The Acoustic Wave Equation and Speed of Sound

Derivation

The Acoustic Wave Equation and Speed of Sound

D-328 Home PU-307 Threads waves · energy · matter Depends on Euler's Equations as the Inviscid Limit, The Continuity Equation

Statement

Linearising the inviscid, compressible Euler momentum equation and the continuity equation about a uniform, quiescent rest state \(\left(\rho_0,\ p_0,\ \mathbf{u}=\mathbf{0}\right)\), the small pressure perturbation \(p'\), density perturbation \(\rho'\), and velocity perturbation \(\mathbf{u}\) each satisfy the second-order linear wave equation \(\dfrac{\partial^2 p'}{\partial t^2}=c^2\nabla^2 p'\), whose signal speed is \(c^2=\left(\dfrac{\partial p}{\partial \rho}\right)_s\), the isentropic derivative of pressure with respect to density evaluated at the rest state.

Why it matters

Sound is the propagation of small mechanical disturbances through a compressible medium, and this derivation is the bridge that turns the raw conservation laws of fluid mechanics into a predictive statement about how fast, and by what law, those disturbances travel. It fixes the speed of sound not as an empirical constant but as a thermodynamic property of the medium: the stiffness of the fluid divided by its inertia, taken at constant entropy.

The same linearisation template — perturb about equilibrium, discard products of small quantities, eliminate variables to reach a single wave equation — recurs across physics, from plasma oscillations to gravitational waves. Getting the acoustic case right, especially the crucial choice of the isentropic rather than isothermal derivative, is what let Laplace correct Newton's estimate of the speed of sound and match observation.

Assumptions

The medium is a continuous, inviscid, non-conducting fluid (ideal Euler fluid).Viscosity and thermal conduction introduce damping and dispersion, so the pure wave equation is replaced by a lossy, frequency-dependent one and the sharp signal speed \(c\) blurs into an attenuation-plus-dispersion relation.

Disturbances are small: \(|\rho'|\ll\rho_0\), \(|p'|\ll p_0\), and \(|\mathbf{u}|\ll c\).Products of perturbations can no longer be neglected; the equations stay nonlinear, wavefronts steepen and shock, and the superposition principle fails.

The rest state is uniform and static: \(\rho_0,\ p_0\) constant in space and time, \(\mathbf{u}_0=\mathbf{0}\).Background gradients (e.g. a stratified atmosphere) or a mean flow add advection and refraction terms; \(c\) becomes position-dependent and rays bend.

The motion is isentropic: each fluid element conserves its specific entropy, so \(p=p(\rho,s)\) with \(s\) held fixed.If instead heat exchange keeps the process isothermal, the relevant derivative is \(\left(\partial p/\partial\rho\right)_T\), giving Newton's too-slow value; a full treatment must carry the entropy equation and thermal conduction.

No body forces (gravity) enter at the perturbation order considered.Gravity reintroduces buoyancy, producing internal gravity waves and an acoustic cutoff frequency, so the dispersion relation acquires a Brunt–Väisälä term.

Derivation

\[ \frac{\partial \rho}{\partial t}+\nabla\cdot\left(\rho\,\mathbf{u}\right)=0,\qquad \rho\left(\frac{\partial \mathbf{u}}{\partial t}+\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}\right)=-\nabla p \]

Start from the prior results: continuity (mass conservation) and the inviscid Euler momentum equation, with no body force. A

\[ \rho=\rho_0+\rho',\qquad p=p_0+p',\qquad \mathbf{u}=\mathbf{0}+\mathbf{u}' \]

Introduce the perturbation ansatz about the rest state; \(\rho_0,p_0\) are constants and \(\rho',p',\mathbf{u}'\) are first-order small. Legal because we are expanding, not yet approximating. A

\[ \frac{\partial \rho'}{\partial t}+\nabla\cdot\left(\left(\rho_0+\rho'\right)\mathbf{u}'\right)=0 \;\Longrightarrow\; \frac{\partial \rho'}{\partial t}+\rho_0\,\nabla\cdot\mathbf{u}'=0 \]

Substitute into continuity. Since \(\rho_0\) is constant \(\partial_t\rho_0=0\); the term \(\nabla\cdot(\rho'\mathbf{u}')\) is second-order small and is dropped. This is the linearised continuity equation. B

\[ \left(\rho_0+\rho'\right)\left(\frac{\partial \mathbf{u}'}{\partial t}+\left(\mathbf{u}'\cdot\nabla\right)\mathbf{u}'\right)=-\nabla\left(p_0+p'\right) \;\Longrightarrow\; \rho_0\,\frac{\partial \mathbf{u}'}{\partial t}=-\nabla p' \]

