Question

The velocity `upsilon` of sound waves in a medium may depend upon modulus of elasticity (E ), density (d) and wavelength `(lambda)` of of the waves. Use method of dimensions to deerive the formula for `upsilon.`

Answer 1

Let `upsilon prop E^a d^b lambda^c … (i)`
where K is dimensionless constant of proportionality.
Writing the dimensions on either side of (i), we get
`[M_0 L^1 T^(-1) = [ ML^(-1)T^^(-2)]^a [ ML^(-3)]^b L^c = M^(a+b) L^(-a - 3b +c) T^(-2a)`
Applying the principle of homogeneity of dimensions, we get
`a +b = 0 .....(ii)`
` - a - 3b +c - 1, `
` - 2a = -1 , a = (1)/(2) (iii)`
From (ii), `b = - a = -(1)/(2)` form (iii), `c =1 +a +3b = 1 +(1)/(2) - (3)/(2) = 0`
Put in (i), `upsilon = KE^(1//2). d^(-1//2). lambda^0 = Ksqrt((E)/(d))`