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Chladni pattern is generated because of the vibration of the membrane. One can produce the vibration from acoustic source or directly vibrating the plate. When I looked at the (equation 1) that is responsible for generating the pattern (the pattern forms because sand settles on the nodal lines and nodal lines occur when the amplitude part of the equation 1 is equal to zero), there is no v or lambda is involved. However, in the general equation of 2-D vibration, v is involved (equation 2).

$$ u(x,y,t) = \sin{(ax)} \sin{(y)} \cos{\sqrt{(a^2 +1)} t} \;\;\;\;\; (1)$$

$$ \frac{\partial^{2}u(x,y,t)}{\partial x^2} + \frac{\partial^{2}u(x,y,t)}{\partial y^2} = \frac{1}{v^2}\frac{\partial^{2}u(x,y,t)}{\partial t^2} \;\;\;\;\;\;\; (2)$$

So does that mean we cannot obtain the speed of the wave from the pattern? Or if we can obtain, how so? Also, if we can find v from the pattern, isn't that v is the same as the speed of sound through the metal medium of the Chladni plate (assuming I am vibrating it using a speaker)?

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    $\begingroup$ When I looked at the equation that is responsible ... Please show which equation you mean. $\endgroup$ Nov 11, 2021 at 19:17
  • $\begingroup$ I have added that equation. Also for a detailed explanation, see this: link $\endgroup$ Nov 11, 2021 at 19:36

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It's helpful to think of a simpler analog. Suppose I tell you the locations of the node along a vibrating string, for example, "there are three nodes, equally spaced, dividing the string into four equal parts." This doesn't tell you anything about the frequency of the vibrations or the speed of sound in the string. However, if I tell you that the string is $L = 100$ cm long and it's being vibrated at a frequency of $f = 400$ Hz, then it's relatively straightforward to calculate the speed of transverse waves in the string; in this case, we would have $c = fL/2$. Note that there are three quantities here (speed, frequency, and size) and we need to know two of them to figure out the third.

Chladni figures work the same way. If you know the frequency of the oscillations and the dimensions of the plate, you could figure out the speed of the waves in the plate. But without knowing two of the pertinent pieces of information, it is not possible to determine the third.

[Is] that $v$ is the same as the speed of sound through the metal medium of the Chladni plate?

No. The vibrational modes responsible for Chladni figures are either flexural or shear waves. The "speed of sound" in a material, in contrast, usually refers to compressional waves. In general, all three of these types of waves (compressional, shear, and flexural) travel at different speeds.

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  • $\begingroup$ Thanks, that analogy is to the point. Also, I added the equations I was talking about after you wrote your answer. But I guess that is irrelevant. Your answer still addresses the question correctly. $\endgroup$ Nov 11, 2021 at 19:39
  • $\begingroup$ Very nice answer, +10 from moi. $\endgroup$
    – Gert
    Nov 11, 2021 at 20:46

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