Substitute into Euler. The convective term \((\mathbf{u}'\cdot\nabla)\mathbf{u}'\) is second-order; \(\rho'\,\partial_t\mathbf{u}'\) is second-order; \(\nabla p_0=\mathbf{0}\). What survives is the linearised momentum equation. B

\[ p=p(\rho,s)\ \Rightarrow\ p'=\left(\frac{\partial p}{\partial \rho}\right)_{s}\bigg|_{0}\,\rho'\equiv c^2\,\rho',\qquad c^2:=\left(\frac{\partial p}{\partial \rho}\right)_{s} \]

Close the system with the isentropic equation of state, Taylor-expanded to first order about \((\rho_0,s_0)\). Isentropic because inviscid, non-conducting flow of an initially uniform fluid conserves each element's entropy, so \(s\) is held fixed. This defines \(c^2\). C

\[ \frac{\partial}{\partial t}\left(\frac{\partial \rho'}{\partial t}+\rho_0\,\nabla\cdot\mathbf{u}'\right)=0 \;\Longrightarrow\; \frac{\partial^2 \rho'}{\partial t^2}+\rho_0\,\nabla\cdot\frac{\partial \mathbf{u}'}{\partial t}=0 \]

Differentiate the linearised continuity equation (Step 3) in time. The divergence and time derivative commute because the fields are smooth. This isolates \(\partial_t\mathbf{u}'\) for elimination. B

\[ \nabla\cdot\left(\rho_0\,\frac{\partial \mathbf{u}'}{\partial t}\right)=\nabla\cdot\left(-\nabla p'\right)\;\Longrightarrow\;\rho_0\,\nabla\cdot\frac{\partial \mathbf{u}'}{\partial t}=-\nabla^2 p' \]

Take the divergence of the linearised momentum equation (Step 4) to produce exactly the quantity \(\rho_0\,\nabla\cdot\partial_t\mathbf{u}'\) that appears in Step 6, now expressed through \(p'\). B

\[ \frac{\partial^2 \rho'}{\partial t^2}-\nabla^2 p'=0 \]

Substitute Step 7 into Step 6, eliminating the velocity field entirely. Only the two scalar perturbations remain. A

\[ \frac{1}{c^2}\frac{\partial^2 p'}{\partial t^2}-\nabla^2 p'=0 \;\Longleftrightarrow\; \frac{\partial^2 p'}{\partial t^2}=c^2\nabla^2 p' \]

Use the equation of state \(p'=c^2\rho'\) (Step 5) with constant \(c^2\) to replace \(\rho'\) by \(p'/c^2\) in Step 8, then multiply through by \(c^2\). The identical equation holds for \(\rho'\) and, taking the gradient, for \(\mathbf{u}'\). A

Result

\[ \boxed{\;\frac{\partial^2 p'}{\partial t^2}=c^2\,\nabla^2 p',\qquad c^2=\left(\frac{\partial p}{\partial \rho}\right)_{s}\;}\]

Reading. Each acoustic perturbation obeys the classical wave equation, so disturbances propagate non-dispersively at a single speed \(c\) set by the isentropic stiffness of the medium, \(\left(\partial p/\partial\rho\right)_s\). Stiffer or springier fluids (large \(\partial p/\partial\rho\)) carry sound faster; denser fluids at fixed stiffness carry it slower, because \(\partial p/\partial\rho=(\partial p/\partial\rho)\) already folds in the inertia \(\rho\). For an ideal gas \(c=\sqrt{\gamma p_0/\rho_0}=\sqrt{\gamma R T/M}\).

Units check. \(\left[\partial p/\partial\rho\right]=\dfrac{\mathrm{Pa}}{\mathrm{kg\,m^{-3}}}=\dfrac{\mathrm{N\,m^{-2}}}{\mathrm{kg\,m^{-3}}}=\dfrac{\mathrm{kg\,m^{-1}s^{-2}}}{\mathrm{kg\,m^{-3}}}=\mathrm{m^2\,s^{-2}}=[c^2]\). In the wave equation, \([\partial_t^2 p']=\mathrm{Pa\,s^{-2}}\) and \([c^2\nabla^2 p']=\mathrm{m^2 s^{-2}}\cdot\mathrm{Pa\,m^{-2}}=\mathrm{Pa\,s^{-2}}\); the two sides match.

