Acoustic_wave_equation

# Acoustic wave equation > *For the general equation shared by strings, water ripples and light, see [[Wave_equation]].* The **acoustic wave equation** is the linear [[Partial_differential_equation|partial differential equation]] that governs small pressure disturbances in a fluid. In one dimension it reads `p_tt = c^2 p_xx`, and in three `p_tt = c^2 (p_xx + p_yy + p_zz)`, where `p` is the acoustic pressure (the small departure from the ambient pressure) and `c` is the [[Speed_of_sound|speed of sound]]. It is the [[Wave|wave]] equation of [[Sound|sound]]: every [[Loudspeaker|loudspeaker]] design, room-acoustics model, [[Sonar|sonar]] calculation and [[Medical_ultrasound|medical-ultrasound]] image starts from it, or from one of its extensions. The equation is not a separate law of nature. It follows from three older statements about a fluid, conservation of mass (the [[Continuity_equation|continuity equation]]), Newton's second law applied to a fluid parcel, and an adiabatic equation of state, once each is linearized for disturbances much smaller than the ambient values. The same derivation that yields the equation also yields its wave speed, `c = sqrt(B/rho)`, so the speed of sound in air, water or steel is a property the equation carries with it rather than an extra input.[^openstax17-2] Its solutions are waves that keep their shape as they travel, that pass through each other and add, and that reflect at walls in a way fixed by the boundary condition. The Wikitube microsim *The acoustic wave equation: a pressure pulse in a room* solves the two-dimensional equation numerically: the reader picks a closed box, a room with one open side, a wall with a slit, or two pulses, and watches the pressure ring spread, bounce and interfere while a virtual microphone records the result. ## Definition in one dimension In one dimension the acoustic wave equation is `d^2p/dx^2 - (1/c^2) d^2p/dt^2 = 0` for the acoustic pressure `p(x, t)` along a tube or in a [[Plane_wave|plane wave]]. It has exactly the form of the linear wave equation that OpenStax derives for a wave on a string, `d^2y/dx^2 = (1/v^2) d^2y/dt^2` (Eq. 16.6), which the textbook calls "one of the most important equations in physics and engineering" and notes applies equally to longitudinal waves.[^openstax16-2] The acoustic version differs only in what is displaced: instead of a string's sideways displacement, the unknown is the pressure, and the same equation holds for the air's [[Density|density]] change and its [[Particle_velocity|particle velocity]]. The scales involved are small. A [[Sound_pressure|sound pressure]] level of 120 dB, painfully loud, corresponds to an rms pressure of 20 Pa against an atmospheric pressure of about 101,325 Pa, a fractional change of 2×10⁻⁴. The corresponding air speed is `u = p/(rho c) = 20/(1.20 × 343) ≈ 0.05 m/s`, a Mach number of about 1.4×10⁻⁴. That is why a linear equation describes almost everything people hear; the nonlinear terms that were dropped become important only near explosions, in sonic booms and in intense focused ultrasound. ### Derivation The derivation follows the route OpenStax takes for the speed of sound in air, a parcel of fluid between `x` and `x + dx` in a pipe of cross-section `A`.[^openstax17-2] Write the density as `rho_0 + rho'`, the pressure as `p_0 + p` and the fluid velocity as `u`, all primed or small quantities being tiny compared with the ambient ones. 1. *Conservation of mass.* The mass flow in must equal the mass flow out plus any increase of mass in the parcel. Keeping only first-order terms, `d rho'/dt + rho_0 du/dx = 0`. 2. *Newton's second law.* The net force on the parcel is the pressure difference across its two faces, so `rho_0 du/dt = -dp/dx`. This is the linearized form of the Euler equation of an inviscid [[Fluid_dynamics|fluid]]. 3. *Equation of state.