One can generate longitudinal waves in the fluid either by displacing the fluid using a piston or by keeping a vibrating tuning fork at one end of the tube.

we shall derive the velocity of waves in two different cases:

1. The velocity of a transverse waves along a stretched string.

2. The velocity of a longitudinal waves in an elastic medium.

Velocity of longitudinal waves in an elastic medium

Consider an elastic medium (here we assume air) having a fixed mass contained in 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 one end of the tube. Let us assume that the direction of propagation of waves coincides with the

Let *ρ* be the density of the fluid which is initially at rest. At *t* = 0, the piston at left end of the tube is 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 ∆*t*, the distance moved by the piston ∆*d *=* u*∆*t*. Now, the distance moved by the* *elastic disturbance is ∆*x = v*∆*t*. Let *m* be the mass of the air that has attained a velocity *v *in a time ∆*t *. Therefore,

*m = ρ A*∆*x = ρ A *(*v*∆*t*)

Then, the momentum imparted due to motion of piston with velocity *u* is

*p = *[*ρ A *(*v t*)]*u*

But the change in momentum is impulse.

The net impulse is

When the sound wave passes through air, the small volume element (Δ*V*) of the air undergoes regular compressions and rarefactions. So, the change in pressure can also be written as

where, V is original volume and B is known as bulk modulus of the elastic medium.

But *V* = *A* ∆*x* = *A v* ∆*t* and

∆*V* = *A* ∆*d* =*A u* ∆*t*

Therefore,

Comparing equation (11.14) and equation (11.15), we get

In general, the velocity of a longitudinal wave in elastic medium is *v* = √E/√P where *E *is the modulus of elasticity of the medium.

(i) one dimension rod (1D)

where *Y* is the Young’s modulus of the material of the rod and *ρ* is the density of the rod. The 1D rod will have only Young’s modulus.

(ii)Three dimension rod (3D) The speed of longitudinal wave in a solid is

where *η* is the modulus of rigidity, K is the bulk modulus and *ρ* is the density of the rod.

Cases: For liquids:

where, *K* is the bulk modulus and *ρ* is the density of the rod.

Calculate the speed of sound in a steel rod whose Young’s modulus *Y* = 2 × 1011 N m-2 and *ρ* = 7800 *kg m*-3.

Therefore, longitudinal waves travel faster in a solid than in a liquid or a gas. Now you may understand why a shepherd checks before crossing railway track by keeping his ears on the rails to safegaurd his cattle.

An increase in pressure of 100 *k*Pa causes a certain volume of water to decrease by 0.005% of its original volume.

(a) Calculate the bulk modulus of water?.

(b) Compute the speed of sound (compressional waves) in water?.

(a) Bulk modulus

(b) Speed of sound in water is

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11th Physics : UNIT 11 : Waves : Velocity of longitudinal waves in an elastic medium |

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