Limiting cases

Ideal gas, isentropic: \(p\propto\rho^{\gamma}\Rightarrow c=\sqrt{\gamma p_0/\rho_0}=\sqrt{\gamma R T/M}\); air at \(20^\circ\mathrm{C}\) gives \(c\approx 343\ \mathrm{m\,s^{-1}}\).
Newton's isothermal error: taking \(\left(\partial p/\partial\rho\right)_T=p_0/\rho_0\) drops the factor \(\gamma\), predicting \(c\approx 290\ \mathrm{m\,s^{-1}}\), about \(16\%\) low — the historical Newton–Laplace discrepancy.
Incompressible limit: \(\left(\partial p/\partial\rho\right)_s\to\infty\Rightarrow c\to\infty\); pressure adjusts instantaneously and the wave equation degenerates to Laplace's equation for \(p'\).
Liquids/solids: \(c=\sqrt{K_s/\rho_0}\) with \(K_s\) the adiabatic bulk modulus; water gives \(c\approx 1480\ \mathrm{m\,s^{-1}}\).
Plane-wave solution: \(p'=A\exp\!\big(i(\mathbf{k}\cdot\mathbf{x}-\omega t)\big)\) satisfies the equation iff \(\omega=c\,|\mathbf{k}|\) — linear, non-dispersive.

Breaks when

Finite amplitude. When \(|\mathbf{u}'|\) is not \(\ll c\), the discarded convective and \(\rho'\mathbf{u}'\) terms matter; wave crests travel faster than troughs, the profile steepens, and shocks form (sonic booms, blast waves), governed by the nonlinear Euler/Burgers dynamics rather than the linear wave equation.
Strong dissipation or high frequency. When viscosity and thermal conduction are not negligible — high ultrasonic frequencies, rarefied gases, small length scales approaching the mean free path — attenuation and dispersion enter, the medium no longer supports a single sharp speed, and the continuum Euler description itself fails.
Non-uniform background. A stratified atmosphere, mean flow, or temperature gradient makes \(c=c(\mathbf{x})\) and adds advection and buoyancy terms, so rays refract and an acoustic cutoff appears; the constant-coefficient wave equation no longer holds globally.

Failure modes

Using the isothermal derivative. Writing \(c^2=p_0/\rho_0\) (constant \(T\)) instead of \(\gamma p_0/\rho_0\) (constant \(s\)); this is precisely Newton's error and gives a speed too low by \(\sqrt{\gamma}\).
Keeping the convective term. Retaining \((\mathbf{u}'\cdot\nabla)\mathbf{u}'\) or \(\rho'\partial_t\mathbf{u}'\) at first order — these are second-order small and must be dropped for a linear theory; keeping some but not all second-order terms is inconsistent.
Forgetting \(\rho_0\) is constant. Treating \(\nabla\rho_0\) or \(\partial_t\rho_0\) as nonzero in a uniform rest state, generating spurious source terms.
Confusing phase and group speed. Because the medium is non-dispersive, \(v_{\text{phase}}=v_{\text{group}}=c\); students who import dispersive intuition invent a nonexistent frequency dependence.
Dimensional slip on \(c^2\). Reading \(c^2=\partial p/\partial\rho\) as \(\partial p/\partial V\) or inserting \(\rho\) twice, corrupting the units.
Perturbing about a moving state. Setting \(\mathbf{u}_0\neq0\) but then dropping the advective \(\mathbf{u}_0\cdot\nabla\) coupling, which is first-order when the background moves.

Discussion

The physical content of \(c^2=\left(\partial p/\partial\rho\right)_s\) is a competition between restoring stiffness and inertia. A compression locally raises the pressure by \(\partial p/\partial\rho\) per unit density excess; that pressure gradient accelerates the neighbouring fluid, whose inertia is \(\rho_0\). The wave equation is the exact bookkeeping of this push-and-return, and its coefficient is a ratio of a "spring constant" (stiffness) to a "mass" (density), just as in a mechanical oscillator \(\omega=\sqrt{k/m}\). Sound is that oscillation propagated in space.

The subscript \(s\) is the whole story of why Newton was wrong and Laplace right. Newton assumed the compressions and rarefactions were slow enough to stay at ambient temperature (isothermal, \(pV=\text{const}\)); Laplace recognised that acoustic oscillations are fast compared to heat diffusion, so each fluid element compresses adiabatically and heats up, stiffening the medium by the factor \(\gamma=c_p/c_v\). The correction \(c_{\text{isentropic}}/c_{\text{isothermal}}=\sqrt{\gamma}\) is not a small print detail — it is a \(16\%\) effect in air and was a genuine crisis in eighteenth-century physics resolved only by taking thermodynamics seriously.

