Transformation of Equation of Motion by Hankel Transform

[the_dialogue]

1. The problem statement, all variables and given/known data

So I've been staring at this problem for hours and I can't figure it out. The idea is to transform a second order equation of motion (depends on 'r' and 't') by the Hankel Transform. I think the purpose is to to avoid using separation of variables which tends to cause trouble when the boundary conditions vary with time. See the first attached image for the equation of motion and the boundary & initial conditions.

2. Relevant equations

I'm trying to understand how they got to the "transformed equation" (see second attached image). All they say in the paper is:

"Transformation of equation (I) and the initial conditions (equations (2) and (3)), and
use of the boundary conditions (equations (4) and (5)) will yield the differential equation..."


3. The attempt at a solution

I can't quite understand through reading the paper how the equation is transformed. I somewhat follow the derivation of the "Hankel Transformation Identity" (third image) but I can't understand how this is used to transform the above mentioned equation.

Update: I now see how they derived the LEFT side of the equation (i.e. the second derivative of y + a^2....). But how about the RIGHT side? Where does all that come from? e.g. 2a^2/Pi*(row*F2-F1) .

Thank you for any help.



Source: FINITE TRANSFORM SOLUTION OF THE VIBRATING ANNULAR MEMBRANE G.R. Sharp Journal of Sound and Vibration (1967)

[gabbagabbahey]

This looks like a pretty straight forward application of the identity you are given (although, the identity you posted uses some unfamiliar notation). In terms of Y(r,t) and its Henkel Transform y(\delta_n,t) the identity is

H=\left(u_n(r),\left[\frac{d^2}{dr^2}Y(r,t)+\frac{1}{r}Y(r,t)\right]\right)=\frac{2}{\pi}\left(\rho_nY(r_2)-Y(r_1)\right)-\delta_n^2y(\delta_n,t)

And your initial conditions tell you what Y(r_2) and Y(r_1) are. (And of course, the Hankel Transform of \frac{d^2}{dt^2}Y(r,t) is just \frac{d^2}{dt^2}y(\delta_n,t))

[the_dialogue]

Thank you for the response.

Unfortunately I'm missing something.

Where does the \delta_n^2y(\delta_n,t) term go in the transformed differential equation?

That is, the left side of the transformed DE looks to be just the transformation of the left side of the original DE, but the right side is the identity without the \delta_n^2y(\delta_n,t) term.

This looks like a pretty straight forward application of the identity you are given (although, the identity you posted uses some unfamiliar notation). In terms of Y(r,t) and its Henkel Transform y(\delta_n,t) the identity is

H=\left(u_n(r),\left[\frac{d^2}{dr^2}Y(r,t)+\frac{1}{r}Y(r,t)\right]\right)=\frac{2}{\pi}\left(\rho_nY(r_2)-Y(r_1)\right)-\delta_n^2y(\delta_n,t)

And your initial conditions tell you what Y(r_2) and Y(r_1) are. (And of course, the Hankel Transform of \frac{d^2}{dt^2}Y(r,t) is just \frac{d^2}{dt^2}y(\delta_n,t))

[the_dialogue]

Nevermind. Got it!

Thank you very much for the help.

[the_dialogue]

Can someone point me to a good explanation/proof of the previously mentioned Henkel identity?