The
wave equation is a differential equation that arises in many applications.
The variable, y, is perhaps most easily visualized as the
vertical displacement of a horizontal string. x measures
the position of the observer along the string. However, y
could be the pressure or displacement in a sound wave, the height of an
ocean wave, or the vertical electric field component in a horizontally
propagating electromagnetic wave.
The wave solution shown travels freely to the right. Suppose that at t=0 an observer is standing next to a wave crest. As time, t increases, the observer must move to a larger value of x to remain at the location of that crest. If both terms in the argument of the cosine function had the same sign, the wave would move freely to the left.
The wave length is the distance between wave crests. If a stationary observer stands next to a wave crest at t=0 then at time t=T the next wave crest will appear at her position. The wave travels one wavelength during one period, T . The wave speed is the ratio of the wavelength to the wave period.
Another solution is obtained by replacing the cosine function by the sine function.