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