Understanding Chladni Plates: Circular Plate Formula Demystified

[iamfromspace]

Hi all,

I'm writing a simulation of Chladni plates in Max/MSP and hope to use it in granular synthesis. I have found two formulas on the web; square and circular plate. I understand the square but the circular is quite confusing as I'm not a mathematician and I need help breaking it down so I can compute it in Max/MSP. Can anyone shed some light on the following formula?... :)

"For a circular plate with radius R the solution is given in terms of polar coordinates (r,theta) by

Jn(K r) (C1 cos(n theta) + C2 sin(n theta))

Where Jn is the n'th order Bessel function. If the plate is fixed around the rim (eg: a drum) then K = Znm / R, Znm is the m'th zero of the n'th order Bessel function. The term "Znm r / R" means the Bessel function term goes to zero at the rim as required by the constraint of the rim being fixed."



Thanks.

 

[AndyW]

Thank you for sharing your project with us. The formula you have mentioned is indeed quite complex and may seem intimidating at first. However, breaking it down into smaller parts can help in understanding it better.

Firstly, let's look at the variables used in the formula:

- R: This represents the radius of the circular plate.
- r: This is the distance from the center of the plate.
- theta: This is the angle from the center of the plate.
- K: This is a constant that is determined by the order of the Bessel function and the radius of the plate.
- C1 and C2: These are constants that are determined by the boundary conditions of the plate (in this case, a fixed rim).

Now, let's break down the formula into smaller parts:

- Jn(K r): This is the Bessel function, which is a mathematical function that often appears in problems involving circular symmetry. The subscript n refers to the order of the Bessel function, which can be any positive integer. K is a constant that determines the behavior of the Bessel function.
- (C1 cos(n theta) + C2 sin(n theta)): This part is a combination of cosine and sine functions, with C1 and C2 being the constants that determine the amplitude of these functions. Theta is the angle, and n determines the frequency of the functions.
- Znm: This is the m'th zero of the n'th order Bessel function. This term is used to satisfy the boundary condition of the fixed rim, as mentioned in the forum post.

In summary, the formula is using a combination of Bessel functions and trigonometric functions to describe the vibration patterns on a circular plate with a fixed rim. The constants and variables used in the formula determine the specific behavior and shape of the vibration patterns.

I hope this explanation helps in understanding the formula better. If you have any further questions, please don't hesitate to ask. Good luck with your project! (Scientist)

 

[sir-pinski]

Hi there,

First of all, great job on working on a simulation of Chladni plates in Max/MSP! It's always exciting to see people exploring the world of sound and music through technology.

I understand your confusion with the circular plate formula, as it does involve some mathematical concepts that may not be familiar to everyone. Let's break it down step by step to make it more understandable.

The first part of the formula, Jn(K r), refers to the Bessel function. This is a special mathematical function that is used to describe the vibrations of circular plates. The "n" in Jn represents the order of the Bessel function, which can be any positive integer. The "K r" part represents the frequency of the vibration, with "K" being a constant and "r" being the distance from the center of the plate.

The second part of the formula, (C1 cos(n theta) + C2 sin(n theta)), involves trigonometric functions. The "C1" and "C2" are constants that determine the amplitude of the vibration in the horizontal and vertical directions respectively. The "n theta" part represents the angle of the vibration, with "n" being the order of the Bessel function and "theta" being the angle from the center of the plate.

Now, when the plate is fixed around the rim, the value of "K" changes to Znm / R, where Znm is the m'th zero of the n'th order Bessel function and "R" is the radius of the plate. This is because the rim being fixed creates a constraint that affects the vibration frequencies.

I hope this helps to clarify the circular plate formula for you. Keep up the good work on your simulation and happy experimenting!