# Paper 2, Section II, C

(a) The function $y(x, t)$ satisfies the forced wave equation

$\frac{\partial^{2} y}{\partial x^{2}}-\frac{\partial^{2} y}{\partial t^{2}}=4$

with initial conditions $y(x, 0)=\sin x$ and $\partial y / \partial t(x, 0)=0$. By making the change of variables $u=x+t$ and $v=x-t$, show that

$\frac{\partial^{2} y}{\partial u \partial v}=1$

Hence, find $y(x, t)$.

(b) The thickness of an axisymmetric drop of liquid spreading on a flat surface satisfies

$\frac{\partial h}{\partial t}=\frac{1}{r} \frac{\partial}{\partial r}\left(r h^{3} \frac{\partial h}{\partial r}\right),$

where $h=h(r, t)$ is the thickness of the drop, $r$ is the radial coordinate on the surface and $t$ is time. The drop has radius $R(t)$. The boundary conditions are that $\partial h / \partial r=0$ at $r=0$ and $h(r, t) \propto(R(t)-r)^{1 / 3}$ as $r \rightarrow R(t)$.

Show that

$M=\int_{0}^{R(t)} r h \mathrm{~d} r$

is independent of time. Given that $h(r, t)=f\left(r / t^{\alpha}\right) t^{-1 / 4}$ for some function $f$ (which need not be determined) and that $R(t)$ is proportional to $t^{\alpha}$, find $\alpha$.

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