 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-1/13c/a/solution

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

With

ξ = x + c t

and

η = x - c t

\partial_{x} = \partial_{ξ} + \partial_{η}, \partial_{t} = c \partial_{ξ} - c \partial_{η},

so the wave equation becomes

u_{tt} - c^{2} u_{xx} = - 4 c^{2} u_{ξ η} = 0.

Thus

u = F (ξ) + G (η)

. At

t = 0

the initial data give

F (x) + G (x) = ϕ (x), c F^{'} (x) - c G^{'} (x) = ψ (x) .

Solving for

F^{'}

and

G^{'}

and integrating gives d'Alembert's formula>

u (x, t) = \frac{ϕ ( x + c t ) + ϕ ( x - c t )}{2} + \frac{1}{2 c} \int_{x - c t}^{x + c t} ψ (s) d s .

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