The wave equation with spherical symmetry may be written

$$\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r \tilde{p})-\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} \tilde{p}=0$$

Find the solution for the pressure disturbance $\tilde{p}$ in an outgoing wave, driven by a timevarying source with mass outflow rate $q(t)$ at the origin, in an infinite fluid.

A semi-infinite fluid of density $\rho$ and sound speed $c$ occupies the half space $x>0$. The plane $x=0$ is occupied by a rigid wall, apart from a small square element of side $h$ that is centred on the point $\left(0, y^{\prime}, z^{\prime}\right)$ and oscillates in and out with displacement $f_{0} e^{i \omega t}$. By modelling this element as a point source, show that the pressure field in $x>0$ is given by

$$\tilde{p}(t, x, y, z)=-\frac{2 \rho \omega^{2} f_{0} h^{2}}{4 \pi R} e^{i \omega\left(t-\frac{R}{c}\right)}$$

where $R=\left[x^{2}+\left(y-y^{\prime}\right)^{2}+\left(z-z^{\prime}\right)^{2}\right]^{1 / 2}$, on the assumption that $R \gg c / \omega \gg f_{0}, h$. Explain the factor 2 in the above formula.

Now suppose that the plane $x=0$ is occupied by a loudspeaker whose displacement is given by

$$x=f(y, z) e^{i \omega t},$$

where $f(y, z)=0$ for $|y|,|z|>L$. Write down an integral expression for the pressure in $x>0$. In the far field where $r=\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2} \gg L, \omega L^{2} / c, c / \omega$, show that

$$\tilde{p}(t, x, y, z) \approx-\frac{\rho \omega^{2}}{2 \pi r} e^{i \omega(t-r / c)} \hat{f}(m, n)$$

where $m=-\frac{\omega y}{r c}, n=-\frac{\omega z}{r c}$ and

$$\hat{f}(m, n)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f\left(y^{\prime}, z^{\prime}\right) e^{-i\left(m y^{\prime}+n z^{\prime}\right)} d y^{\prime} d z^{\prime}$$

Evaluate this integral when $f$ is given by

$$f(y, z)= \begin{cases}1, & -a<y<a,-b<z<b \\ 0, & \text { otherwise }\end{cases}$$

and discuss the result in the case $\omega b / c$ is small but $\omega a / c$ is of order unity.