(5)
If ρ, and therefore k, is constant, the solution of (4) subject to the condition that y = 0 for x = 0 and x = l is
y = B sin kx
(6)
provided
kl = sπ, [s = 1, 2, 3, ...].
(7)
This determines the various normal modes of free vibration, the corresponding periods (2π/n) being given by (5) and (7). By analogy with the theory of the free vibrations of a system of finite freedom it is inferred that the most general free motions of the string can be obtained by superposition of the various normal modes, with suitable amplitudes and phases; and in particular that any arbitrary initial form of the string, say y = ƒ(x), can be reproduced by a series of the type
| ƒ(x) = B1 sin | πx | + B2 sin | 2πx | + B3 sin | 3πx | + ... |
| l | l | l |
(8)