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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 1

With $\xi=x+ct$ and $\eta=x-ct$,
$$
\partial_x=\partial_\xi+\partial_\eta,
\qquad
\partial_t=c\partial_\xi-c\partial_\eta,
$$
so the <wave equation> becomes
$$
u_{tt}-c^2u_{xx}=-4c^2u_{\xi\eta}=0.
$$
Thus $u=F(\xi)+G(\eta)$. At $t=0$ the initial data give
$$
F(x)+G(x)=\phi(x),
\qquad
cF'(x)-cG'(x)=\psi(x).
$$
Solving for $F'$ and $G'$ and integrating gives <D'Alembert formula>[d'Alembert's formula]\>
$$
\boxed{
u(x,t)=\frac{\phi(x+ct)+\phi(x-ct)}2
+\frac1{2c}\int_{x-ct}^{x+ct}\psi(s)\,ds }.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 13C / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 13C / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 13C / a / Solution

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