Application of Bessel function

[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.

Homework Equations

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.

 

[pasmith]

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.

Homework Equations

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.

 

[Luage]

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.