* The compressions in a sound wave are too fast for heat to flow between neighboring parcels, so the process is adiabatic and `p = c^2 rho'`, with `c^2 = (dp/d rho)` at constant entropy. For a fluid this is the bulk modulus over density, `c^2 = B/rho_0`; for an ideal gas it is `c^2 = gamma p_0/rho_0 = gamma R T/M`. Differentiate the first equation with respect to time and the second with respect to position, subtract to eliminate `u`, and substitute the third: `d^2p/dt^2 = c^2 d^2p/dx^2`. The same steps with `rho'` or `u` as the survivor give the same equation for density and velocity. [[Isaac_Newton|Isaac Newton]]'s 1687 estimate of the speed of sound used an isothermal equation of state in step 3 and came out about 15 percent low; Laplace's adiabatic correction multiplies it by `sqrt(gamma) = sqrt(1.4) ≈ 1.18`.[^laplace1816] The assumptions are worth stating because each one names an extension of the equation. Dropping viscosity and heat conduction removes absorption; restoring them gives the attenuation of [[Stokes's_law_of_sound_attenuation|Stokes's law]]. Assuming a fluid at rest removes wind and currents; restoring them gives the convected wave equation. Assuming a uniform medium removes refraction; letting `c` vary with position gives the ray bending of the atmosphere and the ocean. A tube whose area `S(x)` changes slowly obeys Webster's horn equation, `(1/S) d/dx (S dp/dx) = (1/c^2) d^2p/dt^2`, which A. G. Webster published in 1919 for horns and the phonograph.[^webster1919] ### Solution The general solution was found by Jean le Rond d'Alembert in 1747 for the vibrating string, and it carries over unchanged to sound in a tube:[^dalembert1747] `p(x, t) = f(x - c t) + g(x + c t)` Any shape `f` travels to the right at speed `c` without change, any shape `g` travels to the left, and the two pass through each other and add, the [[Superposition_principle|superposition principle]]. A pressure pulse released from rest splits into two half-height copies moving in opposite directions, which is exactly what the microsim shows as a ring in two dimensions. For a single tone, `p = A cos(k x - omega t)` with `omega = c k`, or `f = c/lambda` in the form OpenStax gives as Eq. 17.3.[^openstax17-2] The ratio of pressure to particle velocity in a right-traveling plane wave is `p/u = rho_0 c`, the characteristic [[Acoustic_impedance|acoustic impedance]] of the medium: about 413 Pa·s/m for air at 20 °C and about 1.48×10⁶ Pa·s/m for [[Water|water]], a mismatch of more than three thousand that explains why so little airborne sound enters a lake. Boundaries decide how waves [[Reflection_(physics)|reflect]] and which solutions are allowed. At a rigid wall the fluid cannot move, so `u = 0` and, from the momentum equation, `dp/dx = 0`: the reflected pulse returns with the same sign. At a pressure-release boundary such as the open end of a pipe, `p ≈ 0` and the pulse returns inverted. A tube closed at both ends therefore supports [[Standing_wave|standing waves]] at `f_n = n c/(2L)`, the same series a string fixed at both ends obeys.[^gea12] Factoring the operator gives a useful special case. Because `d^2/dt^2 - c^2 d^2/dx^2 = (d/dt - c d/dx)(d/dt + c d/dx)`, the first-order equation `p_t + c p_x = 0` admits only waves moving toward `+x`. This [[One-way_wave_equation|one-way wave equation]] is the basis of the absorbing boundaries used in numerical acoustics: imposed at the edge of a computational grid, it lets an outgoing wave leave without reflection.[^engquist1977][^mur1981] ## In three dimensions ### Equation In three dimensions the continuity and momentum equations become `d rho'/dt + rho_0 div(u) = 0` and `rho_0 du/dt = -grad(p)`, and the same elimination gives `laplacian(p) - (1/c^2) d^2p/dt^2 = 0` where the Laplacian `laplacian(p) = p_xx + p_yy + p_zz` in Cartesian coordinates. Because the fluid is inviscid and starts at rest, the velocity field has no vorticity and can be written as the gradient of a velocity potential `phi`, with `u = grad(phi)` and `p = -rho_0 d phi/dt`; the potential obeys the same wave equation, and the pressure and velocity follow from it by differentiation.