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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-331/1/b/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 331 / 1 / b / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

Write

ϕ^{'} = Φ (z) e^{ik (x - c t)}

and let

B_{\pm}

be the displacement amplitudes at

z = \pm L

. Use subscripts

I

for the jet interior and

O

for the adjoining exterior. Each interface is a material vortex sheet, so its linearized kinematic boundary conditions are

Φ_{I}^{'} (\pm L) = ik (V - c) B_{\pm}, Φ_{O}^{'} (\pm L) = - ik c B_{\pm} .

The equal mass densities make the hydrostatic displacement terms cancel. The dynamic boundary condition for an inviscid interface therefore gives

(V - c) Φ_{I} (\pm L) = - c Φ_{O} (\pm L), \frac{Φ _{I}^{'} ( \pm L )}{V - c} = \frac{Φ _{O}^{'} ( \pm L )}{- c} = ik B_{\pm} .

The division form assumes

c \neq = 0, V

; the preceding undivided equations remain the proper conditions in those exceptional cases. Potentials themselves need not be continuous, and their vertical derivatives need not be equal: the same material displacement is advected by different tangential base velocities. At infinity the exterior potentials decay. These displacement and pressure conditions, rather than continuity of

Φ

and

Φ^{'}

, determine the equal-density top-hat planar-jet dispersion relation.

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