Thread: Vibrating plate wave reflection problem (Chladni pattern)

As a school project we are investigating the speed of transversal waves in flat circular plates, where the vibration is started in the middle of the plate. There the plate is fixed to the metal rod of a vibration generator, wich is connected to a function generator making sinewaves. This way the frequency of the wave moving through the plate can be controlled.


Because the wavefronts propagate as concentric circles, they reach the edge of the circular plate at the same time and are reflected as concentric circles. Because of this, we could simplify the whole thing to one-dimensional theory and work with waves moving in one or the opposite direction.


When a standing wave occurs, the nodes can be made visible by scattering small grains or powder on the plate. These grains will settle on the nodes, because these are the places where the plate vibrates the least.


In a (one-dimensional) standing wave, the first node appears half a wavelength away from a non-fixed boundary. In our theory we considered both boundaries of the vibrating plate (the edge of the plate and the center, where it is fixed to the rod with a nut) to be non-fixed boundaries. This way the wavelength of a standing wave can be calculated by knowing the amount of nodes of the standing wave and the radius of the plate, with the formula λ = 2r/n (n is the amount of nodes). The position of these nodes can then also be predicted: the first appears at distance λ/4 from the center, and each following node half a wavelength further from the last.


During the experiment however, something strange occurred. The positions of the nodes deviated consequently from the theory near the boundaries of the plate, especially the first node at the center of the plate. For example, in a plate with a radius of 12 cm and 3 nodes the nodes lie at 1.0, 6.1 and 10.5 cm from the centre, and our theory predicts 2.0, 6.0 and 10.0 cm. This deviation gets larger when more nodes appear: the first node consequently lies a lot closer to the nut than expected. The strange thing is that all other nodes appear where the theory predicts.


Could this be because the nut doesn’t act as a non-fixed or a fixed boundary, but something in between? Or is it because the theory of standing waves used in two dimensions differs from standing waves in one dimension?


Thanks in advance for helping, and sorry for the long text

sine waves have nothing to do with circles. Its the sine waves path as was programmed. Which is similar to a circle but not a circle. You know that already.

@Schrodinger's Hat: ignore fiveworlds, he's not the brightest crayon in the box.

And he is still posting irrelevant or incorrect nonsense in the hard science areas despite numerous requests not to and threats of suspension from the mods...

Originally Posted by PhDemon

And he is still posting irrelevant or incorrect nonsense in the hard science areas despite numerous requests not to and threats of suspension from the mods...

thanks for letting me know, I was confused there for a minute

You get used to him after a while, it's like an incontinent house pet leaving small piles of excrement around the forum. Most of us just ignore him but when he responds to new members spouting rubbish with undue confidence we give a heads up

Originally Posted by Schrödingers Hat

As a school project we are investigating the speed of transversal waves in flat circular plates, where the vibration is started in the middle of the plate. There the plate is fixed to the metal rod of a vibration generator, wich is connected to a function generator making sinewaves. This way the frequency of the wave moving through the plate can be controlled.


Because the wavefronts propagate as concentric circles, they reach the edge of the circular plate at the same time and are reflected as concentric circles. Because of this, we could simplify the whole thing to one-dimensional theory and work with waves moving in one or the opposite direction.


When a standing wave occurs, the nodes can be made visible by scattering small grains or powder on the plate. These grains will settle on the nodes, because these are the places where the plate vibrates the least.


In a (one-dimensional) standing wave, the first node appears half a wavelength away from a non-fixed boundary. In our theory we considered both boundaries of the vibrating plate (the edge of the plate and the center, where it is fixed to the rod with a nut) to be non-fixed boundaries. This way the wavelength of a standing wave can be calculated by knowing the amount of nodes of the standing wave and the radius of the plate, with the formula λ = 2r/n (n is the amount of nodes). The position of these nodes can then also be predicted: the first appears at distance λ/4 from the center, and each following node half a wavelength further from the last.


During the experiment however, something strange occurred. The positions of the nodes deviated consequently from the theory near the boundaries of the plate, especially the first node at the center of the plate. For example, in a plate with a radius of 12 cm and 3 nodes the nodes lie at 1.0, 6.1 and 10.5 cm from the centre, and our theory predicts 2.0, 6.0 and 10.0 cm. This deviation gets larger when more nodes appear: the first node consequently lies a lot closer to the nut than expected. The strange thing is that all other nodes appear where the theory predicts.


Could this be because the nut doesn’t act as a non-fixed or a fixed boundary, but something in between? Or is it because the theory of standing waves used in two dimensions differs from standing waves in one dimension?


Thanks in advance for helping, and sorry for the long text

The solution is not a sinusoid, it is a sinusoid multiplied by a Bessel function. As such, the nodes deviate from the ones predicted by a sinusoid.

Thanks for the quick response!

Does it also work when you multiply a bessel function with a standing wave, with equation 2a*cos(ωt)*sin(kx)?
Bessel functions seem a bit difficult for my high-school maths, but differential equations are probably inevitable with this problem so I'll have to look into it.

Originally Posted by fiveworlds

sine waves have nothing to do with circles. Its the sine waves path as was programmed. Which is similar to a circle but not a circle. You know that already.

After repeated warning your are still posting misinformation.

Find a less science or math based forum for your future adventures adventures--perhaps something about middle school level that's more appropriate to your abilities.