Waves & Sound Chapter 15 & 16 Dr. Ray Kwok SJSU

Vibrations & Waves - Dr. Ray Kwok

Topics











Wavelength, frequency and wave speed Traveling & Standing Waves Transverse & Longitudinal Waves Wave Interference Doppler Effect Shock Waves Sound Wave – production and speed Sound Quality – frequency domain Harmonic Beats

Vibrations & Waves - Dr. Ray Kwok

Simple Harmonic Oscillations

pendulum spring – mass system period T = 2π√(L/g) independent of mass

Energy conserve KE ↔ PE

T = 2π√(m/k) Restoring force F = -kx (Hooke’s Law) k is the spring constant

Vibrations & Waves - Dr. Ray Kwok

Sinusoidal Nature of SHO λ

recording paper pen

frequency = 1 / period v=fλ=λ/T

(Time)

Vibrations & Waves - Dr. Ray Kwok

Pendulum – wave nature

Oppenheimer at SF Exploratorium

sand on rolling paper

Vibrations & Waves - Dr. Ray Kwok

Waves radio waves

water waves

tidal waves

light waves

sound waves

Vibrations & Waves - Dr. Ray Kwok

Traveling Waves

direction of propagation

amplitude variation

e.g. sport fans

Vibrations & Waves - Dr. Ray Kwok

Traveling Wave f ( x ± vt ) ≡ f (u )

reverse / forward traveling wave

∂f ∂f ∂u ∂f = = ≡ f ' (u ) ∂x ∂u ∂x ∂u ∂ 2f ∂u = f "(u ) = f " (u ) 2 ∂x ∂x ∂f ∂u = f ' (u ) = ± vf ' (u ) ∂t ∂t ∂ 2f ∂u 2 = ± = vf " ( u ) v f " (u ) 2 ∂t ∂t ∂ 2f 1 ∂ 2f = 2 2 wave equation 2 ∂x v ∂t

note:

x ± vt =

1  2π 2π  x ± vt   λ  k λ

1 (kx ± 2πft ) k 1 = ± (ωt ± kx ) k f ( x ± vt ) = f (ωt ± kx ) r r f ωt − k ⋅ r (3D) =

(

)

Vibrations & Waves - Dr. Ray Kwok

1D - Wave Solution ∂ 2f 1 ∂ 2f = 2 2 2 v ∂t ∂x f ( x , t ) = A cos(kx − ωt + φ) f ( x , t ) = A cos(kx + ωt + φ) 2π k= λ 2π ω = 2πf = T λ ω v = = fλ = T k

propagate to +x direction propagate to -x direction

Fourier Transformation & Superposition

Vibrations & Waves - Dr. Ray Kwok

2 types of traveling waves Longitudinal

Transverse

Vibrations & Waves - Dr. Ray Kwok

Transverse & Longitudinal Waves

(transverse)

Vibrations & Waves - Dr. Ray Kwok

Velocity on string Horizontal F = T (same) only transverse movement

 ∂y  = −  slope F  ∂x  x F2 y  ∂y  = +  F  ∂x  x + ∆x F1y

∑F

y

= ma y

 ∂y   ∂y   F1y + F2 y = F  −    = (µ∆x )&y&  ∂x  x + ∆x  ∂x  x  µ=mass/length  ∂y   ∂y  −     ∂x  x + ∆x  ∂x  x = µ &y& F ∆x ∂2y µ ∂2y = wave equation ∂x 2 F ∂t 2 F T v= = T is the Tension here µ µ

Vibrations & Waves - Dr. Ray Kwok

Periodic waves

Forward transverse traveling wave. Displacement of string :

r u ( x , t ) = yˆA cos(ωt − kx + φ)

Vibrations & Waves - Dr. Ray Kwok

Periodic waves II

Forward longitudinal traveling wave. Displacement of fluid:

r u ( x, t ) = xˆA cos(ωt − kx + φ)

Vibrations & Waves - Dr. Ray Kwok

Wave Reflection




Waves in motion from one boundary (the source) to another boundary (the endpoint) will travel and reflect.

Vibrations & Waves - Dr. Ray Kwok

Boundary Conditions

Vibrations & Waves - Dr. Ray Kwok

Interference 2 waves “add” together

In-phase Constructive interference Larger amplitude

180 deg out-of-phase Destructive interference Zero amplitude

Vibrations & Waves - Dr. Ray Kwok

Periodic Waves with reflection

standing waves

resonance

Vibrations & Waves - Dr. Ray Kwok

Standing wave Assuming same amplitude & phase change is either 0 or 180o

u = u1 + u 2 = u o cos(kx − ωt ) ± u o cos(kx + ωt ) u = u o [(cos kx cos ωt + sin kx sin ωt ) ± (cos kx cos ωt − sin kx sin ωt )] u = 2u o cos kx cos ωt

