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I was reading a paper and I found it hard to comprehend how some of the equations were arrived at, probably because my math rottenness. Anyway I need your help on understanding how these equations were arrived at. The problem goes like this:

We have this PDE in cylindrical co-ordinate:

[itex]\frac{\partial^{2} }{\partial t^{2}} P [/itex] - [itex]c_{0}^{2} \Delta P [/itex] = [itex] \frac{\partial}{\partial t}[/itex] q([itex]\underline{r}[/itex],t)

and q([itex]\underline{r}[/itex],t) = δ(t) Q([itex]\underline{r}[/itex])

Here is a quote from the paper:

Basically my question is, how does the author arrived at equation P(r,z,t) = [itex]∫^{∞}_{0} P_{λ}(z,t) J_{0} (λ r) λ dλ[/itex] and [itex]\frac{\partial^{2} }{\partial t^{2}} P_{λ} [/itex] - [itex]c_{0}^{2} (\frac{\partial^{2} P_{λ}}{\partial z^{2}} - λ^{2} P_{λ} )[/itex] = 0 In axially symmetric case P([itex]\underline{r}[/itex],t) = P(r,z,t), the radial co-ordinate is expressed through a superposition of Bessel function [itex] J_{0} (λ r)[/itex] with continuous (P(r,z,t) = [itex]∫^{∞}_{0} P_{λ}(z,t) J_{0} (λ r) λ dλ[/itex]) or discrete series of λ values.

In both cases for spectrum amplitude [itex] P_{λ}(z,t) [/itex] one get equation:

[itex]\frac{\partial^{2} }{\partial t^{2}} P_{λ} [/itex] - [itex]c_{0}^{2} (\frac{\partial^{2} P_{λ}}{\partial z^{2}} - λ^{2} P_{λ} )[/itex] = 0

And secondly, it is possible to solve the above PDE without using any boundary conditions because I didn't see any BC or IC used in arriving at the two equations (that is, P(r,z,t) = [itex]∫^{∞}_{0} P_{λ}(z,t) J_{0} (λ r) λ dλ[/itex] and [itex]\frac{\partial^{2} }{\partial t^{2}} P_{λ} [/itex] - [itex]c_{0}^{2} (\frac{\partial^{2} P_{λ}}{\partial z^{2}} - λ^{2} P_{λ} )[/itex] = 0)?

Looking forward for your assistance and thank you in advance.

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# Solving Second order non-Homogeneous PDE

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