Sound waves are longitudinal waves. Sound waves travel in all media, namely solids, liquids, and gases. When a longitudinal wave passes through a medium, it produces alternate compression and rarefaction in the medium.
At the portion of compression, the pressure and density slightly increases than the original value. At the portion of rarefaction, the pressure and density are slightly less than the original values.
Thus, the particles in the medium are subjected to changing stresses resulting in strain. Hence, the velocity V of a sound wave can be expressed in terms of modulus of elasticity E of the medium which is given by,
V = √(E/ρ) ................... (1)
where, ρ is the density of the medium.
Equation (1) is called Newton's formula because it was first derived by Newton.
This E is Young's modulus of elasticity Y for solids and the bulk modulus of elasticity K for liquids.
Proof of Newton's formula by the method of dimensions:
If the velocity V of sound in a medium depends upon the elasticity E and density ρ of the medium, then we can write:
V ∝ Ex. ρy
or, V = K . Ex. ρy .......... (2)
where, K is proportionality constant and x and y are numbers, to be determined.
The dimension of V is LT-1, dimension of E (force/area = MLT-2/L2) is ML-1T-2 and dimension of ρ is ML-3. From equation (2) it follows that,
LT-1 = ( ML-1T-2)x . (ML-3)y
or, LT-1 = Mx+y L-x-3y T-2x
Equating the indices of M, L, and T on both sides, we get,
x + y = 0 ......... (3a)
- x - 3y = 1 ......... (3b)
-2x = -1 ......... (3c)
Solving equations (3a), (3b), and (3c), we get,
x = (1/2), y = - (1/2)
Therefore, V = K . E(1/2). ρ-(1/2)
V = K . √(E/ρ) ......... (4)
This method does not give the magnitude of K. But by calculations, it is found that,
V = √(E/ρ) therefore, K = 1.
This gives the velocity of a longitudinal wave in an elastic medium.