Heckl, Körperschall – Physikalische Grundlagen und techniche Anwendungen – 2., völlig neubearbeitete Auflage, Springer, Berlin, 1995

[8]

F. Fahy, Sound and Structural Vibration – Radiation, transmission and Response, Academic Press Inc., London, 1985

[9]

A.J. Berkhout & D. de Vries & M.C. Brink, Array Technology for Bending Wave Field Analysis in Constructions, pre-submitted publication, J. Acoust. Soc. Am.

[10] W. van Rooijen, Distributed Mode Loudspeakers for Wave Field Synthesis, M.Sc. Thesis, Laboratory of Acoustic Imaging and Sound Control, Delft University of Technology, Delft, 2001 [11] J.W.M.A.F. Martens, Analysis of Bending Wave Patterns using Array Technology, M.Sc. Thesis, Laboratory of Acoustic Imaging and Sound Control, Delft University of Technology, Delft, 1998

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