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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-78/2/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 78 / 2 / iii / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Choose interface

z = 0

, source

(0, - h)

in medium 1 and transmitted medium 2 at

z > 0

. Write

μ_{j} = ρ_{j} β_{j}^{2}

, take

p

to be signed tangential slowness, and define

q_{j} = β_{j}^{- 2} - p^{2} > 0, sin θ_{j} = β_{j} p, cos θ_{j} = β_{j} q_{j} .

The propagating range is

∣ p ∣ < 1/ max (β_{1}, β_{2})

; the absolute value is needed if both azimuthal directions are included. Snell law for elastic waves preserves

p

. The incident ray reaches the interface after time

t_{1}

at horizontal position

x_{I}

, with

t_{1} (p) = \frac{h}{β _{1}^{2} q _{1}}, x_{I} (p) = \frac{h p}{q _{1}} .

In the transmitted medium its velocity is

β_{2} (sin θ_{2}, cos θ_{2}) = (β_{2}^{2} p, β_{2}^{2} q_{2})

. Therefore the refraction of a cylindrical SH wavefront gives

x (t, p) = \frac{h p}{q _{1}} + β_{2}^{2} p [t - t_{1} (p)], z (t, p) = β_{2}^{2} q_{2} [t - t_{1} (p)], t \geq t_{1} (p) .

Time is measured from source emission. For a later emission add its time to the clock. These straight transmitted rays and their common-time endpoints determine the refracted wavefront; critical incidence and an evanescent transmitted field are excluded by the strict slowness bound.

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