Two sound waves travel in the same medium along the same direction....

The average power transmitted across a cross-section of area $A$ by a sound wave is given by $P = I A$, where $I$ is the intensity of the wave.

The intensity of a sound wave in terms of its pressure amplitude $\Delta P_0$ is given by $I = \dfrac{(\Delta P_0)^2}{2 \rho v}$, where $\rho$ is the density of the medium and $v$ is the speed of sound in the medium.

Since both waves travel in the same medium, $\rho$ and $v$ are the same for both waves.

It is given that both waves have identical pressure amplitudes, so $\Delta P_0$ is also the same.

Therefore, the intensity $I$ is independent of the wavelength $\lambda$ when the pressure amplitude is constant.

This implies $I_1 = I_2$, and hence the average power transmitted across the same cross-section is equal, giving $P_1 = P_2$.