MATH528

MATH528 Lesson24: Wave equation in 3D, spherical harmonics

Express the 3D wave equation $u_{t t} - c^{2} \nabla^{2} u = 0$ (subscripts denote differentiation) in spherical coordinates

x = r \sin ϕ \cos θ, y = r \sin ϕ \sin θ, z = r \cos ϕ

\frac{1}{c^{2}} u_{t t} = \frac{1}{r^{2}} {(r^{2} u_{r})}_{r} + \frac{1}{r^{2} \sin^{2} ϕ} u_{θ θ} + \frac{1}{r^{2} \sin ϕ} {(\sin ϕ u_{ϕ})}_{ϕ}

Figure 1.

Seek time periodic solutions, and apply separation of variables $u (t, r, θ, ϕ) = a e^{i ω t} R (r) Θ (θ) Φ (ϕ)$ to obtain

^{} - \frac{ω^{2}}{c^{2}} R Θ Φ = R^{''} Θ Φ + \frac{2}{r} R^{'} Θ Φ + \frac{1}{r^{2} \sin^{2} ϕ} R Θ^{''} Φ + \frac{1}{r^{2}} R Θ Φ^{''} + \frac{\cos ϕ}{r^{2} \sin ϕ} R Θ Φ^{'}

- {\frac{Θ^{''}}{Θ}}^{} = \frac{r^{2} \sin^{2} ϕ}{R} (R^{''} + \frac{2}{r} R^{'} + \frac{ω^{2}}{c^{2}} R) + \sin^{2} ϕ (\frac{Φ^{''}}{Φ} + \cot ϕ \frac{Φ^{'}}{Φ}) = m^{2}

\frac{r^{2}}{R} (R^{''} + \frac{2}{r} R^{'} + \frac{ω^{2}}{c^{2}} R) + (\frac{Φ^{''}}{Φ} + \cot ϕ \frac{Φ^{'}}{Φ} - \frac{m^{2}}{\sin^{2} ϕ}) = 0