At a deeper level the isentropic assumption is the leading term of a systematic expansion. Sound carries not only a pressure wave but a weakly coupled entropy (thermal) mode and, in the full Navier–Stokes–Fourier system, a vorticity mode; linearisation about equilibrium block-diagonalises these three. Dissipation couples them at order \(k^2\), giving the Stokes–Kirchhoff attenuation \(\alpha\propto\omega^2\left(\tfrac{4}{3}\mu+\mu_B+\kappa(\gamma-1)/c_p\right)\). That the zeroth-order acoustic mode decouples cleanly, with a real, frequency-independent speed, is precisely what makes the naive wave equation such a good approximation over the audible range, where \(\alpha\) is tiny.

Common misconceptions. Sound speed does not depend on pressure alone: for an ideal gas \(c=\sqrt{\gamma R T/M}\) is set by temperature, not pressure, because \(p_0/\rho_0=RT/M\) — raising pressure at fixed temperature raises density in step and leaves \(c\) unchanged. Nor does sound speed depend on the sound's amplitude or loudness in the linear regime; amplitude only matters once nonlinearity (shocking) sets in. And \(c\) is not the speed of the air molecules themselves (the fluid velocity \(|\mathbf{u}'|\) is far smaller); it is the speed of the disturbance pattern.

Worked examples

Speed of sound in dry air at \(20^\circ\mathrm{C}\). Treat air as an ideal diatomic gas: \(\gamma=1.40\), molar mass \(M=28.97\times10^{-3}\ \mathrm{kg\,mol^{-1}}\), \(R=8.314\ \mathrm{J\,mol^{-1}K^{-1}}\), \(T=293.15\ \mathrm{K}\). A

\[ c=\sqrt{\left(\frac{\partial p}{\partial\rho}\right)_s}=\sqrt{\frac{\gamma p_0}{\rho_0}}=\sqrt{\frac{\gamma R T}{M}} \]

Symbolic form first, using \(p_0/\rho_0=RT/M\) for an ideal gas. B

\[ c=\sqrt{\frac{1.40\times 8.314\ \mathrm{J\,mol^{-1}K^{-1}}\times 293.15\ \mathrm{K}}{28.97\times10^{-3}\ \mathrm{kg\,mol^{-1}}}} \]

Insert numbers with units; \(\mathrm{J=kg\,m^2\,s^{-2}}\) so the radicand is \(\mathrm{m^2\,s^{-2}}\). A

\[ c=\sqrt{\frac{3412.6\ \mathrm{J\,mol^{-1}}}{0.02897\ \mathrm{kg\,mol^{-1}}}}=\sqrt{1.178\times10^{5}\ \mathrm{m^2\,s^{-2}}} \]

\[ c\approx 343\ \mathrm{m\,s^{-1}} \]

Reading. The textbook value for dry air at room temperature, recovered directly from the isentropic sound speed. Had we used the isothermal \(c=\sqrt{RT/M}=290\ \mathrm{m\,s^{-1}}\) we would be \(16\%\) low.

Speed of sound in water and its bulk modulus. Water has adiabatic bulk modulus \(K_s=2.20\times10^{9}\ \mathrm{Pa}\) and density \(\rho_0=1.00\times10^{3}\ \mathrm{kg\,m^{-3}}\). The isentropic derivative for a liquid is \(\left(\partial p/\partial\rho\right)_s=K_s/\rho_0\). A

\[ c=\sqrt{\left(\frac{\partial p}{\partial\rho}\right)_s}=\sqrt{\frac{K_s}{\rho_0}} \]

Because \(K_s\equiv\rho\,(\partial p/\partial\rho)_s\) by definition of the bulk modulus, the density cancels to one power inside the root. B

\[ c=\sqrt{\frac{2.20\times10^{9}\ \mathrm{Pa}}{1.00\times10^{3}\ \mathrm{kg\,m^{-3}}}}=\sqrt{2.20\times10^{6}\ \mathrm{m^2\,s^{-2}}} \]

\(\mathrm{Pa/(kg\,m^{-3})}=\mathrm{m^2\,s^{-2}}\), confirming units. A

\[ c\approx 1.48\times10^{3}\ \mathrm{m\,s^{-1}} \]

Reading. About \(1480\ \mathrm{m\,s^{-1}}\), roughly \(4.3\) times faster than in air: water is far stiffer (\(K_s\) huge) even though it is denser, and stiffness wins. This is why sonar works over long ranges.

