- #1
IridescentRain
- 16
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Hello.
I don't know how to prove that a certain function is a solution to the scalar wave equation in cylindrical coordinates.
The scalar wave equation is
[tex]\left(\nabla^2+k^2\right)\,\phi(\vec{r})=0,[/tex]which in cylindrical coordinates is
[tex]\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial\phi}{\partial\rho}\right)+\frac{1}{\rho^2}\frac{\partial^2\phi}{\partial \varphi^2}+\frac{\partial^2\phi}{\partial z^2},[/tex]where the translation between cartesian and cylindrical coordinates is given by [itex]\rho=\sqrt{x^2+y^2}[/itex], [itex]\varphi=\arctan\left(y/x\right)[/itex], [itex]z=z[/itex].
According to Scattering of electromagnetic waves: theories and applications by Tsang L, Kong J A and Ding K-H, a solution to this is the function
[tex]\phi(\vec{r})=J_n\left(k_\rho \rho\right)\,e^{i\left(n \varphi+k_z z\right)},[/tex]where [itex]k^2=k_\rho^2+k_z^2[/itex], [itex]n\in\mathbb{Z}[/itex], and [itex]J_n[/itex] is the first-kind Bessel function of the [itex]n[/itex]-th order.
I know very little about Bessel functions. I do know, however, that
[tex]J_n(x)=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\,\Gamma(m+n+1)}\left(\frac{x}{2}\right)^{2m+n},[/tex]which, by writing [itex]\Gamma(m+n+1)[/itex] explicitly, becomes[tex]J_n(x)=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\,\int_0^{\infty} t^{m+n}\,e^{-t}\,dt}\left(\frac{x}{2}\right)^{2m+n}.[/tex]I also know that
[tex]\frac{d}{dx}J_n(x)=\frac{1}{2}\left[J_{n-1}(x)-J_{n+1}(x)\right].[/tex]
So I set out to prove that this is indeed a solution to the wave equation in cylindrical coordinates. However, I didn't get very far. Here's what I did:
[tex]\frac{\partial\phi}{\partial\rho}=\frac{k_\rho}{2}\left[J_{n-1}(k_\rho \rho)-J_{n+1}(k_\rho \rho)\right]\,e^{i\,(n \varphi+k_z z)}[/tex][itex]\Rightarrow[/itex][tex]\left(\nabla^2+k^2\right)\,\phi=\frac{1}{\rho}\frac{\partial}{\partial \rho}\left[\frac{k_\rho \rho}{2}J_{n-1}(k_\rho \rho)-\frac{k_\rho \rho}{2}J_{n+1}(k_\rho \rho)\right]\,e^{i\,(n \varphi+k_z z)}-\left(\frac{n^2}{\rho^2}+k_z^2\right)\,J_n(k_\rho \rho)\,e^{i\,(n \varphi+k_z z)}[/tex][itex]\Rightarrow[/itex][tex]\left(\nabla^2+k^2\right)\,\phi=\left[\frac{k_\rho^2}{4}J_{n-2}(k_\rho \rho)+\frac{k_\rho}{2\rho}J_{n-1}(k_\rho \rho)-\left(\frac{k_\rho^2}{2}+k_z^2+\frac{n^2}{\rho^2}\right)\,J_n(k_\rho \rho)-\frac{k_\rho}{2\rho}J_{n+1}(k_\rho \rho)-\frac{k_\rho^2}{4}J_{n+2}(k_\rho \rho)\right]\,e^{i\,(n \varphi+k_z z)}.[/tex]However, I don't know where to go from here.
If I do
[tex]\frac{\partial\phi}{\partial\rho}=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\,\int_0^{\infty} t^{m+n}\,e^{-t}\,dt}\left(\frac{k_\rho}{2}\right)^{2m+n}\,(2m+n)\,\rho^{2m+n-1}\,e^{i\,(n \varphi+k_z z)},[/tex]I get stuck as well.
How should I approach the problem of proving that the above function [itex]\phi(\vec{r})[/itex] is a solution to the wave equation in cylindrical coordinates?
