So far, this is merely a restatement, in mathematical language, of an argument given in the first part of this article. The series (8) may, moreover, be arrived at otherwise, as a particular case of Fourier’s theorem. But if we no longer assume the density ρ of the string to be uniform, we obtain an endless variety of new expansions, corresponding to the various laws of density which may be prescribed. The normal modes are in any case of the type
y = Cu(x)eint
(9)
where u is a solution of the equation
| d²u | + | n²ρ | u = 0. |
| dx² | T |
(10)
The condition that u(x) is to vanish for x = 0 and x = l leads to a transcendental equation in n (corresponding to sin kl = 0 in the previous case). If the forms of u(x) which correspond to the various roots of this be distinguished by suffixes, we infer, on physical grounds alone, the possibility of the expansion of an arbitrary initial form of the string in a series
ƒ(x) = C1u1(x) + C2u2(x) + C3u3(x) + ...
(11)
It may be shown further that if r and s are different we have the conjugate or orthogonal relation