= Solution
Use the <shallow-water approximation>, a hydrostatic vertically mixed layer, a deep motionless ambient, and no ambient <fluid entrainment>. Particle loss changes <reduced gravity> but changes total layer volume only at the discarded dilute-particle order. Likewise <thermal expansion> changes the <equation of state> while the leading <Boussinesq approximation> retains <volume conservation>. Define
$$
b=g(\gamma\phi-\theta),\qquad D=\partial_t+u\partial_x.
$$
Depth integration of <mass conservation>, horizontal <momentum> balance, particle transport, and the heating law gives
$$
\boxed{\begin{aligned}
h_t+(hu)_x&=0,\\
(hu)_t+\left(hu^2+\frac12bh^2\right)_x&=0,\\
(h\phi)_t+(hu\phi)_x&=-W_s\phi,\\
(h\theta)_t+(hu\theta)_x&=\beta h\phi.
\end{aligned}}
$$
Here the integrated excess <hydrostatic pressure> is $bh^2/2$. There is no particle source from the bed. Bed drag and the particle contribution to inertia are neglected at this order. The equivalent <variable-buoyancy shallow water equations> are
$$
Dh=-hu_x,\qquad Du=-bh_x-\frac h2b_x,\qquad
D\phi=-\frac{W_s\phi}{h},\qquad D\theta=\beta\phi.
$$
The term $hb_x/2$ is necessary when heating or particle loss makes the <reduced gravity> vary horizontally.
For $v=(h,u,\phi,\theta)^T$, the <quasilinear system> $v_t+A(v)v_x=S(v)$ has
$$
A=\begin{pmatrix}
u&h&0&0\\
b&u&g\gamma h/2&-gh/2\\
0&0&u&0\\
0&0&0&u
\end{pmatrix},\qquad
S=\left(0,0,-\frac{W_s\phi}{h},\beta\phi\right)^T.
$$
Its <characteristic polynomial> factors as
$$
\det(A-\lambda I)=(u-\lambda)^2\bigl[(u-\lambda)^2-bh\bigr].
$$
Thus the <characteristic curves> have slopes
$$
\boxed{\frac{dx}{dt}=u-\sqrt{bh},\quad u,\quad u,\quad u+\sqrt{bh}}.
$$
On the two repeated contact <characteristic curves>, the particle and heating laws are $D\phi=-W_s\phi/h$ and $D\theta=\beta\phi$. Writing $c_g=\sqrt{bh}$ and $D_\pm=\partial_t+(u\pm c_g)\partial_x$, the gravity-wave compatibility relations are
$$
D_\pm u\ \pm\frac{c_g}{h}D_\pm h=-\frac h2b_x.
$$
For $h>0$ and $b>0$, gravity-wave right <eigenvectors> can be taken as $(1,\pm c_g/h,0,0)^T$. The contact <eigenspace> has dimension two: $\delta u=0$ and $b\delta h+(h/2)\delta b=0$, with independent $\delta\phi,\delta\theta$. Therefore $A$ has a full set of real <eigenvectors>.
\b[The system is hyperbolic throughout strict static stability, but not strictly hyperbolic because the contact speed is repeated.] At $b=0$ the <eigenvalues> coalesce and this diagonalizability is lost; when $b<0$ the gravity-wave speeds become complex, reflecting the loss of a statically stable lower layer.
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