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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 1

Use the <acoustic velocity potential> convention
$$
\mathbf u=\nabla\phi,
\qquad
p'=-\rho_0\phi_t.
$$
For a spherically symmetric outgoing wave with time dependence $e^{i\omega t}$ and $k=\omega/c_0$, write
$$
\widehat\phi(r)=C\frac{e^{-ik(r-a)}}r.
$$
Linearizing the boundary condition at the mean surface $r=a$ gives
$$
\widehat\phi'(a)=i\omega\epsilon.
$$
Since
$$
\widehat\phi'(a)=-\frac{C}{a^2}(1+ika),
$$
we find
$$
C=-\frac{i\omega\epsilon a^2}{1+ika}.
$$
Thus the <outgoing acoustic field of a pulsating sphere> is
$$
\boxed{
\phi(r,t)
=\operatorname{Re}\left[
-\frac{i\omega\epsilon a^2}{1+i\omega a/c_0}
\frac{e^{i\omega t-i(\omega/c_0)(r-a)}}r
\right]
}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / b / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / b / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / b / i / Solution

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