Question

The velocity (v) of water waves may depend upon their wavelength (λ), the density of water (ρ) and the acceleration due to gravity (g). Fine the relation between there quantities by the method of dimensions.

Answer 1

v α λ^a ρ^b g^c ⇒ v = K  λ^a ρ^b g^c

Where K is the dimensionless constant).

Writing the dimensional formulae of various quantities involved in equation (i) we have 

[M⁰LT^-1] = [L]^a [ML^-3]^b [LT^-2]^c

= [M^b L^{a-3b +c} T^-2c]

Comparing the powers of M,L and T, we get a - 3b +c = 1 ...(ii)

-2c = -1

or ,c = 1/2 ....(iii)

b = 0

Again from (iii), (ii) and (iv), we get

a - 3b = 1 -1/2 = 1/2

or, a - 0 = 1/2 ⇔ a = 1/2 ...(iv)

Substituting the values of a, b and c in (i), we get