 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-3/39a/a/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 39A / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Substitution of the monochromatic ansatz gives

w^{''} + k^{2} (N^{2} / ω^{2} - 1) w = 0.

(1)

Thus the outer solutions oscillate with

m = k N_{0}^{2} / ω^{2} - 1

and the middle solutions are combinations of

cosh k z

and

sinh k z

. The incident wave has phase

k x - m z - ω t

; its group velocity is upward, whereas the reflected sign has downward group velocity. The transmitted wave must again use the upward-energy sign. This gives exactly the three forms stated.

 Read the full article

 New to topics? Read the docs here!