Paper 2, Section II, A

Differential Equations | Part IA, 2021

By means of the change of variables η=xt\eta=x-t and ξ=x+t\xi=x+t, show that the wave equation for u=u(x,t)u=u(x, t)

2ux22ut2=0\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial t^{2}}=0

is equivalent to the equation

2Uηξ=0\frac{\partial^{2} U}{\partial \eta \partial \xi}=0

where U(η,ξ)=u(x,t)U(\eta, \xi)=u(x, t). Hence show that the solution to ()(*) on xRx \in \mathbf{R} and t>0t>0, subject to the initial conditions

u(x,0)=f(x),ut(x,0)=g(x)u(x, 0)=f(x), \quad \frac{\partial u}{\partial t}(x, 0)=g(x)

u(x,t)=12[f(xt)+f(x+t)]+12xtx+tg(y)dyu(x, t)=\frac{1}{2}[f(x-t)+f(x+t)]+\frac{1}{2} \int_{x-t}^{x+t} g(y) \mathrm{d} y

Deduce that if f(x)=0f(x)=0 and g(x)=0g(x)=0 on the interval xx0>r\left|x-x_{0}\right|>r then u(x,t)=0u(x, t)=0 on xx0>r+t\left|x-x_{0}\right|>r+t.

Suppose now that y=y(x,t)y=y(x, t) is a solution to the wave equation ()(*) on the finite interval 0<x<L0<x<L and obeys the boundary conditions

y(0,t)=y(L,t)=0y(0, t)=y(L, t)=0

for all tt. The energy is defined by

E(t)=120L[(yx)2+(yt)2]dxE(t)=\frac{1}{2} \int_{0}^{L}\left[\left(\frac{\partial y}{\partial x}\right)^{2}+\left(\frac{\partial y}{\partial t}\right)^{2}\right] \mathrm{d} x

By considering dE/dt\mathrm{d} E / \mathrm{d} t, or otherwise, show that the energy remains constant in time.

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