, or depends on the boundary condition

u = 2u o sin kx sin ωt Standing wave equations

Vibrations & Waves - Dr. Ray Kwok

Resonance


At the right frequency, amplitude adds up fast

Vibrations & Waves - Dr. Ray Kwok

Water wave - interference

Vibrations & Waves - Dr. Ray Kwok

Superposition & Interference

2-dimensional waves

More so in optics…

Vibrations & Waves - Dr. Ray Kwok

Interference in sound wave

constructive

destructive

Vibrations & Waves - Dr. Ray Kwok

Constructive & Destructive Interference

Vibrations & Waves - Dr. Ray Kwok

Doppler Effect

longer T lower f lower pitch

stationary source

shorter T higher f higher pitch

moving source

Vibrations & Waves - Dr. Ray Kwok

Doppler – moving source

Speed of sound v does not depends on vs once the wave leaves the source.

v fs = λs vs v  v  = λ s m s = λ s 1 m s  fs v / λs v  v v fs fL = = = Top sign : approaching (A), higher f, higher pitch λL  vs   vs  Bottom: departing (B), lower f, lower pitch λ s 1 m  1 m  v  v  λ L = λ s m vs Ts = λ s m

Vibrations & Waves - Dr. Ray Kwok

Doppler - Audio

e.g. red shift & blue shift in astronomy

Vibrations & Waves - Dr. Ray Kwok

Moving Observer

v fs = λs fL =

Same effect. Which end would hear a higher pitch? “Motion is relative”

wave travels at a higher speed w.r.t. the observer

v ± vL v v   v  = 1 ± L  = f s  1 ± L  λs λs  v  v  

Top sign : approaching, higher f, higher pitch Bottom: departing, lower f, lower pitch

Vibrations & Waves - Dr. Ray Kwok

Doppler - summary fL =

fs  vs  1 m  v 

Moving source

 v  f L = f s 1 ± L  Moving observer v    vL  1 ±   v ± vL  v    f L = fs = f s  combined v m vs   vs   1 m   Top sign : approaching, higher f, higher pitch v   Bottom: departing, lower f, lower pitch

Vibrations & Waves - Dr. Ray Kwok

Radar Gun f

receive Double shift!!!

moving observer f + ∆f

reflection moving source

Vibrations & Waves - Dr. Ray Kwok

Source traveling at high speed

Mach 1 High pressure wave front e.g. speed boat

Vibrations & Waves - Dr. Ray Kwok

Fastest Plane


Commercial (still running) Concorde, Mach 2.02 (1,330 mph) [ Boeing 747 cruise speed 550 mph ]










Lockheed’s Blackbird SR-71 Mach 3.2 (2,094 mph, Dec 1964) USAF X-15 (rocket engine) Mach 6.85 (4,520 mph) USAF X-43 (unmanned, launched in air) Mach 9.8 (7,000 mph, Nov 2004)

Vibrations & Waves - Dr. Ray Kwok

Shock Wave

Vibrations & Waves - Dr. Ray Kwok

Sonic Boom

Thunder is a type of natural sonic boom, created by the rapid heating and expansion of air in lightning discharge.

Vibrations & Waves - Dr. Ray Kwok

Sound Wave – tuning fork

compression

rarefaction

Vibrations & Waves - Dr. Ray Kwok

Sound Wave

Vibrations & Waves - Dr. Ray Kwok

Pickup pressure (sound) wave (amplifier)

with hair cells in fluid

Vibrations & Waves - Dr. Ray Kwok

Speed of sound

Sound travels faster in denser material !!

Vibrations & Waves - Dr. Ray Kwok

Much louder

Bypass air, middle ear bones picks up vibration Faster sound speed with bigger amplitude.

Vibrations & Waves - Dr. Ray Kwok

Quality of Sound

Vibrations & Waves - Dr. Ray Kwok

Tuning Fork

frequency Fourier Transformation

Vibrations & Waves - Dr. Ray Kwok

Same note, different instrument

Vibrations & Waves - Dr. Ray Kwok

Different composition

4th

Vibrations & Waves - Dr. Ray Kwok

Normal modes (string) even after the driving force is removed !

Fundamental, L = λ/2, f1 = v/λ λ = v/2L

Fourth Harmonic, L =2λ λ, f4 = v/λ λ = 2v/L = 4f1

Second Harmonic, L = λ, f2 = v/λ λ = v/L = 2f1

Third Harmonic, L =1.5λ λ, f3 = v/λ λ = 3v/2L = 3f1

Vibrations & Waves - Dr. Ray Kwok

Tube open at both ends pressure wave

v is the speed of sound in air

Vibrations & Waves - Dr. Ray Kwok

Tube closed at one end

Vibrations & Waves - Dr. Ray Kwok

Sound signature

Vibrations & Waves - Dr. Ray Kwok

The logarithmic decibel scale of loudness dB = 10 log(I/I ) Io =

10-12

o 2 W/m

3dB = 10 log(2) -3dB = 10 log(1/2)

Vibrations & Waves - Dr. Ray Kwok

Beats

Vibrations & Waves - Dr. Ray Kwok

Beat - superposition y1 = A sin ω1t y 2 = A sin ω2 t a+b a−b sin a + sin b = 2 sin cos 2 2 ( ( ω1 + ω2 )t ω1 − ω2 )t y total = y1 + y 2 = 2A sin cos 2 2 y total = 2A sin[(f1 + f 2 )πt ]cos[(f1 − f 2 )πt ]

far away from f1 or f2

beat frequency = f1 – f2

Vibrations & Waves - Dr. Ray Kwok

Unequal spacing

Visual equivalent to beating in sound