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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-3/15d/solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 15D / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

In the sense of distribution theory,

H^{'} (t) = δ (t)

. Since

sin (α t)

vanishes at zero,

ψ^{'} (t) = H (t) cos (α t), ψ^{''} (t) = δ (t) - α H (t) sin (α t),

and therefore

ψ^{''} + α^{2} ψ = δ (t) .

Let

G_{ret} (x, t; y)

solve

(\partial_{t}^{2} - c^{2} \nabla^{2}) G_{ret} = δ (t) δ^{(3)} (x - y), G_{ret} = 0 (t < 0) .

The spatial Fourier transform turns this into the preceding oscillator equation with

α = c k

, so

G (k, t) = H (t) \frac{s i n ( c k t )}{c k} .

Inverting with the supplied radial integral gives the three-dimensional retarded Green function

G_{ret} (x, t; y) = \frac{H ( t )}{4 π c ∣ x - y ∣} δ (∣ x - y ∣ - c t) = \frac{δ ( t - ∣ x - y ∣/ c )}{4 π c ^{2} ∣ x - y ∣} .

For zero initial displacement and initial velocity

f

, convolution with this kernel gives the Kirchhoff formula

u (x, t) = \int_{R^{3}} G_{ret} (x, t; y) f (y) d^{3} y = \frac{t}{4 π} \int_{S^{2}} f (x + c t ω) d Ω = t ⟨ f ⟩_{t} .

Thus the value at $(x, t)$ depends only on the initial data on the sphere of radius $c t$ , not on its interior. This is the Huygens principle and describes propagation at the finite wave speed $c$ .

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