The velocity of sound waves 'v' through a medium may be assumed to depend on:
1. the density of the medium 'd' 2. the modulus of elasticity 's'
Deduce by the method of dimension for formula for the velocity of sound.(take dimensional constant k=1)
Hi ! proceed as follows . Let the velocity of the sound wave in the medium be (v).
Let V = f(ρ,s). where (ρ) stands for the density of the medium and (s) stands for the moduli of elasticity.
Dimentional analysis proves to be beneficial only if the dependance is of product type...assuming that the function f(ρ,s) is of product type we have :
V α ρ^{x}s^{y}
=> V = kρ^{x}s^{y} .....(1) { where K is a dimentionless constant }
The dimensional formalism of the above equation is :
[M^{o}L^{1}T^{1}] = K[M^{1}L^{3}T^{o} ]^{x}[M^{1}L^{1}T^{2}]^{y}
=> [M^{o}L^{1}T^{1}] = K[M^{x+y}L^{3x2y}T^{2y}]
Comparing the corresponding exponents of M,L and T the above equation can be deconstructed into three individual equations
x+y = 0 ......(2)
3x2y = 1 ........(3)
2y = 1 .......(4)
solving which yields x = 1/2 and y = 1/2
substituting the value of (x) and (y) in equation (1) we get
