What is the period of the wave?

Answer 1

Given a wave f(x, t), we call period T the interval of time we need to wait before the form of the wave is repeated. Obviously, we must choose a constant x and observe the wave passing trough it.

For exemple, for f(x, t) = ei(kx - wt) it's f(x, t + T) = ei(kx - w(t + T)), so that we have 0 = f(x, t + T) - f(x, t) = ei(kx - wt) (e-iwT - 1) → e-iwT = 1 → T = n 2π/w, with n element of Z. If we are considering two near peaks of f (that is a good way to define T) n = 1, and we have the famous T = 2π/w.