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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-79/2/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 79 / 2 / ii / Solution by

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

The angles from part (i) have sines

sin θ_{P} = \frac{α R}{s ^{2} + α ^{2} R ^{2}}, sin θ_{z} = \frac{α _{f} R}{s ^{2} + α ^{2} R ^{2}} .

(1)

By Snell law for elastic and acoustic waves,

sin θ_{f} / α_{f} = sin θ_{P} / α

. Both angles lie in

[0, π /2)

, so $θ_{f} = θ_{z}$ . The outgoing fluid ray tilts away from the inward-tilted normal by exactly its tilt from vertical; it therefore travels vertically upwards. This is hyperbolic interface collimation of P-waves.

The source-to-interface distance satisfies

r (R) = R^{2} + Z (R)^{2} = \frac{α D + α _{f} s}{d ^{2}} .

(2)

One can verify this by squaring, using

D^{2} - s^{2} = d^{2} R^{2}

. Consequently

\frac{r}{α} - \frac{Z}{α _{f}} = - \frac{s}{α α _{f}} = \frac{z _{0}}{α} - \frac{z _{0}}{α _{f}} .

(3)

Adding the vertical fluid travel time from the interface to height

z

gives

τ (R, z) = \frac{r ( R )}{α} + \frac{z - Z ( R )}{α _{f}} = \frac{z _{0}}{α} + \frac{z - z _{0}}{α _{f}} .

(4)

Thus surfaces of equal arrival time in the fluid are horizontal planes. Their leading wave amplitude may vary with

R

: a planar wavefront does not imply a spatially uniform plane-wave solution at every order.

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