ECE/TAM 473
Homework Assignment #1
Due: Friday, August 26, 2016
1. Let x e t ( A1 cos t + A2 sin t ) . Show
that x ( t ) Ae t cos (t + ) . Evaluate the constants A and
=
=
in terms of A1 and A2 , then define (using A and ) u(t) and a(t) (speed and accele
ECE 473
Homework Assignment #7
Due: Friday, October 14, 2016
1. a) A 20 kHz SONAR source is producing sound underwater (assume a B/A = 5 and 1.1 atm
ambient pressure at the depth of the source at 20C). If fully developed shock wave
formation occurs at a d
ECE 473
Homework Assignment #9
Due: Friday, October 28, 2016
1. A pulsating sphere of radius a = 0.15 m radiates spherical waves into air at a frequency of
100 Hz. (a) If the intensity at a distance of 0.5 m from the center of the spherical source is 10
2
Chapter 9 Cavities and Waveguides
(9.2) Rectangular Cavity
Consider a rectangular cavity
z
Lz
y
Ly
Lx
x
This cavity (or room) has perfectly smooth, rigid walls. This box could approximate a living room,
an auditorium or approximate a concert hall.
The aco
(7.8) Simple Line Array (N sources)
kL
Recall for an in-phase line source, H ( ) = sinc sin
2
z
y
L
x
r
p(r,t)
Now, consider N point (simple) sources, each with same source strength and phase, and separated
by a center-to-center distance d, where L = N
(7.4) Radiation from a Plane Circular Piston
The plane circular piston is of particular interest in acoustics because it is a model for a number of
sources, i.e. loudspeakers, open ended organ pipes, ventilation ducts, and many types of single
element tra
Chapter 7 Radiation and Reception of Acoustic Waves
(7.1) Will show that a small (compared to ) source of arbitrary shape and velocity distribution, a
simple source, produces the same field as a small spherical source.
Lets consider a radially oscillating
Lets compare the far-field response with the on axis response. From the near-field on axis response
kr
a 2
p ( r , 0 ) = 2 o cU o sin 1 + 2 1
r
2
The axial pressure amplitude for the far field is given by (for = 0)
p ( r , , t ) =
jo a 2U o j (t
ECE/TAM 473
Homework Assignment #4
Due: Friday, September 16, 2016
1. Problem 5.2.1a in Kinsler er al.
2. For an acoustic pressure p =105cos [10(600t + 2y) ] Pa, where time is in seconds and y is in meters, in a
material with o = 1,250 kg/m3, find the spe
Homework # 2 Solutions
1.
m = 20 kg
x0 = 0.1 m
= 1/ = 0. 333 s
= 3 s-1
Rm = 2 m = 2 (20) (3) = 120 N.s/m
0 =
s
200
=
= 3.16 rad/s
m
20
d = 02 2 = 10 9 = 1 rad/s
x = A e-t cos (dt + ) = A e-3t cos (t + )
u = A e-3t (-1) sin (t + ) 3A e-3t cos (t + )
Appl
ECE 473
Homework Assignment #2
Due: Friday, September 2, 2016
1. A mass of 20 kg hanging on a spring is pulled down a distance of 0.1 m and then released. The mass
then oscillates with decreasing amplitude to 1/e of its initial value in 0.333 s. If s = 20
Chapte 1 - Fund
er
damentals of Vibration
s
(1.2) Simp Harmonic Oscillator (SHO): A simple mode for all wav phenomen
ple
cs
r
s
el
ve
na.
If we unde
erstand the SHO, then we will be abl to quickly understand acoustic wa propagat
S
le
y
ave
tion.
If a mass
Cha
apter 2 The Vibrat
T
ting String
g
(2.1) For simple harmo oscillator (one mass the goal w to find t single fun
s
onic
s),
was
the
nction x(t) th
hat
would desc
cribe the ent history of the motion
tire
o
n.
For a finite number of N masses co
e
f
onnect
Nonlinear Acoustic Propagation
In the previous derivation of the acoustic wave equation we considered only linear disturbances. Lets
look briefly at what happens when you include nonlinear terms.
In this derivation we are going to consider Nonlinear Propa
ECE 473
Homework Assignment #8
Due: Friday, October 21, 2016
1. For plane wave reflection from a fluid fluid interface it is observed that at normal incidence
the pressure amplitude of the reflected wave is one-half that of the incident wave (no phase
inf
(6.3) Traansmission
n Through a Layer: Normal
N
Inccidence
ble to:
Applicab
(1) Effect of waalls, barrierrs, etc.
l
acoustic
a
windows
(2) Seeparation layers
(3) Matching
M
laayers
mpared to 2L c2 thiss looks likee two separrate interfaaces. For
For puls
(Appendix 1) Solids
We will divide our discussion of waves in solids into two different conditions: (1) bulk propagation,
(2) bar propagation. Each of these will be considered in order below.
