Spherical waves

We now consider spherically symmetric solutions to the wave equation

$$\nabla^2 \xi - \frac{1}{c_s^2 } \frac{\partial^2 \xi}{\partial t^2} = 0 \,.$$ (15.40)

where $\xi(\vec{r},t)=\xi(r,t)$ depends only on the distance $r$ from the origin. It is now convenient to use the spherical polar coordinates instead of $(x,y,z)$, and the wave equation becomes

$$\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\... ... \frac{1}{c_s^2 } \frac{\partial^2 \xi}{\partial t^2} = 0 \,.$$ (15.41)

Substituting $\xi(r,t)=u(r,t)/r$ we have

which gives us the wave equation in a single variable $r$,

$$\frac{\partial^2 u}{\partial r^2} - \frac{1}{c_s^2 } \frac{\partial^2 u}{\partial t^2} = 0 \,.$$ (15.43)

whose solutions are already known. Using these we have

$$\xi(r,t)=\frac{f_1(r+c t)}{r}+\frac{f_1(r-c t)}{r}$$ (15.44)

which is the most general spherical wave solution. The first part of the solution $f_1(r+c t)$ represents a spherical wave travelling towards the origin and the second part $f_1(r-c t)$ represents a wave travelling out from the origin as shown in the left and right panels of Figure 15.7 respectively. In both cases the amplitude varies as $1/r$ and the solution is singular at $r=0$.

Figure 15.7: Spherical waves