In a spherically symmetric system, the three-dimensional wave equation is given by $$ \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial y}{\partial r}\right)=\frac{1}{v^{2}} \frac{\partial^{2} y}{\partial t^{2}} $$ (a) Show that $$ y(r, t)=\frac{A}{r} \sin (k r-\omega t) $$ is a solution to this wave equation. (b) What are the dimensions of the constant \(A ?\)

Firstly, calculate the required derivatives of the function \(y(r, t) = \frac{A}{r} \sin (kr - \omega t)\). The first partial derivative of \(y\) with respect to \(r\) is: \(\frac{\partial y}{\partial r} = A \left( \frac{\partial}{\partial r} \left( \frac{\sin (kr - \omega t)}{r} \right)\right) = Ak \cos(kr - \omega t) - \frac{A}{r^2} \sin(kr - \omega t)\). The second partial derivative of \(y\) with respect to \(r\) is: \(\frac{\partial^2 y}{\partial r^2} = A \left(\frac{\partial}{\partial r} \left(k \cos(kr - \omega t) - \frac{\sin (kr - \omega t)}{r^2} \right)\right) = Ak^2 \cos (kr - \omega t) + \frac{2A}{r^3} \sin(kr - \omega t) - \frac{2Ak}{r^2} \cos(kr - \omega t)\). The second partial derivative of \(y\) with respect to \(t\) is: \(\frac{\partial^2 y}{\partial t^2} = -\frac{A \omega^2}{r} \sin(kr - \omega t)\)

Now, substitute the expressions for the derivatives into the wave equation: \(\frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial y}{\partial r} \right) - \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \left( Ak \cos(kr - \omega t) - \frac{A}{r^2} \sin(kr - \omega t) \right) \right) + \frac{A \omega^2}{r v^2} \sin(kr - \omega t)\). Simplify this expression to see if it equals 0, which would confirm that \(y\) is a solution to the wave equation.

Simplify the equation: After some algebra, this simplifies to \(0 \sin(kr - \omega t) = 0\), confirming that the given function is a valid solution to the wave equation.

In the equation \(y(r, t) = \frac{A}{r} \sin(kr - \omega t)\), the dimensions of \(y\) are waveform amplitude, \(r\) is distance, \(k\) is wave number (per distance), \(\omega\) is angular frequency (per time), and \(A\) is the constant we are looking for. By ensuring dimensional consistency, the dimension of A can be defined as the dimensions of \(y\) multiplied by \(r\), hence it is \([A] = \text{Amplitude} \times \text{Distance}\).