(1) Propagation of bulk waves in a solid. Bulk waves have the m
1. e
t
( A1 cos t + A2 sin t )
Homework # 1 Solutions
Letting A1 = A cos and A2 = A sin then
e t ( A1 cos t + A2 =
sin t ) e t ( A cos cos t A sin sin t )
1
1
sin t
cos (t ) + cos (t + ) and sin =
cfw_cos (t ) cos (t + )
cfw_
2
2
1
Giving e t ( A cos cos
Homework # 4 Solutions
P
= for the adiabatic case and relating to the condensation gives
1. Starting with
P0 0
P
= (1 + s ) . Expanding in a series and keeping only the linear terms in s assuming s <1 then
P0
P
1 + s P P0 + sP0 .
P0
Relating this back
Homework # 5 Solutions
1.
a) M av = 44 ( 0.3) + 28 ( 0.3) + 32 ( 0.4 ) = 34.4
=
c
b) =
o
c) =
e
(1.35) (8314 J/kg o K )( 263.16o K )
g RTo
=
M av
34.4
= 293 m/s
g Po
=
c2
(1.35)( 400 Pa
)
=
2
( 293 m/s )
po
=
c2
1 0.003557 Pa
2.93 108 kg/m3
=
2
2 ( 293 m/
ECE 473
Homework Assignment #3
Due: Friday, September 9, 2016
1. A string of density 0.0200 kg/m is stretched with a tension of 1.28 N from a rigid support at x = 0.55
m to a device producing transverse periodic vibrations at the input to the string (x =
Homework # 2 Solutions
1.
m = 20 kg
x0 = 0.1 m
= 1/ = 0. 333 s
= 3 s-1
Rm = 2 m = 2 (20) (3) = 120 N.s/m
=
0
s
=
m
200
=
20
d=
02 2=
3.16 rad/s
10 9=
1 rad/s
x = A e cos (dt + ) = A e-3t cos (t + )
u = A e-3t (-1) sin (t + ) 3A e-3t cos (t + )
Apply ini
Homework # 3 Solutions
1. (a) /2 = 0.10 m or = 0.20 m
1/2
1/2
1/2
c = (T/L) = (1.28/0.02) = (64) = 8 m/s
f = c/ = 8/0.20 = 40 Hz
F sin k ( L x )
cos t
kT
cos kL
with k = 2/ = 2/0.20 = 31.4 rad/m
kL = (/0.10)(0.55) = 5.5
cos (kL) = cos(5.5) = 0.0000
This
ECE/TAM 473
Homework Assignment #4
Due: Friday, September 16, 2016
1. Problem 5.2.1a in Kinsler er al.
2. For an acoustic pressure p =105cos [10(600t + 2y) ] Pa, where time is in seconds and y is in meters, in a
material with o = 1,250 kg/m3, find the spe
ECE 473
Homework Assignment #2
Due: Friday, September 2, 2016
1. A mass of 20 kg hanging on a spring is pulled down a distance of 0.1 m and then released. The mass
then oscillates with decreasing amplitude to 1/e of its initial value in 0.333 s. If s = 20
(5.10) Specific Acoustic Impedance
Specific acoustic impedance
z
p Pa s
or ray1
u m
For plane waves
z 0 c for going waves
The product 0c is the characteristic impedance. It is specified by the properties of the material.
The characteristic impedance i
Chapter 5 The Acoustic Wave Equation and Simple Solutions
(5.1) In this chapter we are going to develop a simple linear wave equation for sound propagation in
fluids (1D). In reality the acoustic wave equation is nonlinear and therefore more complicated t
(5.7) Harmonic Plane Waves
The classical wave equation is of the form:
r
2 p% ( r , t )
r
= c 2 2 p% ( r , t )
2
t
r
where r = rx x + ry y + rz z is the position vector.
If we assume that the solution is of the form:
r
r
p% ( r , t ) = g% ( r ) h% ( t )
Cha
apter 2 The
T Vibratting Stringg
(2.1) For simple
s
harmo
onic oscillator (one masss), the goal w
was to find tthe single funnction x(t) thhat
would desccribe the enttire history of
o the motion
n.
For a finitee number off N masses co
onnected by vari