Small displacements $\mathbf{u}(\mathbf{x}, t)$ in a homogeneous elastic medium are governed by the equation

$$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+2 \mu) \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{u})-\mu \boldsymbol{\nabla} \wedge(\boldsymbol{\nabla} \wedge \mathbf{u})$$

where $\rho$ is the density, and $\lambda$ and $\mu$ are the Lamé constants.

(a) Show that the equation supports two types of harmonic plane-wave solutions, $\mathbf{u}=\mathbf{A} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)]$, with distinct wavespeeds, $c_{P}$ and $c_{S}$, and distinct polarizations. Write down the direction of the displacement vector A for a $P$-wave, an $S V$-wave and an $S H$-wave, in each case for the wavevector $(k, 0, m)$.

(b) Given $k$ and $c$, with $c>c_{P}\left(>c_{S}\right)$, explain how to construct a superposition of $P$-waves with wavenumbers $\left(k, 0, m_{P}\right)$ and $\left(k, 0,-m_{P}\right)$, such that

$$\mathbf{u}(x, z, t)=e^{i k(x-c t)}\left(f_{1}(z), 0, i f_{3}(z)\right)$$

where $f_{1}(z)$ is an even function, and $f_{1}$ and $f_{3}$ are both real functions, to be determined. Similarly, find a superposition of $S V$-waves with $\mathbf{u}$ again in the form $(*)$.

(c) An elastic waveguide consists of an elastic medium in $-H<z<H$ with rigid boundaries at $z=\pm H$. Using your answers to part (b), show that the waveguide supports propagating eigenmodes that are a mixture of $P$ - and $S V$-waves, and have dispersion relation $c(k)$ given by

$$a \tan (a k H)=-\frac{\tan (b k H)}{b}, \quad \text { where } \quad a=\left(\frac{c^{2}}{c_{P}^{2}}-1\right)^{1 / 2} \quad \text { and } \quad b=\left(\frac{c^{2}}{c_{S}^{2}}-1\right)^{1 / 2}$$

Sketch the two sides of the dispersion relationship as functions of $c$. Explain briefly why there are infinitely many solutions.