Pressure, particle velocity, and acoustic power

For a right-moving plane wave in a uniform lossless fluid, substitute the pressure sinusoid into the linear momentum equation. Integration gives particle velocity in phase with pressure, with amplitude divided by equilibrium density times sound speed. Their ratio is the characteristic specific acoustic impedance. Its units are pascal seconds per metre. It is a material-and-propagation relation for this wave, rather than an additional force law. The fluid particles move slowly even though the pressure pattern travels at sound speed.

A left-moving wave has particle velocity opposite to the positive coordinate direction when pressure is positive, so its signed pressure-to-velocity ratio is negative. A superposition can have a position-dependent, complex local impedance. Also distinguish specific impedance, pressure divided by particle velocity, from acoustic impedance defined using volume velocity. For a uniform duct flow over cross-sectional area A, volume velocity is A times particle velocity, so pressure divided by volume velocity is smaller by A.

Instantaneous acoustic intensity in one dimension is perturbation pressure times particle velocity, to leading nonzero order. Both first-order signals have zero mean, but their product need not. For sinusoidal peak amplitudes P and U with phase difference φ, averaging the product over a cycle produces one half PU cosφ. The cosine is the acoustic power factor. In-phase pressure and forward velocity transport positive mean energy; a quarter-cycle phase offset gives zero active mean intensity despite nonzero oscillations.

Root-mean-square amplitudes absorb the factor of one half: average intensity becomes pressure RMS times velocity RMS times cosφ. For a single traveling plane wave in a lossless medium, substitute U=P/Z₀ to obtain P²/(2Z₀). A pressure-only estimate using that formula is invalid in an arbitrary standing or near field. There, pressure and velocity need not have the traveling-wave amplitude ratio or phase relationship. A large pressure measurement can coexist with little net energy crossing the chosen surface.

At a rigid termination, particle velocity vanishes exactly at the wall while pressure can be large. Slightly away from it, a standing field stores and returns compressional and kinetic energy during a cycle. Its average transport can vanish without the sound field being absent. Calling every reduction in active intensity absorption would confuse reversible storage or reflection with dissipation. To identify absorbed power, compare incoming and outgoing fluxes and account for boundaries or losses that convert acoustic energy to other forms.

The waves lab displays two same-direction sinusoids and their algebraic sum. Use a ratio of one and predict the sum amplitude as phase changes from zero to 90° to 180°. That experiment helps you reason about phase-sensitive addition, but its two curves are not automatically pressure and particle velocity. It has no impedance or power readout. Calculate the pressure–velocity product separately, and do not interpret the displayed sum as an energy flux or a simulation of a standing acoustic near field.