Question

Velocity of sound in air depends on pressure (p), density (ρ) and acceleration due to gravity (g).
Derive the equation for velocity of sound using homogeneity principle.​

Answer 1

Explanation:

V∝P^a ρ^b g^c

Here,P=Pressure

ρ=density

g=acceleration due to gravity

From the relation, using dimensional analysis,

Dimension of LHS= Dimension of RHS

∴LT^−1=[ML^−1 T^−2]^a [ML−3 ]^b[LT^-2]

⇒LT^−1 =M^(a+b) L^(−a−3b+c) T^(−2a-2c)

Comparing LHS and RHS,

a+b=0

−a−3b+c=1

−2a-2c=−1

On solving the equations,

a=1/2

b=-1/2

c=0

∴V∝√P\√ρ