Problems

(A) Temperature scaling. By what percentage does the speed of sound in air increase when the temperature rises from \(0^\circ\mathrm{C}\) to \(30^\circ\mathrm{C}\), assuming ideal-gas behaviour?
Solution
For an ideal gas \(c\propto\sqrt{T}\) with \(T\) in kelvin. So \(\dfrac{c_2}{c_1}=\sqrt{\dfrac{303.15}{273.15}}=\sqrt{1.1098}=1.0535\). The increase is about \(5.3\%\). (Concretely \(c_1\approx331\ \mathrm{m\,s^{-1}}\to c_2\approx349\ \mathrm{m\,s^{-1}}\).)
(A) Isentropic vs isothermal. Show that for an ideal gas the isentropic sound speed exceeds the isothermal one by exactly the factor \(\sqrt{\gamma}\), and evaluate the ratio for air (\(\gamma=1.40\)).
Solution
Isentropic: \(p\propto\rho^\gamma\Rightarrow (\partial p/\partial\rho)_s=\gamma p/\rho\), so \(c_s=\sqrt{\gamma p/\rho}\). Isothermal: \(p\propto\rho\Rightarrow(\partial p/\partial\rho)_T=p/\rho\), so \(c_T=\sqrt{p/\rho}\). Ratio \(c_s/c_T=\sqrt{\gamma}\). For air \(\sqrt{1.40}=1.183\), i.e. the isentropic value is \(18.3\%\) higher.
(B) Plane-wave verification. Verify by direct substitution that \(p'(x,t)=A\cos(kx-\omega t)\) solves \(\partial_t^2 p'=c^2\partial_x^2 p'\) and derive the dispersion relation. Is the medium dispersive?
Solution
\(\partial_t^2 p'=-\omega^2 A\cos(kx-\omega t)\); \(\partial_x^2 p'=-k^2 A\cos(kx-\omega t)\). Substituting: \(-\omega^2=c^2(-k^2)\Rightarrow \omega^2=c^2k^2\Rightarrow\omega=ck\). Phase speed \(v_p=\omega/k=c\) and group speed \(v_g=d\omega/dk=c\) are equal and independent of \(k\), so the medium is non-dispersive.
(B) Velocity amplitude of everyday sound. A sound wave in air (\(\rho_0=1.21\ \mathrm{kg\,m^{-3}}\), \(c=343\ \mathrm{m\,s^{-1}}\)) has pressure amplitude \(p'_0=1.0\ \mathrm{Pa}\) (about \(94\ \mathrm{dB}\), loud). Using the linearised momentum equation for a plane wave, find the fluid-velocity amplitude \(u'_0\) and confirm \(u'_0\ll c\).
Solution
For a plane wave \(p'=p'_0\cos(kx-\omega t)\), \(u'=u'_0\cos(kx-\omega t)\), the relation \(\rho_0\partial_t u'=-\partial_x p'\) gives \(\rho_0\omega u'_0=k p'_0\), and with \(\omega=ck\), \(u'_0=\dfrac{p'_0}{\rho_0 c}\). Numerically \(u'_0=\dfrac{1.0\ \mathrm{Pa}}{1.21\ \mathrm{kg\,m^{-3}}\times343\ \mathrm{m\,s^{-1}}}=2.4\times10^{-3}\ \mathrm{m\,s^{-1}}\). Since \(u'_0/c\approx7\times10^{-6}\ll1\), linearisation is superbly justified even for a loud sound. The quantity \(Z=\rho_0 c=415\ \mathrm{Pa\,s\,m^{-1}}\) is the acoustic impedance of air.
(C) Helium voice. A person inhales helium (\(\gamma=1.667\), \(M=4.00\times10^{-3}\ \mathrm{kg\,mol^{-1}}\)) at \(20^\circ\mathrm{C}\). Compute the speed of sound and, given that resonant frequencies of the vocal tract scale as \(f\propto c\), find the factor by which formant frequencies shift relative to air (\(c=343\ \mathrm{m\,s^{-1}}\)). Does pitch (fundamental of the vocal folds) change?
Solution
\(c_{\text{He}}=\sqrt{\dfrac{\gamma R T}{M}}=\sqrt{\dfrac{1.667\times8.314\times293.15}{4.00\times10^{-3}}}=\sqrt{1.016\times10^{6}}=1.01\times10^{3}\ \mathrm{m\,s^{-1}}\). The formant shift factor is \(c_{\text{He}}/c_{\text{air}}=1008/343\approx2.94\), so vocal-tract resonances rise by nearly a factor of three, producing the characteristic "chipmunk" timbre. The pitch (fundamental frequency set by the vibrating vocal folds) is essentially unchanged, because it is governed by fold tension and mass, not by the gas sound speed; helium alters the resonant filtering (formants), not the source frequency — a common misconception.