A plane longitudinal harmonic wave propagates in a medium with density `rho`. The velocity of the wave propagation si `v` . Assuming that the density variations of the medium, induced by the propagating wave, `Delta rho lt lt rho` , demonstrate that ltbr. `(a)` the pressure increment in the medium `Deltap=- rho v^(2)( delta xi//delta x), ` where` delta xi // delta x ` is the relative deformation,
`(b) ` the wave intensity is defined by Eq. `(4.3i)`

Explanation

`(a)` Let us consider the motion of an element of the medium of thikness `dx` os unit area of cross `-` section. Let `xi=` displacement of the particles of the medium at location `x`. Then by the equation of motion.
`rho dx dot(xi)=-dp`
where `dp` is the pressure increment over the length `dx`
Recalling the wave equation
`ddot(xi)=v^(2)(delta^(2)xi)/( delta x^(2))`
we can write the foregoing equation as
`rho v^(2)(delta^(2)xi)/(deltax^(2))dx=-dp `
Integrating thie equation, we get
`Delta p=` surplus pressure `=- rho v^(2) ( deltaxi)/( deltax)+ Const.`
In the absence of a deformation `(` a wave `),` the surplus pressure is `Delta p=0`. So ' Const ' `=0` and ltbr. `Deltap=-rho v^(2)( deltaxi)/( deltax)`.
`(b)` We have found earlier that
`w=w_(k)+ w_(p)=` total energy density
`w_(k)=(1)/(2)rho((deltaxi)/( delta))^(2), w_(p) =(1)/(2)E((deltaxi)/( deltax))^(2)=(1)/(2) rho v^(2)((deltaxi)/(deltax))^(2)`
It is easy to see that the space `-` time average of both densities is the same and the space time average of total energy density is then
`lt w gt = lt rho v^(2)((deltaxi)/(deltax))^(2)`
The intensity of the wave is
`I=vlt w ge lt ((Deltap)^(2))/(rhov) gt`
Using `lt (Deltap)^(2) gt =(1)/(2) (Deltap)_(m)^(2)` we get `I=((Deltap)_(m)^(2))/(2 rho v)`