Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 39B / b

Solution

Fix

ω = k c

. For a P-wave,

m_{P}^{2} = \frac{ω ^{2}}{c _{P}^{2}} - k^{2} = k^{2} (\frac{c ^{2}}{c _{P}^{2}} - 1) .

a = \frac{m _{P}}{k} = \frac{c ^{2}}{c _{P}^{2}} - 1 .

\pm m_{P}

then give, after an irrelevant common normalization,

u_{P} = A e^{ik (x - c t)} (cos (ak z), 0, ia sin (ak z)) .

Thus

f_{1}

is even and both scalar functions multiplying

1

and

i

are real.

Similarly, put

b = \frac{m _{S}}{k} = \frac{c ^{2}}{c _{S}^{2}} - 1 .

(m_{S}, 0, - k)

and

(m_{S}, 0, k)

for the two vertical directions gives

u_{S V} = B e^{ik (x - c t)} (b cos (bk z), 0, - i sin (bk z)) .

This has the same required parity and reality properties.

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