JEE Main · PhysicsHard

$A transverse wave is described by $y = A \sin(kx - \omega t + \phi)$. If the wave travels through a medium where the density varies as $\rho(x) = \rho_0(1 + \alpha x)$, what is the change in the wave amplitude as it propagates?$

Show correct answer & step-by-step solution

Correct answer: A — $A \propto (1 + \alpha x)^{-1/4}$

Solution

The intensity

I = \frac{1}{2} ρ ω^{2} A^{2} v

is constant. Since

v = T / ρ

, we have

I \propto ρ A^{2} ρ^{- 1/2} = ρ^{1/2} A^{2}

. Thus

A \propto ρ^{- 1/4} = (1 + α x)^{- 1/4}