Thanks! :)
I don't know how to prove that a certain function is a solution to the scalar wave equation in cylindrical coordinates.
The scalar wave equation is
[tex]\left(\nabla^2+k^2\right)\,\phi(\vec{r})=0,[/tex]which in cylindrical coordinates is
[tex]\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial\phi}{\partial\rho}\right)+\frac{1}{\rho^2}\frac{\partial^2\phi}{\partial \varphi^2}+\frac{\partial^2\phi}{\partial z^2},[/tex]where the translation between cartesian and cylindrical coordinates is given by [itex]\rho=\sqrt{x^2+y^2}[/itex], [itex]\varphi=\arctan\left(y/x\right)[/itex], [itex]z=z[/itex].
According to Scattering of electromagnetic waves: theories and applications by Tsang L, Kong J A and Ding K-H, a solution to this is the function
[tex]\phi(\vec{r})=J_n\left(k_\rho \rho\right)\,e^{i\left(n \varphi+k_z z\right)},[/tex]where [itex]k^2=k_\rho^2+k_z^2[/itex], [itex]n\in\mathbb{Z}[/itex], and [itex]J_n[/itex] is the first-kind Bessel function of the [itex]n[/itex]-th order.
I know very little about Bessel functions. I do know, however, that
[tex]J_n(x)=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\,\Gamma(m+n+1)}\left(\frac{x}{2}\right)^{2m+n},[/tex]which, by writing [itex]\Gamma(m+n+1)[/itex] explicitly, becomes[tex]J_n(x)=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\,\int_0^{\infty} t^{m+n}\,e^{-t}\,dt}\left(\frac{x}{2}\right)^{2m+n}.[/tex]I also know that
[tex]\frac{d}{dx}J_n(x)=\frac{1}{2}\left[J_{n-1}(x)-J_{n+1}(x)\right].[/tex]
So I set out to prove that this is indeed a solution to the wave equation in cylindrical coordinates. However, I didn't get very far. Here's what I did:
[tex]\frac{\partial\phi}{\partial\rho}=\frac{k_\rho}{2}\left[J_{n-1}(k_\rho \rho)-J_{n+1}(k_\rho \rho)\right]\,e^{i\,(n \varphi+k_z z)}[/tex][itex]\Rightarrow[/itex][tex]\left(\nabla^2+k^2\right)\,\phi=\frac{1}{\rho}\frac{\partial}{\partial \rho}\left[\frac{k_\rho \rho}{2}J_{n-1}(k_\rho \rho)-\frac{k_\rho \rho}{2}J_{n+1}(k_\rho \rho)\right]\,e^{i\,(n \varphi+k_z z)}-\left(\frac{n^2}{\rho^2}+k_z^2\right)\,J_n(k_\rho \rho)\,e^{i\,(n \varphi+k_z z)}[/tex][itex]\Rightarrow[/itex][tex]\left(\nabla^2+k^2\right)\,\phi=\left[\frac{k_\rho^2}{4}J_{n-2}(k_\rho \rho)+\frac{k_\rho}{2\rho}J_{n-1}(k_\rho \rho)-\left(\frac{k_\rho^2}{2}+k_z^2+\frac{n^2}{\rho^2}\right)\,J_n(k_\rho \rho)-\frac{k_\rho}{2\rho}J_{n+1}(k_\rho \rho)-\frac{k_\rho^2}{4}J_{n+2}(k_\rho \rho)\right]\,e^{i\,(n \varphi+k_z z)}.[/tex]However, I don't know where to go from here.
If I do
[tex]\frac{\partial\phi}{\partial\rho}=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\,\int_0^{\infty} t^{m+n}\,e^{-t}\,dt}\left(\frac{k_\rho}{2}\right)^{2m+n}\,(2m+n)\,\rho^{2m+n-1}\,e^{i\,(n \varphi+k_z z)},[/tex]I get stuck as well.
How should I approach the problem of proving that the above function [itex]\phi(\vec{r})[/itex] is a solution to the wave equation in cylindrical coordinates?
Thanks! :)