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Particle Velocity and Wave Velocity

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

Umax=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)