4.7 Homework Exercises |
111 |
function [pressure,velocity]=getStandingWave(A,w,k,Time,XDir,phi)
% ps(t,x)=P+(t,x)+p-(t,x) |
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=A cos(wt-kx+phi)+A cos(wt+kx+phi) |
(Form 2) |
%=2A cos(wt+phi)*cos(kx) --- see the homework Problem 3.4
%modify the following two lines for the pressure and velocity
%assume loc=0
pressure=0; % pressure of standing wave |
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velocity=0; % velocity of standing wave |
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end |
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% from Project 9.1.1 |
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function [A, phi]=get_A_and_phi(Ac, As) |
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% p(t) = Ac cos(wt) + As sin(wt) |
(Form 1) |
% = A cos(wt + phi) |
(Form 2) |
A=(Ac^2+As^2)^(1/2); |
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phi=atan2(As,Ac); |
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end |
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4.7Homework Exercises
Exercise 4.1 (Traveling Wave with Known Velocity)
Use one of the harmonic motions shown below (chosen by the instructor) as the velocity u(t) to answer every question of this exercise (a–c):
i.uðtÞ ¼ U0 sin ωt π2 ; U0 is a real number.
ii.u(t) ¼ Re (U0e j(ωt π)); U0 is a real number.
iii.uðtÞ ¼ U0 cos ωt þ π2 ; U0 is a real number.
iv. uðtÞ ¼ Re U0e jðωtþπ2Þ ; U0 is a real number.
A large, rigid wall is vibrating with a velocity u(t) perpendicular to its planar surface. The motion of the wall creates plane waves that propagate into the air. Consider the particle velocity of the plane wave on the surface of the wall (x ¼ 0) to be equal to the velocity of the wall:
= 0
a)Obtain the expressions for the flow velocity at any point in space (not just on the wall surface) of the plane wave radiating from the wall in terms of the wall vibration velocity.
112 |
4 Acoustic Intensity and Specific Acoustic Impedance |
b)Obtain the expressions for the sound pressure at any point in space (not just on the wall surface) of the plane wave radiating from the wall in terms of the wall vibration velocity.
c)What is the root-mean-square (RMS) pressure at any point in space if the wall velocity U0 ¼ 0.01 m/s.
Use 415 rayls for the characteristic impedance of air. (Answers):
i, ii |
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(a) Uo sin |
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; (b) ρoc Uo sin |
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ð4:15Þ2 |
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iii, iv |
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(a) Uo cos |
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kx |
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π |
; (b) ρoc Uo cos |
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ð4:15Þ2 |
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Exercise 4.2 (Traveling Wave with Known Velocity)
Use the following harmonic motion u(t) as velocity to answer every question of this exercise:
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uðtÞ ¼ |
Re 0:02 e jðωtþ2Þ |
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A large, rigid wall is vibrating with a velocity u(t) perpendicular to its planar surface. The motion of the wall creates plane waves that propagate into the air. Considering the particle velocity of the plane wave on the surface of the wall (x ¼ 0) would be equal to the velocity of the wall:
= 
a)Obtain the expressions for the flow velocity at any point in space (not just on the wall surface) of the plane wave radiating from the wall in terms of the wall vibration velocity.
b)Obtain the expressions for the sound pressure at any point in space (not just on the wall surface) of the plane wave radiating from the wall in terms of the wall vibration velocity. Use 415 rayls for the characteristic impedance of air.
c)What is the root-mean-square (RMS) pressure at any point in space?
Use 415 rayls for the characteristic impedance (ρoc) of air.
4.7 |
Homework Exercises |
113 |
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(Answers): (a) 0:02 cos ωt kx þ π2 ms ; (b) |
8:3 cos ωt kx þ π2 ½Pa&; |
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(c) |
ð8:23Þ2 |
½Pa& |
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Exercise 4.3 (Traveling Wave with Known Pressure)
Use one of the traveling waves shown below (chosen by the instructor) to answer every question of this exercise (a–e):
p ðx, tÞ ¼ P0 cos ωt þ kx þ |
π |
2 |
pþðx, tÞ ¼ P0 cos ðωt kx þ πÞ pþðx, tÞ ¼ P0 sin ðωt kx πÞ
a.Obtain the expressions for the particle velocity at any point in space.
b.What is the RMS pressure of the wave at any point in space?
c.What is the RMS flow velocity of the wave at any point in space?
d.What is the acoustic intensity of the wave at any point in space?
e.What is the specific acoustic impedance of the wave at any point in space? (Answers):
i. (a) 1 |
P0 cos |
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kx |
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1 |
Po |
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Po ; (d) 1 |
Po2 |
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p2ρoc j |
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p2 j |
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π ; (b) |
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ii. (a) |
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ρoc |
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p2ρoc j |
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iii. (a) |
sin |
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p2 j |
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π ; (b) 1 |
Po ; (c) |
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Exercise 4.4 (Standing Wave Pattern)
Use one of the combinations of a forward traveling wave p+(x, t) and a backward traveling wave p (x, t) shown (chosen by the instructor) to answer every question of this exercise (a–f):
pþðx, tÞ ¼ A cos ðωt kxÞ ¼ |
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Ae jðωt kxÞ þ Ae jðωt kxÞ |
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p ðx, tÞ ¼ A cos ðωt þ kxÞ ¼ |
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Ae jðωtþkxÞ þ Ae jðωtþkxÞ |
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pþðx, tÞ ¼ A cos ðωt kxÞ ¼ |
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Ae jðωt kxÞ Ae jðωt kxÞ |
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p ðx, tÞ ¼ A cos ðωt þ kxÞ ¼ |
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Ae jðωtþkxÞ þ Ae jðωtþkxÞ |
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jAe jðωt kxÞ þ jAe jðωt kxÞ |
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pþðx, tÞ ¼ A sin ðωt kxÞ ¼ |
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