Equation for a Progressive Wave

The simplest type of wave is the one in which the particles of the medium are set into simple harmonic vibrations as the wave passes through it. The wave is then called a simple harmonic wave.

Consider a particle O in the medium. The displacement at any instant of time is given by

Where A is the amplitude, w is the angular frequency of the wave. Consider a particle P at a distance x from the particle O on its right. Let the wave travel with a velocity v from left to right. Since it takes some time for the disturbance to reach P, its displacement can be written as

Where f is the phase difference between the particles O and P.

We know that a path difference of l corresponds to a phase difference of 2p radians. Hence a path difference of x corresponds to a phase difference of 

Substituting equation (1.5) in equation (1.4)

We get,

Similarly, for a particle at a distance x to the left of 0, the equation for the displacement is given by

Differential equation for wave motion

Differentiating equation (1.7) with respect to x,

We get,

The velocity of the particle whose displacement y is represented by equation (1.7), is obtained by differentiating it with respect to t, since velocity is the rate of change of displacement with respect to time.

Comparing equations (1.8) and (1.9) we get,

Particle velocity = wave length x slope of the displacement curve or strain

Differentiating equation (1.8),

Differentiating equation (1.9)

Comparing equation (1.11) and (1.12) we get,

This is the differential equation of wave motion.