[^pierce2019] Two variants carry most applied work. For a single frequency, write `p = Re[P(x) e^(-i omega t)]`; the wave equation becomes the Helmholtz equation `laplacian(P) + k^2 P = 0` with `k = omega/c`, the starting point for room modes, [[Acoustic_resonance|acoustic resonance]] and scattering. With sources present, a term on the right-hand side represents them: a small pulsating sphere (a monopole) injects volume, a vibrating piston or loudspeaker cone acts through a boundary condition on the normal velocity, and turbulence in a jet acts as a distributed quadrupole source.[^kinsler2000] In spherical coordinates with no angular dependence, the equation reduces to `d^2(r p)/dr^2 = (1/c^2) d^2(r p)/dt^2`, so the product `r p` obeys the one-dimensional equation. That observation drives the whole of the next section. ### Solution **Plane waves.** The simplest solutions are plane waves `p = A cos(k·x - omega t)` with wave vector `k` of magnitude `omega/c` pointing along the direction of travel. Any sound field in a uniform medium can be assembled from them by [[Fourier_analysis|Fourier]] superposition, which is how array processing and acoustic holography treat it. **Spherical waves.** Because `r p` obeys the one-dimensional equation, an outgoing spherical wave has the form `p = f(t - r/c)/r`. The pressure amplitude falls as `1/r` and the intensity as `1/r^2`, which is the [[Inverse-square_law|inverse-square law]] for sound: doubling the distance from a small source in open air lowers the level by 20 log₁₀ 2 ≈ 6 dB. A pulse emitted in three dimensions stays a pulse; once the shell has passed, the pressure returns exactly to zero. In two dimensions this sharp form of Huygens's principle fails, and the pressure behind the ring decays only gradually, leaving a wake. The microsim is two-dimensional, and its spec says so; the wake visible behind its rings is a real property of the 2-D equation, not a numerical artifact. **Rooms and boxes.** In a rigid rectangular room of sides `L_x`, `L_y`, `L_z`, the Helmholtz equation with `dp/dn = 0` on every wall has standing-wave solutions with frequencies `f = (c/2) sqrt((l/L_x)^2 + (m/L_y)^2 + (n/L_z)^2)` for non-negative integers `l, m, n`. A 5 m × 4 m × 3 m room has its lowest mode (1, 0, 0) at 343/(2 × 5) ≈ 34 Hz, and the modes crowd together rapidly above a few hundred hertz. These are the room modes of [[Room_acoustics|room acoustics]], and their decay sets [[Reverberation|reverberation]]. **Numerical solution.** For real geometries the equation is solved on a grid. The finite-difference time-domain (FDTD) method, introduced by Kane Yee in 1966 for Maxwell's equations, updates pressure and velocity on staggered grids in alternating half-steps.[^yee1966] It is stable only if the time step obeys the Courant–Friedrichs–Lewy condition, `c Δt/Δx ≤ 1/sqrt(2)` in two dimensions for the standard scheme.[^cfl1928] The microsim's bakes were computed this way offline: a second-order 2-D scheme at Courant number 0.45 on a 240 × 144 grid, with `c = 1` in a box of 10 × 6 units, rigid (`dp/dn = 0`) walls and, in the open case, a first-order Mur absorbing boundary on the right-hand side.[^mur1981] The four rooms in the sim each isolate one property of the solution. In the closed box, the ring reaches each rigid wall and returns with the same sign, and the microphone trace fills with echoes that never quite die, since the model has no absorption. With the right side open, the one-way condition lets the ring leave and the room quietens as the remaining walls hand their reflections to the open side. With a slit 1.2 units wide, comparable to the pulse, the gap becomes a new point source and a fresh circular wave fans out behind the wall by [[Diffraction|diffraction]]. With two pulses, linearity means the rings simply add; along the midline the field is mirror-symmetric, exactly as if a rigid wall stood there, which is the method of images in action. Scaled to a 10 m room of air, the readout converts the dimensionless time to milliseconds: the ring crosses the room in about 29 ms. *Try: set **room** to closed box and watch the microphone trace fill with same-sign echoes; switch to open right side and see them stop; choose slit wall to watch the gap radiate a new circular wave; pick two pulses to see the rings add where they cross; slow the **playback speed** to follow a single reflection, and narrow the **colour range +/- p** slider to see the faint 2-D wake behind each ring.