Paper 2, Section II, C

Differential Equations | Part IA, 2009

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

2yx22yt2=4\frac{\partial^{2} y}{\partial x^{2}}-\frac{\partial^{2} y}{\partial t^{2}}=4

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

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

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

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

ht=1rr(rh3hr),\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)h=h(r, t) is the thickness of the drop, rr is the radial coordinate on the surface and tt is time. The drop has radius R(t)R(t). The boundary conditions are that h/r=0\partial h / \partial r=0 at r=0r=0 and h(r,t)(R(t)r)1/3h(r, t) \propto(R(t)-r)^{1 / 3} as rR(t)r \rightarrow R(t).

Show that

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

is independent of time. Given that h(r,t)=f(r/tα)t1/4h(r, t)=f\left(r / t^{\alpha}\right) t^{-1 / 4} for some function ff (which need not be determined) and that R(t)R(t) is proportional to tαt^{\alpha}, find α\alpha.

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