Fluid transients – Speed of pressure wave (celerity) The celerity (wave speed) a of a small pressure/disturbance wave in a fluid is given in terms of bulk modulus K and density ρ by which relation?

Introduction:
The propagation speed of small pressure disturbances in a fluid (acoustic speed) is a key parameter in water-hammer analysis and compressible-flow acoustics. It depends on the compressibility of the medium (bulk modulus) and its density.

Given Data / Assumptions:

• Linear, small-amplitude disturbances.
• Isentropic or isothermal assumptions reduce to the same form for defining the celerity using an appropriate K.
• Homogeneous fluid with bulk modulus K and density ρ.

Concept / Approach:

The classical derivation from continuity and momentum linearized about a uniform state yields the wave equation with speed a satisfying a^2 = K / ρ, where K = −V * (dP/dV) is the bulk modulus. Thus a = sqrt(K / ρ). In liquid–pipe systems, pipe wall elasticity can be incorporated by using an effective modulus that reduces a below the pure-fluid value.

Step-by-Step Solution:

Verification / Alternative check:

For water with K ≈ 2.2 * 10^9 Pa and ρ ≈ 1000 kg/m^3, a ≈ sqrt(2.2 * 10^6) ≈ 1483 m/s—consistent with tabulated sound speeds in water.

Why Other Options Are Wrong:

K / ρ, ρ / K, K * ρ: Miss the necessary square root or have wrong dimensions.sqrt(ρ / K): Inverts the ratio; would give unphysical behavior.

Common Pitfalls:

Forgetting to include pipe wall elasticity in water-hammer problems; the effective a can drop substantially in flexible pipes.

Final Answer:

a = sqrt(K / ρ)