Question

Corresponding to displacement equation, y =A sin (kx+omegat) of a longitudinal wave make its pressure and density wave also. Bulk modulus of the medium is B and density is p.

Answer 1

Given:

Displacement equation of a longitudinal wave:

y =A sin (kx+ωt)

Bulk modulus of the medium = B and

Density = p

To Find:

Pressure wave and density wave.

Solution:

We know,

Pressure (P) equation is given by,

• ΔP = - B ( $\frac{\delta y}{\delta x}$ )  -- (a)

Density equation is given by,

• Δd = $\frac{p}{B}$ ( ΔP) = $\frac{p}{B}$ ( -B ( $\frac{\delta y}{\delta x}$ ) ) = -p  $\frac{\delta y}{\delta x}$ --- (b)

Given,

• y ( x, t) = A sin ( kx + ωt )

Partial differentiation with respect to x gives,

• $\delta$y/$\delta$x = Ak cos ( kx + ωt ) -- (c)

Now for the pressure equation of the wave substitute (c) in (a)

• ΔP = -B( Ak cos (kx + ωt))
• ΔP = -BAk cos ( kx + ωt )

For the density equation of the wave substitute (c) in (b)

• Δd = -pAk cos (kx + ωt )

The pressure wave is ΔP =   -BAk cos ( kx + ωt ) and density wave is

Δd = -pAk cos (kx + ωt )