You can find the lecture notes at this link
or at the Wiki.
Week 10 - Strings and Membranes [9] [11]
Summary
How can we generalize our techniques to systems with an infinite number of
degrees of freedom (fields)?
Problem Set - 25 November 2005 Answers
Problem 1 - Steel Drum 
A steel drum consists of the sawed off bottom of a steel barrel. The
bottom of the barrel is hammered into a portion of a sphere.
Bumps of various sizes are hammered into the spherical surface.
Check out
this link
for more information.
What changes would you make to the analysis of membranes that we
presented in class?
Problem 2 - A Real Drum
In class we analyzed the modes of oscillation of a drum with a square
membrane. In reality most drums are round. Write the equation of
motion of the membrane in cylindrical coordinates and substitute a
trial solution of
z(r,θ t) = R(r) f(θ) g(t)
You should get a differential equation for the three functions.
For the square-top drum head the three functions were all sines and cosines.
Here you will get something different.
To solve the differential equations, divide both sides of the equation
of motion by the trial solution z(r,θ t). You should find that
the second derivative of f divided by f is a constant (not a function
of f) - similarly for g. What types of functions satisfy these
formulae? Now you know f(θ) and g(t).
For the final function we have
1
R(r)
|
/
|
\
|
d2 R(r)
d r2
|
+ |
1
r
|
d R(r)
d r
|
\
|
/
|
= -A2 +
|
n2
r2
|
This differential equation has the solution
R(r) = Jn ( A r )
where J is a Bessel Function of the First Kind.
What is the significance of the constant A? If I hit the drum
in the exact center, what is the ratio of the frequency of the second
and first harmonic to the fundamental? Is it 5:3:1 like when I pluck a string in the middle or hit a square drum in the middle?
What if I hit the drum in a random spot?
Use the website above (and here) to figure this out.
Draw nodal diagrams for the fundamental and first two overtones in both cases.
Reading List
- ``Classical Mechanics''
REF: Goldstein, H. 1980, Chapter 12 up to page 562.
Last modified: Wednesday, 30 November 2005 12:14:25
| 
