# Application of Bessel function

• Luage
In summary: Unfortunately I have no idea how to do that.In summary, the problem is that when trying to solve for the radial component of the standing waves, the proposed function by my calculator does not seem to be a correct solution. However, by replacing the input of the functions with x^2 \frac{d^2 R}{dx^2} + x\frac{dR}{dx} + (x^2 - n^2)R = 0, the problem becomes trivial to solve for.
Luage

## Homework Statement

This is not exactly a homework problem. It is just a bump in my own spare time calculations that i can't seem to get through.

When trying to model a drum membrane (the physical details are not important) I came up with the following equation for the radial component of the standing waves:

$R''+\frac{1}{r}R'+\left(\lambda -\frac{1}{r^2}n^2\right)R=0$
where R is a function of rThis is almost Bessel's differential equation except that \lambda≠1.
My calculator tells me that the result is found by multiplying r by √λ in the input of the bessel functions, and I thought that this would be easy to show by substitution. But somehow I can't get this to work.

## The Attempt at a Solution

I tried working backwards. The final result is:
$R(r)=C_1 J_n\left(r\sqrt{\lambda}\right)+C_2 Y_n\left(r\sqrt{\lambda}\right)$

The differential equation that i know this must be a solution of is(r^2 has been multiplied through from the first equation and substituded for r√λ):
$(r\sqrt{\lambda})^2\frac{d^2 R(r\sqrt{\lambda})}{d(r\sqrt{\lambda})^2}+(r\sqrt{\lambda})\frac{d R(r\sqrt{\lambda})}{d(r\sqrt{\lambda})}+\left((r\sqrt{\lambda})^2-n^2\right)R(r\sqrt{\lambda})=0$

$r^2\frac{d^2 R(r\sqrt{\lambda})}{dr^2}+r\frac{d R(r\sqrt{\lambda})}{dr}+\left(r^2\lambda-n^2\right)R(r\sqrt{\lambda})=0$

So it looks like that this is the solution except that the input of all the funktions and funktion derivatives are multiplied by a factor. How do I proceed from here. For now I would be satisfied if I could at least be able to show that the funktion proposed by my calculator is a correct solution. If there is an elegant way then it would of course also be great to know how to spot these kinds of substitutions in the future, but those problems are often related.

Last edited:
Luage said:

## Homework Statement

This is not exactly a homework problem. It is just a bump in my own spare time calculations that i can't seem to get through.

When trying to model a drum membrane (the physical details are not important) I came up with the following equation for the radial component of the standing waves:

$R''+\frac{1}{r}R'+\left(\lambda -\frac{1}{r^2}n^2\right)R=0$
where R is a function of r

This is almost Bessel's differential equation except that \lambda≠1.
My calculator tells me that the result is found by multiplying r by √λ in the input of the bessel functions, and I thought that this would be easy to show by substitution. But somehow I can't get this to work.

## The Attempt at a Solution

I tried working backwards.

That can be done, but it is much simpler to work forwards. You know you want to replace
$$r^2 \frac{d^2 R}{dr^2} + r\frac{dR}{dr} + (\lambda r^2 - n^2)R = 0$$
with
$$x^2 \frac{d^2 R}{dx^2} + x\frac{dR}{dx} + (x^2 - n^2)R = 0.$$
This suggests $x = r\sqrt{\lambda}$ so that
$$\frac{dR}{dr} = \frac{dR}{dx} \frac{dx}{dr} = \sqrt{\lambda}\frac{dR}{dx}$$
and so forth.

pasmith said:
That can be done, but it is much simpler to work forwards. You know you want to replace
$$r^2 \frac{d^2 R}{dr^2} + r\frac{dR}{dr} + (\lambda r^2 - n^2)R = 0$$
with
$$x^2 \frac{d^2 R}{dx^2} + x\frac{dR}{dx} + (x^2 - n^2)R = 0.$$
This suggests $x = r\sqrt{\lambda}$ so that
$$\frac{dR}{dr} = \frac{dR}{dx} \frac{dx}{dr} = \sqrt{\lambda}\frac{dR}{dx}$$
and so forth.

Well I tried this but it didn't help me. From your notation it is not clear, but very quickly the same problem arises. Here you can see where i get stuck:

First I have to rewrite your last equation because right now it is a little ambiguous. is it R(x) or R(r)?

$$\frac{dR(x)}{dr} = \frac{dR(x)}{dx} \frac{dx}{dr} = \sqrt{\lambda}\frac{dR(x)}{dx}$$
and for compleetness:
$$\frac{d^2R(x)}{dr^2} = \lambda \frac{d^2R(x)}{dx^2}$$

Then we have
$$r^2 \frac{d^2 R(r)}{dr^2} + r \frac{dR(r)}{dr} + (\lambda r^2 - n^2)R = 0$$
⇔(THIS SUBSTITUTION IS UNJUSTIFIED. HELP ME JUSTIFY IT. THEN THE PROBLEM IS SOLVED)
$$r^2 \lambda \frac{d^2R(x)}{dx^2} + r \sqrt{\lambda}\frac{dR(x)}{dx} + (\lambda r^2 - n^2)R = 0$$

$$\left(\frac{x}{\sqrt{\lambda}}\right)^2 \lambda \frac{d^2R(x)}{dx^2} +\frac{x}{\sqrt{\lambda}} \sqrt{\lambda}\frac{dR(x)}{dx} + \left(\lambda \left(\frac{x}{\sqrt{\lambda}}\right)^2 - n^2\right)R = 0$$
This last equation trivially simplifies to Bessel's equation. If we can justify how we arrive at it then the problem is solved.

## What is a Bessel function and what is it used for?

A Bessel function is a special mathematical function that is commonly used in many areas of science and engineering. It is particularly useful in solving problems involving waves, vibrations, and heat transfer.

## What are some real-world applications of Bessel functions?

Bessel functions have a wide range of applications in various fields such as physics, engineering, and mathematics. Some examples include analyzing the vibrations of a drum, calculating the temperature distribution in a circular metal plate, and modeling the behavior of electromagnetic waves.

## How is the Bessel function related to other mathematical functions?

Bessel functions are closely related to other special functions, such as Legendre functions, Hermite functions, and Laguerre functions. They can also be expressed in terms of more familiar functions, such as trigonometric functions and exponential functions.

## How are Bessel functions calculated and represented graphically?

Bessel functions can be calculated using various numerical methods, such as series expansions and approximations. They can also be represented graphically using different techniques, such as plotting their values on a graph or visualizing them as contour plots.

## Are there different types of Bessel functions and how are they classified?

Yes, there are several types of Bessel functions, including Bessel functions of the first kind, Bessel functions of the second kind, and modified Bessel functions. They are classified based on their order, which is a measure of their oscillatory behavior, and their argument, which is the independent variable in the function.

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