A longitudinal sound wave travels in the positive x-direction through a fluid of equilibrium density ρ₀ and bulk modulus B. The longitudinal displacement of the particles is s = sₘ cos(kx-ωt). Which expression correctly represents the excess pressure Δp produced by the sound wave?
Options

Question

A longitudinal sound wave travels in the positive x-direction through a fluid of equilibrium density ρ₀ and bulk modulus B. The longitudinal displacement of the particles is s = sₘ cos(kx-ωt). Which expression correctly represents the excess pressure Δp produced by the sound wave?

Options

G1 Physics · Accepted Answer

For a small-amplitude longitudinal sound wave, Δp = −B(∂s/∂x). Given, s = sₘ cos(kx−ωt). Differentiate with respect to x: ∂s/∂x = −ksₘ sin(kx−ωt). Therefore, Δp = −B[−ksₘ sin(kx−ωt)] = Bksₘ sin(kx−ωt). Thus pressure variation is displaced in phase by π/2 relative to particle displacement. Correct option: B.

A longitudinal sound wave travels in the positive x-direction through a fluid of equilibrium density ρ₀ and bulk modulus B. The longitudinal displacement of the particles is s = sₘ cos(kx−ωt). Which expression correctly represents the excess pressure Δp produced by the sound wave?