# Standing waves on a circular plate.

1. May 16, 2012

### ImaLooser

Suppose I have a rigid circular metal plate that takes sound X microseconds to cross. What frequency would I have to vibrate that plate to get standing waves that form seven concentric circles? The obvious answer is 14X but I'm not sure.

BTW, this is not homework.

2. May 16, 2012

### sophiecentaur

Don'y forget that standing waves have two antinodes per wavelength.

3. May 16, 2012

### nasu

It's not so obvious in several respects.
First, if x is a time, the frequency cannot be 14x.
Second, the answer depends on the boundary conditions (free edge, clamped edge, etc).
And then the answer depends on the plate's thickness too (not only diameter).
For a plate with clamped edges, the mode with 1 nodal circle has a frequency given by
f_01=0.47*c*h/a^2
where c is the speed of longitudinal waves, h is the thickness and a is the radius.
Your x would be 2a/c, I suppose (you didn't say which way is the sound going in X seconds) so you can eliminate either c or a from the formula but still have h dependence.
For the mode with two circular nodes:
f_02 is 3.89 f_01.
I don't have the value for f_06 but you can see that the relationship between mode frequency is not harmonic (integer multiples).