There is nothing arbitrary in the use of a circular function to represent the waves. As a general rule this is the only kind of wave which can be propagated without a change of form; and, even in the exceptional cases where the velocity is independent of wave-length, no generality is really lost by this procedure, because in accordance with Fourier’s theorem any kind of periodic wave may be regarded as compounded of a series of such as (1), with wave-lengths in harmonical progression.
A well-known characteristic of waves of type (1) is that any number of trains of various amplitudes and phases, but of the same wave-length, are equivalent to a single train of the same type. Thus
| ΣA cos { | 2π | (Vt − x) + α } = ΣA cos α·cos | 2π | (Vt − x) − ΣA sin α·sin | 2π | (Vt − x) |
| λ | λ | λ |
| = P cos { | 2π | (Vt − x) + φ } |
| λ |
(3),
where
P2 = (ΣA cos α)2 = Σ(A sin α)2
(4),
| tan φ = | Σ(A sin α) |
| Σ(A cos α) |
(5).