Write the relation for velocity of longitudinal waves in,
(i) One dimensional rod
(ii) Three dimensional rod
(iii) Liquid medium

Explanation

Consider an elastic medium (here we assume air) having a fixed mass contained inn a long tube (cylinder) whose cross sectional area is A and maintained under a pressure P. One can generate longitudinal waves in the fluid either by displacing the fluid using a piston or by keeping a vibrating tuning fork at oe end of hte tube. Assume that the direction of propogation of waves coincides wit the axis of the cylinder. Let `rho` be the density of the fluid which is initially at rest. At t=0, the piston at left end of the tube in set in motion toward the right with a speed u. Let u be the velocity of the piston and v be the velocity of the elastic wave. In time interval `Delta t`, the distance moved by the piston `Delta d=u Delta t`. Now, the distance moved by the eleastic disturbance is `Deltax = v Delta t`. Let `Delta m` be the mass of the air that has attained a velocity v in a time `Delta t`. Therefore,
`Delta m= rho A Delta x= rho A(v Delta t)`
Thus, the momentum imparted due to motion of piston with velocity `mu` is
`Delta p =[rho A(v Deltat)mu`
But the change in momentum is impulse.
or `(Delta P A)Delta t =[rho A(v Delta t)]mu`
`delta P= rho v u`........(1)
When the sound wave passes through air,the small volume element `(Delta V)` of the air undergoes regular compression and rarefraction. So, the change in pressure can also be written as
`Delta P= B(DeltaV)/V`
Where, V is original volume and B is known as bulk modulus of the elastic medium.
But V= A `Delta x= Av Delta t` and `Delta V= A Delta d= A u Deltat`
Therefore,
`Delta P=B(Au Delta t)/(Av Deltat) = B u/v`............(2)
Comparing equation (1) and equation (2), we get
`rho v u = B u/v` or `v^(2) = B/rho`........(2)
`rArr v= sqrt(B/rho)`.........(3)
In general, the velocity of a longitudinal wave in elastic medium is `v= sqrt(E/rho)`, where E is the modulus of elasticity of the medium.
Cases: For a solid:
(i) One dimensional rod (ID)
`v= sqrt(Y/rho)`........(4)
where Y is the Young's modulus of the material of the rod and `rho` is the density of the rod. The ID rod will have only Young's modulus.
(ii) Three dimension rod (3D): The speed of longitudinal wave in a solid is
`v= sqrt((K + 4/3 eta)/rho)`........(5)
where `eta` is the modulus of rigidity, K is the bulk modulus and `rho` is the density of the rod.
Cases: For liquids:
`v= sqrt(K/rho)`........(6)
where, K is the bulk modulus and `rho` is the density of the rod.