# Mechanical Waves

The waves which needs material medium to propagate is called mechanical waves. Example: water waves, sound waves, etc.

Mechanical waves are of two types:
1. Longitudinal Waves
2. Transverse waves

### Speed of Sound Wave in any medium:

Experimentally it is found that speed of sound wave in any medium depends on two factors; elasticity of medium and density of medium.
Let 'V' be the speed of sound in any medium, 'E' be the elasticity and 'p' be the density of the medium. Then,

V \alpha   E^a-------------(1)
V \alpha   \rho^b ---------(2)

Combining equation (1) and (2) we get,
V \alpha  E^a\rho^b
V = KE^a\rho^b-----------(3)

Writing the dimension of each terms in eqn (3) we get,
[M^0L^1T^-1]=[M^1L^-1T^-2]^a [M^1L^-3T^0]^b
[M^0L^1T^-1]=[M^(a+b)L^(-a-3b)T^(-2a)]

Equating the power of M, L and T we get,
a+b=0, -a-3b=1 and -2a=-1

Solving these equation we get,
a=\frac{1}{2}   and   b=\frac{-1}{2}

Putting the value of a and b in equation (3) we get,
V = K E^\frac{1}{2} \rho^\frac{-1}{2}
V = K \sqrt\frac{E}{\rho}

Experimentally the value of K is found to be 1 so,

V = \sqrt\frac{E}{\rho}   which is the required expression for the velocity of sound wave in any medium.

### Velocity of sound in solid:

Elasticity(E)= Young's Modulus of Elasticity(Y)
\therefore velocity of sound in solid(v_s)= \sqrt\frac{Y}{\rho}
For steel, Y = 2\times10^11 Nm^(-2)
\rho = 7800 kgm^(-3)
\therefore velocity of sound in steel (V_steel)= \sqrt\frac{2\times10^11}{7800}
= 5064m/s

### Velocity of sound in liquid

Elasticity(E) = Bulk modulus of Elasticity (\beta)
\therefore velocity of sound in liquid (v_l)= \sqrt\frac{\beta}{\rho}
For water, \beta = 2.04\times10^9Nm^(-2)
\rho = 1000kgm^(-3)
\therefore velocity of sound in water (v_w) = \sqrt\frac{2.04\times10^9}{1000}
= 1428 m/s

### Velocity of sound in gaseous medium

i) Newton's formula for the velocity of sound in gas:
Elasticity(E) =Bulk modulus of elasticity(\beta) = Pressure exerted by gas (P)
\therefore velocity of sound in gas (v_g) = \sqrt\frac{P}{\rho}
At NTP, P =  1.013 \times 10^5 N/m^-2  and \rho = 1.29 kg/m^3
\therefore velocity of sound in gas (v_g) =\sqrt\frac{1.013\times10^5}{1.29}
= 280 m/s which does not match with the experimental result.

ii) Laplace correction for the velocity of sound in gas:
Elasticity(E)= Bulk modulus of Elasticity (\beta) = \gammaP
\therefore velocity of sound in gas (v_g) = \sqrt\frac{\gammaP}{\rho}
At NTP, P =  1.013 \times 10^5 N/m^-2 , \rho = 1.29 kg/m^3 and \gamma=1.4
\therefore velocity of sound in gas (v_g) =\sqrt\frac{1.4\times1.013\times10^5}{1.29}
= 332 m/s which match with the experimental result.