Sound speed measures a material response

Imagine compressing a small fluid parcel by a tiny fractional volume change. Bulk modulus is the pressure increase divided by the fractional volume decrease in the limit of an infinitesimal change. Because mass stays fixed, decreasing volume increases density. The corresponding density derivative converts bulk modulus into the coefficient that closes the acoustic equations. A large modulus means that even a small compression produces a substantial restoring pressure, while equilibrium density sets the mass that must accelerate.

The derivative needs a thermodynamic constraint. The subscript s means fixed specific entropy, not fixed total entropy of an arbitrarily changing collection of material. Ordinary linear sound in a weakly dissipative fluid is approximately reversible and adiabatic over each oscillation, so this isentropic stiffness is appropriate. Near boundaries or at sufficiently small length scales, thermal exchange can change the effective closure. A material number copied without its thermal conditions can therefore predict the wrong speed.

For a calorically perfect ideal gas with constant heat-capacity ratio γ, an isentropic change obeys pressure proportional to density raised to γ. Differentiating gives c²=γp₀/ρ₀. Substituting the ideal-gas equation of state gives γRT when R is the gas-specific constant. At fixed composition and γ, raising absolute temperature by a factor of four doubles sound speed. Celsius temperature cannot be inserted into that scaling because its zero is not the thermodynamic zero.

An isothermal ideal-gas perturbation instead has derivative p₀/ρ₀ and predicts a speed smaller by the square root of γ. Neither expression is selected by preference: compare the disturbance period with the time required for a parcel to exchange heat with its surroundings. Thermal boundary layers in narrow passages can make the distinction important. At extreme frequencies or temperatures, internal molecular relaxation can also make γ frequency dependent and introduce attenuation, outside the constant-γ approximation.

For water-like numbers, take a bulk modulus of 2.25 billion pascals and density of 1000 kilograms per cubic metre. Their ratio is 2.25 million square metres per square second, giving a speed of 1500 metres per second. The units are a useful guardrail: modulus divided by density must have dimensions of speed squared. This example explains the size of a result without pretending that real water has one immutable modulus independent of temperature, pressure, and composition.

The wave lab cannot vary bulk modulus or density. It can isolate the superposition assumption used after a constant sound speed has already been chosen. Predict the sum at equal amplitudes with zero phase difference, then change only the phase to 180°. The cancellation concerns perturbations about equilibrium, not the disappearance of the equilibrium fluid pressure. Keep the thermodynamic speed calculation on paper; the lab's phase and amplitude controls do not test which thermal derivative applies.