Standing waves on a circular plate.

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SUMMARY

The discussion centers on calculating the frequency required to produce standing waves with seven concentric circles on a rigid circular metal plate. The initial assumption of a frequency of 14X is incorrect, as the actual frequency depends on boundary conditions and the plate's thickness. For a clamped edge plate, the frequency for the first mode is calculated using the formula 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. The relationship between mode frequencies is non-harmonic, indicating that frequencies do not simply follow integer multiples.

PREREQUISITES
  • Understanding of standing wave theory
  • Knowledge of boundary conditions in wave mechanics
  • Familiarity with the properties of circular plates
  • Basic grasp of wave speed and frequency calculations
NEXT STEPS
  • Research the effects of boundary conditions on standing wave formation
  • Learn about the calculation of frequencies for different modes of vibration in circular plates
  • Explore the relationship between wave speed, thickness, and frequency in materials
  • Investigate non-harmonic frequency relationships in wave mechanics
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Physicists, mechanical engineers, and acoustics researchers interested in wave behavior on circular structures.

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

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).
 

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