Normal modes of square membrane

[bon]

Homework Statement

Please see question attached

Homework Equations

The Attempt at a Solution

Ok so I've been able to do the first few parts and have derived that Wm,n = c pi / L (m^2 + n^2)

I've thus been able to show that the second lowest freq is a factor of root(5/2) times larger than the first and that there are two modes that have this frequency. But how do you do the next two parts?

How do I show that by combining them it is possible to have a node along either diagonal of the square? I guess I have to add the two solutions and subtract, but then how do I show that these nodes are along the diagonal?

Also what about the last part? How am i meant to deduce the fundamental mode for a triangular membrane?

Thanks!

 

[SammyS]

Homework Statement

Please see question attached

Homework Equations

The Attempt at a Solution

OK so I've been able to do the first few parts and have derived that Wm,n = c pi / L (m^2 + n^2)

I've thus been able to show that the second lowest freq is a factor of root(5/2) times larger than the first and that there are two modes that have this frequency. But how do you do the next two parts?

How do I show that by combining them it is possible to have a node along either diagonal of the square? I guess I have to add the two solutions and subtract, but then how do I show that these nodes are along the diagonal?

Also what about the last part? How am i meant to deduce the fundamental mode for a triangular membrane?

Thanks!

It looks like what you have so far is:

u_m,n(x,y,t) = A_m,n·sin(mπx/L)·sin(nπy/L)·cos(ω_m,nt), where ω_m,n = (cπ/L)·√(m² + n²).

You actually had ω_m,n = (cπ/L)·(m² + n²), but since you got the correct ratio, I assume you inadvertently left out the radical. I also took the liberty to include subscripts on u(x,y,t) to indicate what values of m & n are used.

Notice, that if u_m,n(x,y,t) is a solution to the equation, then, of course, so is u_n,m(x,y,t).

Not only that, u_m,n(x,y,t) ± u_n,m(x,y,t) are also solutions.

In particular, look at u_1,2(x,y,t) - u_2,1(x,y,t).

Use the double angle identity for sine. Do some factoring, then use the identity:

cos(θ) - cos(φ) = -2·sin((θ+φ)/2)·sin((θ-φ)/2)

That will get the node along the y = x diagonal.


As for the last question: What does it mean for there to be a node along a diagonal?

Edited to add:
It occurs to me that to check for a node along the diagonal y = x, for instance, it may be simpler to simply plug in x for y in u_m,n(x,y,t) ± u_n,m(x,y,t), and see if the result is zero.

The other diagonal occurs along the line: y = 1 - x.