Chapter 5 – Reflection, Transmission and Standing Waves Slides to accompany lectures in
Vibro-Acoustic Design in Mechanical Systems © 2012 by D. W. Herrin Department of Mechanical Engineering University of Kentucky Lexington, KY 40506-0503 Tel: 859-218-0609
[email protected]
Normal Incidence Against a Rigid Barrier ux = 0
x
ux = ui e (
j ω t−kx )
+ ur e (
j ω t+kx )
at x = 0 j ωt ui e jωt + ur e ( ) = 0
• The amplitude of the reflected wave is equal to that of the incident wave. • The intensity of both waves is equal.
ur = −ui = ui e jπ Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Normal Incidence Against a Rigid Barrier ux = 0
ux = ui e (
j ω t−kx )
+ ur e (
ux = ui e (
− ui e (
j ω t−kx )
j ω t+kx )
x
j ω t+kx )
ux = ui e jωt ( e− jkx − e jkx ) = −2 jui e jwt sin ( kx ) u ( x, t ) = 2ui sin ( kx ) sin (ω t ) p ( x, t ) = 2ui ρ0 c cos ( kx ) cos (ω t ) Dept. of Mech. Engineering University of Kentucky
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First Node (where p(x,t) is zero) π kx0 = − 2 2π f 2π kx0 = x0 = x0 c λ λ x0 = − 4 ME 510 Vibro-Acoustic Design
Standing Waves
Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Two Elastic Media Medium 1
Medium 2
Z1 = ρ1c1
Z 2 = ρ2 c2
pi ( x, t ) = pi e (
pt ( x, t ) = pt e (
j ω t−k2 x )
j ω t−k1x )
ui ( x, t ) =
pi j(ωt−k1x) e ρ1c1
ut ( x, t ) =
pr ( x, t ) = pr e (
j ω t+k1x )
ur ( x, t ) = −
pt j(ωt−k2 x) e ρ2 c2
x
pr j(ωt+k1x) e ρ1c1
Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Boundary Conditions at x = 0 pi ( x = 0, t ) + pr ( x = 0, t ) = pt ( x = 0, t ) ui ( x = 0, t ) + ur ( x = 0, t ) = ut ( x = 0, t ) pi + pr = pt pi pr pt − = ρ1c1 ρ1c1 ρ2 c2 Then Dept. of Mech. Engineering University of Kentucky
pi − pr pi + pr = ρ1c1 ρ2 c2 6
ME 510 Vibro-Acoustic Design
The Reflection Coefficient pi − pr pi + pr = ρ1c1 ρ2 c2
1− R 1+ R = ρ1c1 ρ2 c2
where
pr R= pi
ρ1c1 ρ2 c2 − ρ1c1 ρ2 c2 R= = ρ2 c2 + ρ1c1 1+ ρ1c1 ρ2 c2 1−
R is always real for real specific impedances. Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
The Transmission Coefficient pt T= pi
2 ρ2 c2 2 T= = ρ2 c2 + ρ1c1 1+ ρ1c1 ρ2 c2
T is always real for real specific impedances.
Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Three Special Cases ρ1c1 1− ρ2 c2 R= ρc 1+ 1 1 ρ2 c2 For
ρ1c1 < ρ2 c2
0 < R ρ2 c2
−1 < R < 0 pi and pr are out of phase
ME 510 Vibro-Acoustic Design
Complex Specific Impedance Medium 2 typically has losses or is limited in its extent.
p p Z= = u ⋅ n un Medium 1 Wi
Medium 2 Wt
Wr
x Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Complex Specific Impedance Z 2 − ρ1c1 R= Z 2 + ρ1c1
Medium 1 Wi
Medium 2 Wt
Wr
x Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
The Absorption Coefficient Wt Wi − Wr Wr α= = = 1− Wi Wi Wi
α = 1+
I x, r I x, i
pˆ r2 2 = 1− 2 = 1− R pˆ i
Medium 1 Wi
Medium 2 Wt
Wr
x Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Propagation of Plane Waves in 3D Space ω k= c k = k ⋅ n = k x ex + k y ey + kz ez j (ω t−k x x−k y y−kz z ) p ( r, t ) = pe
Wave Fronts
n
λ
k = k x2 + k y2 + kz2
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ME 510 Vibro-Acoustic Design
Oblique Incidence between Two Fluid Media
Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Blackstock, 2000
Oblique Incidence between Two Fluid Media c1Δt = Δysin θ i c2 Δt = Δysin θ t c1 c2 = sin θ i sin θ t
The incident, reflected, and transmitted waves have the same periodicity.
λi λ λ = r = t sin θ i sin θ r sin θ t Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Case 1 Case 1 For c1 > c2
c2 sin θ t = sin θ i c1 θt is redirected to the normal if θi is 90°
θ tmax = arcsin
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c2 c1
ME 510 Vibro-Acoustic Design
Case 1
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ME 510 Vibro-Acoustic Design
Case 2 Case 2 For c1 < c2
c1 sin θ i = sin θ t c2 θt > θi if θt is 90°
θ icutoff = arcsin
Dept. of Mech. Engineering University of Kentucky
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c1 c2
ME 510 Vibro-Acoustic Design
Case 2
Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Oblique Incidence between Two Fluid Media The Reflection Coefficient
pr ρ2 c2 cosθ i − ρ1c1 cosθ t R= = pi ρ2 c2 cosθ i + ρ1c1 cosθ t The Transmission Coefficient
T=
Dept. of Mech. Engineering University of Kentucky
pt 2 ρ2 c2 cosθ i = pi ρ2 c2 cosθ i + ρ1c1 cosθ t
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ME 510 Vibro-Acoustic Design
Locally Reacting Surface Every point on the surface is considered to be completely isolated from all other points.
p Z= u⊥
Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Locally Reacting Surface Continuity of Sound Pressure
pˆ i + pˆ r e jδr = pˆ r e jδt Continuity of Particle Velocity
pˆ i pˆ r jδr pˆ t jδt − e = e ρ1c1 ρ1c1 Z 2 Eliminate
pˆ t e jδt
pˆ r jδr Z 2 cosθ i − ρ1c1 R= e = pˆ i Z 2 cosθ i + ρ1c1 Dept. of Mech. Engineering University of Kentucky
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ME 510 Vibro-Acoustic Design
Locally Reacting Surface The locally reacting assumptions is applicable for: 1. Anisotropic medium (see below) 2. Medium with significant losses like fibers or foams 3. Medium with c2
