Mathematical Description of Compressional waves

Compression (Longitudinal) waves propagate with their particles moving parallel to the direction of the wave. However these waves involve changes in volume, hence pressure. They move by something like expanding and contracting. The region under high pressure is called Condensation while the one associated with low pressure is known as rarefaction. In Earthquake seismology it is those waves known as P - waves that move with this manner while A good example in our daily life is sound waves that we used to hear from different sources.

This post will tell you how the wave Equation associated with these waves is derived.

Since wave of a moving sound, obey harmonic function with sinusoidal shape , their displacement can be shown as,

X = A coswt 

Then wave function ,  y (X, t) = A cos (wt + Q) 

Where Q - phase angle

 For sinusoidal wave moving in + X - direction (right)

y ( X, t) =A cos [kX - wt]

Consider the equation 1, since we are dealing with Longitudinal waves we can use s rather than y in order to differentiate this from that of transverse waves. Since here amplitude is due to change in pressure it is known as Pressure Amplitude (∆P)

Then s ( X, t) = sm cos( kX - wt)...................(1)

Remember pressure variation in a gas is given as

∆P = - B (∆V/V).......................(2)

If a Volume of a layer thickness ∆x and cross section area A, then 

V = A∆x, while the volume that associated with pressure change ∆s, at area A, then

∆V = A∆s

Then put two volumes above into equation 2

∆P = - B (∆V/V) = - B (A∆s/A∆x)

Then A canceled out, the equation remained as 

∆P = - B∆s/∆x.....................(3)

As ∆s approaching zero, then ∆s/∆x = ∂s/∂x

Then equation 3, form, ∆P = - B (∂s/∂x)...........(4)

But equation 1, recalls 

s ( X, t) = sm cos( kX - wt).

Then put it into equation 4 above, then

∆P = - B ∂[smcos( kX - wt)]/∂x

∆P = - B [- ksm sin(kX - wt)]= B smk sin(kX - wt )

∆P = B smksin(kX - wt)..................(5)

Since Bulk modulus B = dV2  also w = kV where d - density

∆P = dwVsmsin(kX - wt)...................(6)

Since ∆Pm = dwsmV is Maximum pressure Amplitude.

Then, ∆P = ∆Pmsin(kX - wt).................(7)

Equation 7 is the Compressional Wave Equation

It shows pressure variation during Longitudinal wave propagation through medium

Problem! 

Consider a Longitudinal (Compressional) wave of wavelength h traveling with a speed V, along the x direction through a medium of density d. The displacement of the molecules of the medium from their equilibrium position is given by

s = sm sin(kx - wt)

Show that the pressure variation in the medium is given by

P = - (2πdV2sm/h) cos(kx - wt)

Soln.

We have to show the pressure variation as given in our problem above 

From s( X, t) = sm sin( kX - wt).

By using equation 4 above ∆P= - B (∂s/∂x)

Then if you substitute s = sm sin(kx - wt)

P = - B ∂ [sm sin(kx - wt)]/∂x

Then, P = - Bksmcos(kx - wt)

But B = dV2 and k =2π/wavelength (h), substitute these into equation above

P = - (2πdV2sm/h) cos (kx - wt)

Hence done! 

Thanks for your Attention!

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