Acoustic_impedance - Wikiparency

# Acoustic impedance **Acoustic impedance** is the opposition that a medium or an acoustic system presents to the flow of sound: the ratio of the [[Sound_pressure|sound pressure]] applied to it to the flow of the medium that the pressure produces. A high-impedance medium needs a large pressure to set it moving; a low-impedance one moves easily under a small pressure. The idea was borrowed from electrical engineering, where impedance is the ratio of [[Voltage|voltage]] to [[Electric_current|current]], and it carries the same bookkeeping: an acoustic impedance has a resistive part that absorbs energy and a reactive part that stores and returns it. Two closely related quantities share the name. The *acoustic impedance* proper divides pressure by volume flow, and describes ducts, horns and resonators; the *specific acoustic impedance* divides pressure by the [[Particle_velocity|particle velocity]], and describes a medium or a surface. For a [[Plane_wave|plane wave]] in an open medium the specific impedance reduces to the product of [[Density|density]] and [[Speed_of_sound|sound speed]], ρc, the medium's *characteristic impedance*. That single number decides what happens at a boundary: when a sound wave meets a medium of different ρc, part of it is sent back. Air and water differ by a factor of about 3,600, so sound in air barely enters a lake, and ultrasound scanners need a coupling gel between probe and skin. The Wikitube microsim for this article sends a wave at an interface: the reader picks the two media, tilts the beam and adds a quarter-wave matching layer, and watches the reflected and transmitted waves and the split of energy between them. ## Mathematical definitions ### Acoustic impedance For an acoustic system with a single port, such as the mouth of a tube or the throat of a horn, the acoustic impedance is the ratio of the sound pressure p at the port to the volume flow rate Q through it. In the general time-dependent case the ratio is taken between Laplace transforms, `Z(s) = p(s) / Q(s)`, which makes Z a [[Transfer_function|transfer function]] in the sense of [[Linear_time-invariant_system|linear systems]]; in the steady state at a single angular frequency ω it becomes a complex number, `Z(ω) = R(ω) + iX(ω)`.[^kinsler] The real part R, the acoustic resistance, is the part in phase with the flow and accounts for energy that leaves the port for good, radiated away or turned to heat. The imaginary part X, the acoustic reactance, is a quarter-cycle out of phase: positive reactance behaves like a mass of air that must be accelerated, negative reactance like a volume of air that is compressed like a spring. The [[Laplace_transform|Laplace transform]] form matters because real acoustic systems are frequency-dependent. A tube a quarter of a wavelength long has very different impedance from one half a wavelength long, and the impedance of a room, a loudspeaker enclosure or a musical instrument's bore is a curve across the [[Frequency_domain|frequency domain]], not one value. Arthur Gordon Webster set out the concept in these terms in 1919, in a paper on horns and the phonograph that defined acoustical impedance as the complex ratio of pressure to volume flow and used it to calculate how a horn loads a vibrating diaphragm.[^webster] The electrical analogy he drew on is exact for small signals: pressure maps to voltage, volume flow to current, an enclosed volume of air to a capacitor and a short plug of air in a narrow neck to an inductor. A [[Helmholtz_resonance|Helmholtz resonator]] is, in that picture, a series LC circuit, and the [[Lumped-element_model|lumped-element model]] of acoustics is built on it. ### Specific acoustic impedance The specific acoustic impedance z is the ratio of sound pressure to particle velocity v at a point, `z = p / v`. Where the acoustic impedance describes a whole port, the specific impedance describes a material or a surface, independent of its area.[^kinsler] It is the quantity that enters the boundary conditions at an interface, and the quantity that sets the [[Sound_intensity|intensity]] carried by a wave of given pressure. OpenStax derives the intensity of a sound wave as I = Δp²/(2ρv), with Δp the pressure amplitude, ρ the density and v the wave speed; the denominator is twice the characteristic impedance.