Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 39A / a

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- Solution a

Solution

The linear homentropic acoustic equations are

ρ_{t}^{'} + ρ_{0} \nabla \cdot u = 0, ρ_{0} u_{t} = - \nabla p^{'}, p^{'} = c_{0}^{2} ρ^{'} .

Differentiate the first equation in time, take the divergence of the second, and use the third. This eliminates

ρ^{'}

and

u

and gives the acoustic pressure wave equation

p_{tt}^{'} = c_{0}^{2} \nabla^{2} p^{'} .

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