 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017-resit/11/b/i/solution

For a uniform string, let

c = T / μ

. A separated mode obeys

Y^{''} + k^{2} Y = 0

ω = c k

. Fixed ends require

Y (0) = Y (L) = 0

, so

k_{n} = \frac{nπ}{L}, y_{n} (x, t) = A_{n} sin \frac{nπ x}{L} cos (c k_{n} t + ϕ_{n}), n = 1, 2, \dots .

The first three normal modes have one, two and three half-waves respectively. Their interior nodes of a standing wave are at

x = j L / n

for

j = 1, \dots, n - 1

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