Paper 2, Section II, B

Differential Equations | Part IA, 2014

The so-called "shallow water theory" is characterised by the equations

ζt+x[(h+ζ)u]=0ut+uux+gζx=0\begin{aligned} &\frac{\partial \zeta}{\partial t}+\frac{\partial}{\partial x}[(h+\zeta) u]=0 \\ &\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+g \frac{\partial \zeta}{\partial x}=0 \end{aligned}

where gg denotes the gravitational constant, the constant hh denotes the undisturbed depth of the water, u(x,t)u(x, t) denotes the speed in the xx-direction, and ζ(x,t)\zeta(x, t) denotes the elevation of the water.

(i) Assuming that u|u| and ζ|\zeta| and their gradients are small in some appropriate dimensional considerations, show that ζ\zeta satisfies the wave equation

2ζt2=c22ζx2,\frac{\partial^{2} \zeta}{\partial t^{2}}=c^{2} \frac{\partial^{2} \zeta}{\partial x^{2}},

where the constant cc should be determined in terms of hh and gg.

(ii) Using the change of variables

ξ=x+ct,η=xct\xi=x+c t, \quad \eta=x-c t

show that the general solution of ()(*) satisfying the initial conditions

ζ(x,0)=u0(x),ζt(x,0)=v0(x)\zeta(x, 0)=u_{0}(x), \quad \frac{\partial \zeta}{\partial t}(x, 0)=v_{0}(x)

is given by

ζ(x,t)=f(x+ct)+g(xct)\zeta(x, t)=f(x+c t)+g(x-c t)


df(x)dx=12[du0(x)dx+1cv0(x)]dg(x)dx=12[du0(x)dx1cv0(x)]\begin{aligned} &\frac{d f(x)}{d x}=\frac{1}{2}\left[\frac{d u_{0}(x)}{d x}+\frac{1}{c} v_{0}(x)\right] \\ &\frac{d g(x)}{d x}=\frac{1}{2}\left[\frac{d u_{0}(x)}{d x}-\frac{1}{c} v_{0}(x)\right] \end{aligned}

Simplify the above to find ζ\zeta in terms of u0u_{0} and v0v_{0}.

(iii) Find ζ(x,t)\zeta(x, t) in the particular case that

u0(x)=H(x+1)H(x1),v0(x)=0,<x<u_{0}(x)=H(x+1)-H(x-1), \quad v_{0}(x)=0, \quad-\infty<x<\infty

where H()H(\cdot) denotes the Heaviside step function.

Describe in words this solution.

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