Wave
Wave forms

The form or shape of F in d'Alembert's formula involves the argument x − vt. Constant values of this argument correspond to constant values of F, and these constant values occur if x increases at the same rate that vt increases. That is, the wave shaped like the function F will move in the positive x -direction at velocity v (and G will propagate at the same speed in the negative x -direction).

In the case of a periodic function F with period λ, that is, F (x + λ − vt) = F (x − vt), the periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F (x − v (t + T)) = F (x − vt) provided vT = λ, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ / v.
combination:two_responses