ME 610
Chap. 8 Assignments
1. For the closed waveguide example discussed in class (see handout notes, Chapter 8),
the sound pressure at a distance x from the source end was stated to be
 jz U 
p~( x )    o o  cos kL(1  x / L)
 sin kL 
(a) Verify that this equation is correct by applying the B.C.’s to the expression for the
total sound pressure and simplifying to obtain the above result.
(b) Using MATLAB or Excel, plot the magnitude of the sound pressure (normalized by
zoUo) vs x/L from 0 to 1 when kL = 8, a typical value. Note the slope of the standing
wave at x = 0 and particularly at x = L. Why is the slope at x = L zero?
(c) Repeat (b) for a frequency such as kL = 3.14 which is very close to the first
resonance frequency (the resonance frequencies occur when sin kL = 0). From the
standing wave, what can you conclude about the particle velocity at x = 0 at this
frequency? The sound pressure at this frequency?
2. (a) For the closed waveguide in Problem 1, plot on a single graph the magnitude of the
sound pressure (normalized by zoUo) vs. kL from 0.1 to 15 when x/L = 0 and then
again when x/L = 0.5. (For this graph plot the sound pressure on a log scale – the
command for this in MATLAB is semilogy.)
(b) What can you conclude about the differences in the sound pressure spectra at these
two positions?
3. (a) For the closed waveguide in Problem 1, show that the mean active sound intensity at
x = 0 is zero (i.e., the average sound power supplied by the source to the waveguide
is zero).
(b) Denote the phase difference between the sound pressure and particle velocity at any
point x in the waveguide by θ. Show that the mean active sound intensity can be
expressed by:
1~
I p
( x ) u~( x ) cos 
2
(c) What is the explanation for the fact that the sound power is zero while the sound
pressure is non-zero?
4. In the class notes, the following expression was derived for the normalized specific
acoustic impedance at the entrance (excitation point, x = - L) of a tube:
z / (  L) 
zt/  j tan kL
1  jzt/ tan kL
(a) Show that the impedance at the entrance is a cotangent function when the
termination (at x = 0) is rigid.
(b) Using the results for the sound pressure from problem (1), find the impedance at the
entrance from its definition.
(c) For a conservative system, the natural frequencies are those values of frequency that
make the impedance zero. What are the natural frequencies of the tube with a rigid
termination?
(d) What are the natural frequencies for a tube that has the boundary condition p(0) = 0?
5. In the class notes, the stiffness K of a volume of fluid Vo was given as
K
oc 2Sb2
Vo
where Sb is the area over which a force is acting to compress the volume. The stiffness
is defined as K = -F/x where Sbx is an incremental volume change due to an
incremental force change F = SbP (the minus sign is needed in this definition because
a positive pressure outside the volume results in a negative volume change).
Derive the formula for the stiffness K given above. (One way to do this is to consider a
volume of constant mass m = V, from which we can get Vo + oV = 0. Then, using
the adiabatic relationship between small changes in density  and pressure P, K may
be found.)
6. (a) Starting with the expression in the class notes for c for the side branch muffler and
the specific acoustic impedance zb for a Helmholtz resonator, show that the TL for a
Helmholtz resonator used as a side branch is:



TL  10 log10 1 



 1   2 c   S  
b
 
 
  




2

L

S



n


2
  



 1


 n 


2







(b) Using the numerical values for the Helmholtz resonator given in the class notes, plot
the TL and compare your plot with the one in the notes. Does your plot of TL become a
maximum at the natural frequency?
(c) A co-op student points out that as long as the natural frequency is kept the same, a
much smaller Helmholtz resonator would be just as effective as a larger one. To find out
if this is true, reduce the area of the neck and the volume each by a factor of five without
changing any other parameters (thus, keeping the natural frequency the same); plot the
new TL curve on the same graph as used for part (b). Is the co-op student correct, or is
there a significant difference in the TL of the two resonators?