Consider a two-dimensional stratified fluid of sufficiently slowly varying background density $\rho_{b}(z)$ that small-amplitude vertical-velocity perturbations $w(x, z, t)$ can be assumed to satisfy the linear equation

$$\nabla^{2}\left(\frac{\partial^{2} w}{\partial t^{2}}\right)+N^{2}(z) \frac{\partial^{2} w}{\partial x^{2}}=0, \quad \text { where } N^{2}=\frac{-g}{\rho_{0}} \frac{d \rho_{b}}{d z}$$

and $\rho_{0}$ is a constant. The background density profile is such that $N^{2}$ is piecewise constant with $N^{2}=N_{0}^{2}>0$ for $|z|>L$ and with $N^{2}=0$ in a layer $|z|<L$ of uniform density $\rho_{0}$.

A monochromatic internal wave of amplitude $A_{I}$ is incident on the intermediate layer from $z=-\infty$, and produces velocity perturbations of the form

$$w(x, z, t)=\widehat{w}(z) e^{i(k x-\omega t)}$$

where $k>0$ and $0<\omega<N_{0}$.

(a) Show that the vertical variations have the form

$$\widehat{w}(z)= \begin{cases}A_{I} \exp [-i m(z+L)]+A_{R} \exp [i m(z+L)] \quad \text { for } z<-L \\ B_{C} \cosh k z+B_{S} \sinh k z & \text { for }|z|<L \\ A_{T} \exp [-i m(z-L)] & \text { for } z>L\end{cases}$$

where $A_{R}, B_{C}, B_{S}$ and $A_{T}$ are (in general) complex amplitudes and

$$m=k \sqrt{\frac{N_{0}^{2}}{\omega^{2}}-1}$$

In particular, you should justify the choice of signs for the coefficients involving $m$.

(b) What are the appropriate boundary conditions to impose on $\widehat{w}$at $z=\pm L$ to determine the unknown amplitudes?

(c) Apply these boundary conditions to show that

$$\frac{A_{T}}{A_{I}}=\frac{2 i m k}{2 i m k \cosh 2 \alpha+\left(k^{2}-m^{2}\right) \sinh 2 \alpha},$$

where $\alpha=k L$.

(d) Hence show that

$$\left|\frac{A_{T}}{A_{I}}\right|^{2}=\left[1+\left(\frac{\sinh 2 \alpha}{\sin 2 \psi}\right)^{2}\right]^{-1}$$

where $\psi$ is the angle between the incident wavevector and the downward vertical.