91Ó°ÊÓ

Let's first substitute the assumed product form \(\Psi = R(r) \Theta(\theta) \Phi(\phi) T(t)\) into the scalar wave equation: $$ \frac{1}{R(r) \Theta(\theta) \Phi(\phi) T(t)} \left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial (R \Theta \Phi T)}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial (R \Theta \Phi T)}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 (R \Theta \Phi T)}{\partial \phi^2} - \frac{1}{c^2}\frac{\partial^2 (R \Theta \Phi T)}{\partial t^2}\right] = 0 $$

Next, we will perform the derivatives in the equation: $$ \frac{1}{R \Theta \Phi T} \left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial R}{\partial r}\Theta \Phi T\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta R\frac{\partial \Theta}{\partial \theta}\Phi T\right) + \frac{1}{r^2\sin^2\theta} R \Theta \frac{\partial^2 \Phi}{\partial \phi^2} T - \frac{1}{c^2} R \Theta \Phi \frac{\partial^2 T}{\partial t^2}\right] = 0 $$

Now, we will separate the variables by dividing each term in the equation by the corresponding product of functions: $$ \frac{T''(t)}{c^2 T(t)} = \frac{1}{R(r)}\frac{\partial}{\partial r}\left(r^2\frac{\partial R(r)}{\partial r}\right) + \frac{1}{\Theta(\theta)}\frac{1}{\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial \Theta(\theta)}{\partial \theta}\right) + \frac{1}{\Phi(\phi)}\frac{1}{\sin^2\theta}\frac{\partial^2 \Phi(\phi)}{\partial \phi^2} $$ Since the left side of this equation depends only on \(t\) and the right side depends only on \(r, \theta, \phi\), both sides must be equal to a constant. We can write this as: $$ \frac{T''(t)}{c^2 T(t)} = k $$ And: $$ k = \frac{1}{R(r)}\frac{\partial}{\partial r}\left(r^2\frac{\partial R(r)}{\partial r}\right) + \frac{1}{\Theta(\theta)}\frac{1}{\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial \Theta(\theta)}{\partial \theta}\right) + \frac{1}{\Phi(\phi)}\frac{1}{\sin^2\theta}\frac{\partial^2 \Phi(\phi)}{\partial \phi^2} $$

Following the separation of variables, we can now solve for each function separately, using the given radial functions (spherical Bessel functions) and polar-angle functions (associated Legendre polynomials): 1. For the radial function \(R(r)\), we will use the spherical Bessel functions \(z_l(\kappa r)\). 2. For the polar-angle function \(\Theta(\theta)\), we will use the associated Legendre polynomials \(P_l^m(\cos\theta)\). 3. For the azimuthal function \(\Phi(\phi)\), we can use the trigonometric functions \(e^{im\phi}\), where \(m\) is an integer. With these functions, the general solution to the scalar wave equation in spherical coordinates becomes: $$ \Psi(r, \theta, \phi, t) = \sum_{l=0}^{\infty}\sum_{m=-l}^{l}z_l(\kappa r) P_l^m(\cos\theta)e^{im\phi} T(t) $$ By solving for the time function \(T(t)\) using the constant \(k\) and applying the appropriate boundary and initial conditions, we can determine the general solution of the scalar wave equation in spherical coordinates.