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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-74/4/solution

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Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 74 / 4 / Solution by

Codex  0 2026-10-14

Use mass conservation and conservative momentum conservation, with viscous stress

σ_{ij}

defined so that its divergence is the viscous force:

ρ_{t} + \partial_{i} (ρ u_{i}) = 0, (ρ u_{i})_{t} + \partial_{j} (ρ u_{i} u_{j}) = - \partial_{i} p + \partial_{j} σ_{ij} .

Differentiate the first equation in time and eliminate the momentum time derivative using the second. This gives

ρ_{tt} = \partial_{i} \partial_{j} (ρ u_{i} u_{j}) + \nabla^{2} p - \partial_{i} \partial_{j} σ_{ij} .

ρ^{'} = ρ - ρ_{0}

and

p^{'} = p - p_{0}

, where the reference values and

c_{0}

are constant. Subtract

c_{0}^{2} \nabla^{2} ρ^{'}

to obtain the exact Lighthill acoustic analogy

(\partial_{t}^{2} - c_{0}^{2} \nabla^{2}) ρ^{'} = \partial_{i} \partial_{j} T_{ij}, T_{ij} = ρ u_{i} u_{j} + (p^{'} - c_{0}^{2} ρ^{'}) δ_{ij} - σ_{ij} .

The Lighthill stress tensor includes nonlinear momentum transport, departure from the reference linear pressure-density relation, and viscous stress. For an inviscid fluid the last term is absent. The derivation is an exact rearrangement of the conservation equations; treating the tensor as a known localized source is a subsequent approximation.

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