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Variation of Velocity and Pressure in a Progressive Wave

A plane progressive wave is represented by the equation

 

 

y =a sin 2π/λ (vt-x)

 

and its graphical representation of variation of displacement y is shown in Figure 2.1

 

The particle velocity is given by

 

u = dy/dt=2πav/λ cos2π/λ (vt-x)

 

and its variation is shown in Figure 2.2.

The volume strain in the medium is dy/dx , given by

 

dy /dt =- 2πa/λ cos 2π/λ(vt-x)

 

The modulus of elasticity (K) of the medium is defined as

 

K = change in pressure /  volume strain = -dp/(dy/dx)

 

dp = – k dy/dx

 

dp = -k (–dy /dx)

 

dp = -k(-dy/dx)

 

In a region where dy/dx is -ve, so that dP +ve i.e., it is a region of compression. If dy/dx then dP is -ve i.e., a region of rarefaction.

 

Using equations (3) and (4); we have

The variation of dP is shown in Figure 2.3, where P0 is the normal pressure of the medium when wave is not propagating.