# Sound wave inside a closed cylinder - Bessel function

In summary, the conversation discusses finding the normal modes and lowest frequencies of a sound wave in a cylinder with length L and radius R. The approach involves using the wave equation for 3D and deriving an expression for the wave function ψ. Initial conditions are not used to determine the normal modes, but rather to determine the energy distribution among them. The next step is to use the wave equation to find a relation between the different coefficients for a given normal mode.

## Homework Statement

The question is as follows, there is a cylinder with length L and radius R, there is a sound wave with a phase velocity v, they ask for the normal modes and the 5 lowest frequencies when L=R

## Homework Equations

Wave equation for 3D, (d^2/dt^2)ψ=v^2*(∇^2)ψ

## The Attempt at a Solution

I have used the wave equation and derived an expression for ψ as function of t,ϑ,z,ρ using bessel formula.
the expression I derived is: Ψ=e^(-iωt)e^(inϑ)e^(ikz)*J(κρ)
I wouldn't like to post all the derivation here (it will take hours to type everything), but the idea was clear, assume that ψ is a function of R,Θ,Z,T which are all independent of one another, and insert into the wave equation.
Im not sure how to continue, what initial conditions should I use?
Hopefully that was clear enough, thats my first post, so I hope i did it well :)

Normal modes themselves are only the patterns in which the system naturally oscillates. You don't use initial conditions to determine the normal modes themselves; instead, initial conditions would tell you how the system's energy is distributed among the normal modes.

As far as how to continue, it seems that you haven't actually used the 3D wave equation yet. You can tell that you have to use that at some point because (for one thing) the answer has to depend on the phase velocity of the wave somehow, and the only place that comes in is the wave equation. Right now the coefficients $\omega$, $n$, $k$, and $\kappa$ are arbitrary; using the wave equation should give you a relation between those coefficients for a given normal mode.

## 1. What is a sound wave inside a closed cylinder?

A sound wave inside a closed cylinder refers to the propagation of sound waves within a cylindrical container, such as a pipe or tube, where the ends are closed off. This creates a resonant system where the sound waves bounce back and forth between the two ends, resulting in a unique pattern of vibrations known as Bessel functions.

## 2. What are Bessel functions and how are they related to sound waves in a closed cylinder?

Bessel functions are a type of special mathematical function that describes the behavior of waves in cylindrical systems, such as the sound waves inside a closed cylinder. They are used to calculate the amplitude and frequency of the sound waves and can help predict the resonant frequencies of the system.

## 3. How do the dimensions of the closed cylinder affect the sound wave pattern?

The dimensions of the closed cylinder, such as the length and diameter, play a significant role in the sound wave pattern. The resonant frequencies of the system are determined by the dimensions, and different dimensions can result in different Bessel function patterns. Additionally, the material and shape of the cylinder also impact the sound wave pattern.

## 4. Can Bessel functions be used to analyze sound waves in other systems besides closed cylinders?

Yes, Bessel functions can be applied to sound waves in various systems, such as spherical or conical containers, as long as the system has a cylindrical symmetry. They are also used in other fields of science, such as electromagnetics and quantum mechanics, to describe wave behavior.

## 5. How do Bessel functions contribute to our understanding of sound and acoustics?

Bessel functions are an essential tool for understanding and analyzing sound waves in closed cylindrical systems. They help predict the resonant frequencies and amplitude of the waves, which is crucial for designing and optimizing acoustical systems. Furthermore, understanding Bessel functions can also aid in the study and development of other scientific fields involving wave phenomena.

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