Q. The speed of a longitudinal wave in a metallic bar is 400 m/s. If the density and Young's modulus of the bar material are increased by 0.5% and 1%, respectively then the speed of the wave is changed approximately to ______ m/s.

Correct Answer: 401

Solution

The speed of a longitudinal wave in a solid is given by:

v = √( Y / ρ )

Taking small percentage changes:

Δv / v = ½ ( ΔY / Y − Δρ / ρ )

Given:

ΔY / Y = 1% ,   Δρ / ρ = 0.5%

Substituting:

Δv / v = ½ ( 1 − 0.5 ) = 0.25%

Original speed:

v = 400 m/s

Change in speed:

Δv = 0.25% of 400 = 1 m/s

New speed:

v' = 400 + 1 = 401 m/s

Hence, the speed of the wave becomes approximately 401 m/s.

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