5.2 Spherical waves

Spherical waves

To find spherical solutions to the wave equation it is natural to use spherical coordinates. $$x=r\sin \theta \cos \phi$$ $$y=r\sin \theta \sin \phi$$ $$z=r\cos \theta$$

There is a very nice discussion of Spherical Coordinates at:

(External Link)

There is also a nice discussion of Cylindrical coordinates at the same site

(External Link)

Beware the confusion about $\theta$ and $\phi$ . We are calling the polar angle $\theta \text{.}$ All other mathematical disciplines get it wrong and call it $\phi \text{.}$

The Laplacian can be written in spherical coordinates, but where does that come from?looking at just the $x$ term $$\frac{\partial }{\partial x}=\frac{\partial r}{\partial x}\frac{\partial }{\partial r}+\frac{\partial \theta }{\partial x}\frac{\partial }{\partial \theta }+\frac{\partial \phi }{\partial x}\frac{\partial }{\partial \phi }$$ Then you take the second derivative to get $\frac{{\partial }^2}{\partial x^2}$ which as you can imagine is a tremendously boring and tedious thing to do.Since this isn't a vector calculus course lets just accept thesolution.In the case of spherical waves it is not so difficult since the $\theta$ and $\phi$ derivative terms all go to $0$ .

$${\displaystyle \begin{array}{c}{\nabla }^2\psi (r)=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial \psi }{\partial r}\right)\\ =\frac{{\partial }^2\psi }{\partial r^2}+\frac{2}{r}\frac{\partial \psi }{\partial r}\\ =\frac{1}{r}\frac{\partial }{\partial r}\left(\psi +r\frac{\partial \psi }{\partial r}\right)\\ =\frac{1}{r}\frac{{\partial }^2\left(r\psi \right)}{\partial r^2}\end{array}}$$ Thus for spherical waves, we can write the wave equation: $$\frac{1}{r}\frac{{\partial }^2\left(r\psi \right)}{\partial r^2}=\frac{1}{{\text{v}}^2}\frac{{\partial }^2\psi }{\partial t^2}$$ Now we can multiply both sides by $r$ and since $r$ does not depend upon $t$ write $$\frac{{\partial }^2\left(r\psi \right)}{\partial r^2}=\frac{1}{{\text{v}}^2}\frac{{\partial }^2}{\partial t^2}(r\psi )$$ This is just the one dimensional wave equation with a harmonic solution $$r\psi (r,t)=A{e}^{ik\left(r\mp \text{v}t\right)}$$ or $$\psi (r,t)=\frac{A}{r}{e}^{ik\left(r\mp \text{v}t\right)}$$