PH 404 Acoustics Winter 2001-2002

M. J. Moloney        CL-109        Office 812 877 8302         Home 812 466 1328          Revised    11/21/01

Homework Assignments

Approximate material in Kinsler, Frey, Coppens and Sanders Fundamentals of Acousics, 4th Ed.

One midterm exam, one final exam.
Last 10 days to two weeks devoted to a project
Some possible project ideas

Animation of pulse on a string fixed at both ends (maple code)
Animation of pulse on a string ; we are not told the boundary conditions
Driven damped oscillator (graphs, maple code, problem)
Animation and maple code for simple vibrating membrane
Transverse waves on rods and bars (chart, graphs, boundary conditions, maple code)

Elasticity and Bending of Beams

This animation shows a pulse traveling on a string fixed at both ends.

Press here to download the maple file which generated the animation above of a pulse traveling at 4 m/s along a 2-m string fixed at both ends.  (You will want to do a 'Save File', then open it later in Maple.)

The contents of this maple file are :

strpulse.mws pulse on a string via eigenfunction expansion
Use notation of Meeks, p. 47

An cos n pi ct/L + Bn sin n pi ct/L
An = 2/L integral 0 to L sin n pi x/L y(x,0)
Bn = 2/m pi c integral 0 to L v(x,0) cos n pi x/L

pulse centered at x=7/8 @t=0 pulse isn't really 0 at boundaries, but close to it.
string from 0 to 2. 11/17/97

c:=4; # pulse speed (which must be string wave speed!)
animate(y,x=0..2,t=0..1/2,frames=20,color=black); # this pulse runs off end of the string!

N:=30; # of terms in series
L:=2; # length of string
for m from 1 to N do
A[m]:=evalf(2/L* Int(subs(t=0, y)*sin(m*Pi*x/L),x=0..L)); od:

for m from 1 to N do
str:=str+(A[m]*cos(m*Pi*c*t/L)+B[m]*sin(m*Pi*c*t/L))*sin(m*Pi*x/L); od: animate(str,x=0..2,t=0..1,frames=30,color=black);

Return to Contents

What kind of string is this pulse travelling on?      Can you modify the program above to give this behavior?

[This string is not fixed at both ends!]

Take-home problem for Exam 1 - modify previous pulse problem to give this behavior. (1 problem of 4, other 3 in class.)

Return to Contents

The response curve for a driven damped oscillator is shown below, both for amplitude vs frequency, and phase vs frequency. The phase shown is the amount by which the driver leads the response of the driven oscillator.

Click here to download the maple file which generates these graphs. After it comes up, you will want to do a 'Save File', and later open the file in Maple.

The commands in this file are as follows:

drivnosc.mws 9/22/97

Plot the phase and amplitude response of a driven, damped oscillator as a function of frequency.

It is interesting that very heavy damping (b=10, k=10, m=1) crushes the response away from zero frequency.

assume(k,real);assume(m,real);assume(A,real); assume(omega>0);assume(t,real);assume(b,real);assume(phi,real); eqn:=A*exp(I*(omega*t+phi))-k*x(t)-b*diff(x(t),t) = m*diff(x(t),t,t); > s:=dsolve(eqn,x(t));
steady:=op(1,xs); # keep only the oscillating terms for steady state response ev:=evalc(subs(t=0,phi=0,steady));

The conventional phase response is the driver phase - the responding phase, so the amplitude response will be found as exp(-I alph)

cosal:=op(1,ev); sinal:=-op(2,ev)/I;
plot(alph+Pi*Heaviside(-alph),omega=2..4,title=` phase [0->Pi] vs. omega`);

The driver is in phase with the driven system at very low frequencies, and pulls ahead until it is Pi/2 ahead at resonance, and goes on to be Pi ahead (180 degrees out of phase) at very high frequencies.

The Heaviside function forces the inverse tangent to run between 0 and Pi, rather than between -Pi/2 and +Pi/2.

xcc:=subs(I=-I,steady); # get complex conjugate
xmag:=simplify(steady*xcc); # get magnitude squared
plot(sqrt(xev),omega=0..6,title=`Response amplitude vs freq.`);

Return to Contents

Homework problem P6

Turn in this problem instead of 1.12 on Tues 12/9/97

a) Figure out the theoretical Q value for the oscillator in the problem above.

b) Re-plot the amplitude response curve in Maple from, say, 3 to 4 sec, and determine the Q 'experimentally' by clicking on the plot at points where the amplitude is 0.707 of the max amplitude. Show your calculations for Q.

c) Double the Q of the system (be sure to tell me how you did this). Re-plot and 'experimentally' check that the new parameters resulted in the correct value of Q.

Here is an animated vibrating rectangular membrane, in its lowest mode

Click here to download the maple file which does this animation. Do a 'Save File', then open the file in Maple.

Here is the code in the maple file:

Lx:=2; Ly:=1;

c:=100; # wave speed on membrane
omega:=c*sqrt((Pi/Lx)^2+(Pi/Ly)^2); # lowest mode angular freq
Nframes:=10; tfac:=2*Pi/(omega*Nframes);

for i from 0 to Nframes-1 do
display([plt.(0..Nframes-1)],insequence=true); # this is the animation

Return to Contents

Transverse waves on bars and rods.

They are solutions of y'''' = k^4 y = w^2 /(kap^2 c^2),

where c = sqrt(Y/rho), and kap = sqrt(int r^2 dA/ int dA).

frequencies:     w = k^2 kap c.

same frequencies for free-free and clamped-clamped bars, but different functions!

free-free bar 

y''( +/- L/2)=0

y'''( +/- L/2)=0


y( +/- L/2) = 0

y'( +/- L/2)=0





solve tan kL/2 = -tanh kL/2

kL =. 3, 7, 11, etc. Pi/2


solve tan kL/2 = -tanh kL/2

kL =. 3, 7, 11, etc Pi/2

all solutions : solve 

cos kL  cosh kL = -1


solve tan kL/2 = +tanh kL/2

kL = . 5, 9, 13, etc. Pi/2


solve tan kL/2 = +tanh kL/2

kL = . 5, 9, 13, etc. Pi/2

kL =. 1, 3, 5, 7, etc Pi/2
lowest even mode 

lowest even mode 

lowest mode 

lowest odd mode 

lowest odd mode 

next-lowest mode 

Here is the maple file which produced the graphs shown above.     Do a 'Save File, then open later in Maple

Return to Contents

Some theory on beam bending. .

Examine the neglect of Ic alpha in the derivation of equation [2] above, and then obtain an improved equation for the bar, which enables us to make frequency corrections at high frequencies.

Return to Contents

Vibration frequencies and equations - free-free, clamped-clamped, and clamped-free.

Return to Contents



Return to Contents