A wave propagates through a medium with a constant speed, known as a wave velocity. It is different from the speed of the particles of the medium, which is not constant. In addition, the velocity of the medium is perpendicular to the velocity of the wave. The variable speed of the particles of the medium implies that there must be acceleration associated with it.
The velocity of the particles can be obtained by taking the partial derivative of the position equation with respect to time. We can then take the partial derivative of the velocity equation with respect to time, to obtain the acceleration of the medium. We can also calculate the slope and curvature of the wave by taking the first and second partial derivative of the position equation with respect to the position, respectively.
These wave parameters can be used to obtain the linear wave equation, which is the ratio of acceleration to the curvature of a wave. The linear wave equation can describe both transverse and longitudinal waves. Interestingly, if two wave functions are individual solutions to the linear wave equation, then the sum of the two linear wave functions is also a solution to it.
This text is adapted from Openstax, University Physics Volume 1, Section 16.2: Mathematics of Waves.