* ## See also - [[Wave_equation]] - [[One-way_wave_equation]] - [[Wave]] - [[Sound]] - [[Speed_of_sound]] - [[Acoustic_resonance]] ## References [^openstax17-2]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax. §17.2 "Speed of Sound," pp. 810–817, Eqs. 17.3–17.7 and the derivation of the speed of sound in air from the continuity equation and the force on a fluid parcel. https://openstax.org/details/books/university-physics-volume-1 (Portal Books 077). [^openstax16-2]: Ling, Sanny & Moebs (2016). *University Physics Volume 1*. OpenStax. §16.2 "Mathematics of Waves," Eq. 16.6 (the linear wave equation), p. 764. (Portal Books 077.) [^gea12]: Gea-Banacloche, Julio (2019). *University Physics I: Classical Mechanics*. University of Arkansas Open Educational Resources. Ch. 12 "Waves in one dimension," §12.1.3 "The wave velocity" (c = sqrt(B/ρ₀), Eq. 12.12) and the standing-wave frequencies f_n = nc/2L (Eq. 12.18). https://scholarworks.uark.edu/oer/3 (Portal Books 076.) [^dalembert1747]: d'Alembert, Jean le Rond (1747). "Recherches sur la courbe que forme une corde tenduë mise en vibration." *Histoire de l'Académie Royale des Sciences et Belles Lettres de Berlin* 3: 214–219 (volume for 1747, printed 1749). Volume and page as commonly cited; not checked against a scan for this article. [^laplace1816]: Laplace, Pierre-Simon (1816). "Sur la vitesse du son dans l'air et dans l'eau." *Annales de chimie et de physique* 3: 238–241. Volume and page as commonly cited; not checked against a scan for this article. Newton's isothermal estimate is in *Philosophiæ Naturalis Principia Mathematica* (1687), Book II, Section VIII. [^webster1919]: Webster, Arthur Gordon (1919). "Acoustical impedance, and the theory of horns and of the phonograph." *Proceedings of the National Academy of Sciences* 5 (7): 275–282. https://doi.org/10.1073/pnas.5.7.275 [^engquist1977]: Engquist, Björn; Majda, Andrew (1977). "Absorbing boundary conditions for the numerical simulation of waves." *Mathematics of Computation* 31 (139): 629–651. https://doi.org/10.1090/S0025-5718-1977-0436612-4 [^mur1981]: Mur, Gerrit (1981). "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations." *IEEE Transactions on Electromagnetic Compatibility* EMC-23 (4): 377–382. https://doi.org/10.1109/TEMC.1981.303970 [^pierce2019]: Pierce, Allan D. (2019). *Acoustics: An Introduction to Its Physical Principles and Applications* (3rd ed.). Springer / ASA Press. Ch. 1 (the linear acoustic equations and the velocity potential). https://doi.org/10.1007/978-3-030-11214-1 [^kinsler2000]: Kinsler, Lawrence E.; Frey, Austin R.; Coppens, Alan B.; Sanders, James V. (2000). *Fundamentals of Acoustics* (4th ed.). Wiley. ISBN 9780471847892. Ch. 5 "The acoustic wave equation and simple solutions" and Ch. 7 "Radiation and reception of acoustic waves." [^yee1966]: Yee, Kane S. (1966). "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media." *IEEE Transactions on Antennas and Propagation* 14 (3): 302–307. https://doi.org/10.1109/TAP.1966.1138693 [^cfl1928]: Courant, Richard; Friedrichs, Kurt; Lewy, Hans (1928). "Über die partiellen Differenzengleichungen der mathematischen Physik." *Mathematische Annalen* 100: 32–74. https://doi.org/10.1007/BF01448839  **Microsim — three.js (Wikitube framework):** *Acoustic wave equation* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Acoustic_wave_equation.html" data-title="Acoustic wave equation"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Acoustic_wave_equation.json`; part of the [[PORTAL_Acoustics|Acoustics portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Acoustic_wave_equation) : [Wikitube](https://en.wikitube.io/wiki/Acoustic_wave_equation) - skeleton pinned to revision 1363019273 (2026-09-11).