Deriving the Speed of Sound in a Liquid

The speed of sound in a liquid depends on the bulk modulus and the density of the fluid. The bulk modulus measures the fluid's resistance to compression, while density represents its mass per unit volume. These two properties together determine how quickly a mechanical wave propagates through the liquid medium.

The impulse-momentum theorem relates the impulse created by gauge pressure changes to the momentum imparted to a fluid column. By calculating the impulse from pressure, area, and duration, and equating it to the momentum change of the disturbed fluid, the speed of the wave can be derived mathematically from the bulk modulus and density.

The bulk modulus quantifies how much a fluid resists compression when pressure is applied. Since sound waves involve alternating compression and rarefaction of the medium, a higher bulk modulus means the fluid resists deformation more strongly, allowing sound to propagate faster through the liquid.

When a mechanical wave disturbs a fluid at the origin, a column of fluid within distance v-t from the source experiences the disturbance after time t. The column's original volume decreases due to compression, creating a gauge pressure change that can be calculated using the bulk modulus of the fluid.

The speed of sound in liquids depends on bulk modulus and density, while in solids it depends on Young's modulus and density. Despite using different elastic moduli, the mathematical relationship between wave speed and these properties remains the same in both materials, reflecting a fundamental principle of wave propagation.

Sound waves propagate so rapidly through gases that compressed and rarefied gas elements cannot exchange heat with surrounding elements in time. This adiabatic condition allows the derivation of sound speed in terms of gas temperature, providing a complete understanding of how mechanical waves travel through gaseous media.

Fluid density represents the inertia of the medium—its resistance to motion. While a higher bulk modulus increases wave speed, higher density increases inertia and slows propagation. The speed of sound results from the balance between these two competing effects, with speed proportional to the square root of bulk modulus divided by density.