The equation for a simple harmonic wave is given by

Here v is the velocity of wave and y is the displacement of the particle.

The velocity of the particle u = dy dt

So differentiating equation (1) with respect to time t

The equation for a simple harmonic wave is given by

The maximum value of the particle velocity is

U_{max}=2πav/λ

maximum particle velocity =2πa/λ wave velocity

To find the particle acceleration, differentiate equation (2) with respect to time

The acceleration is maximum when y = a

The negative sign shows that the acceleration of the particle is directed towards its mean position.

Differentiating equation (1) with respect to x

Dy/dt represents the slope of the displacement curve

From equations (2) and (6)

. . (Particle velocity at any instant)