[^ost-173] A pressure amplitude of 1 Pa therefore carries about 1.2 mW/m² in air (ρc ≈ 413 rayl) but only about 0.34 µW/m² in water (ρc ≈ 1.48 × 10⁶ rayl). The same pressure moves the water about 3,600 times less, and carries about 3,600 times less power. ### Acoustic ohm The SI unit of acoustic impedance is the pascal-second per cubic meter, Pa·s/m³, sometimes called the acoustic ohm. The unit of specific acoustic impedance is the pascal-second per meter, Pa·s/m, called the rayl after Lord Rayleigh, whose *Theory of Sound* laid much of the foundation of the subject.[^kinsler][^rayleigh] An older CGS rayl, 1 dyn·s/cm³, equals 10 SI rayl, so tables need care. Solids and liquids have characteristic impedances in the millions of rayl, and ultrasound texts quote them in megarayl (MRayl). Soft tissue is about 1.6 MRayl, liver and blood about 1.65 MRayl and bone about 5 MRayl, against about 0.0004 MRayl for air.[^us-187] ### Relationship For a plane wave traveling along a duct of cross-sectional area A, the volume flow is the particle velocity times the area, Q = vA, so the two impedances are related by `Z = z / A`.[^kinsler] A narrow duct presents a higher acoustic impedance than a wide one filled with the same air. For a round tube 25 mm in diameter, A ≈ 4.9 × 10⁻⁴ m², and the acoustic impedance of the air column is 413 / 4.9 × 10⁻⁴ ≈ 8.4 × 10⁵ Pa·s/m³. A third quantity, the mechanical impedance, divides force by velocity and equals zA; it is the one used for [[Vibration|vibrating]] plates, pistons and loudspeaker cones. Because all three are ratios of an effort to a flow, converting between them is a matter of multiplying or dividing by the area, and the choice is set by what the problem measures. ## Characteristic acoustic impedance ### Characteristic specific acoustic impedance In an unbounded fluid carrying a plane wave, pressure and particle velocity rise and fall together and their ratio is a real constant of the medium: `z₀ = ρ₀c₀`, the density times the speed of sound.[^kinsler] It depends only on the material, which is why it is tabulated alongside density and sound speed, and it is the number that governs reflection and transmission at a boundary between two media. For a plane wave arriving at normal incidence from medium 1 onto medium 2, the ratio of reflected to incident pressure is `R = (Z₂ − Z₁) / (Z₂ + Z₁)`, and the fractions of intensity reflected and transmitted are R² and 1 − R².[^us-181] The reflection depends on the *mismatch* of the impedances, not on which medium is denser or faster: a wave going from a medium of 1 MRayl into one of 4 MRayl reflects the same fraction of its energy as a wave going the other way. What changes with direction is the sign. Going from low to high impedance, R is positive and the reflected pressure is upright; going from high to low, R is negative and the reflection is inverted, the acoustic counterpart of OpenStax's string tied to a fixed wall or to a free ring.[^ost-165] The microsim computes these numbers from the media table of the portal's acoustics library, and the table below repeats the cases it offers. | Interface (medium 1 → medium 2) | Z₁ (MRayl) | Z₂ (MRayl) | R (pressure) | Energy reflected | Energy transmitted | |---|---|---|---|---|---| | Air → water | 0.000413 | 1.48 | +0.9994 | 99.89% | 0.11% (−29.5 dB) | | Soft tissue → air | 1.63 | 0.000413 | −0.9995 | 99.90% | 0.10% | | Soft tissue → bone | 1.63 | 6.65 | +0.61 | 36.8% | 63.2% | | Fat → muscle | 1.34 | 1.70 | +0.12 | 1.4% | 98.6% | | PZT transducer → soft tissue | 30 | 1.63 | −0.90 | 80.4% | 19.6% | | Water → steel | 1.48 | 46.8 | +0.94 | 88.1% | 11.9% | The table explains [[Medical_ultrasound|medical ultrasound]] in three rows. Between soft tissues the mismatches are small, a percent or so of the energy returns from each boundary, and those faint echoes are what the scanner images. At bone a third of the energy bounces back, and very little reaches what lies behind it. At any trapped air almost everything reflects, which is why the probe cannot image through the lung or a bowel gas bubble, and why "the air between the tissue and the transducer inhibits the propagation of the ultrasound beam, [so] a conducting gel is usually applied between them."[^us-25] At oblique incidence the transmitted wave bends according to [[Refraction|Snell's law]], sin θ₂ / sin θ₁ = c₂ / c₁, and the reflection coefficient becomes `R = (Z₂ cos θ₁ − Z₁ cos θ₂) / (Z₂ cos θ₁ + Z₁ cos θ₂)`.[^kinsler] At the sim's default of soft tissue to bone at 20°, the transmitted beam bends to about 51°, and the pressure reflection rises from 0.61 at normal incidence to about 0.72, so 51.6% of the energy returns. Past sin θ₁ = c₁ / c₂, about 26° for this pair, no transmitted wave exists and the reflection is total. The sim treats bone and steel as fluids, ignoring the shear waves a real solid carries, and says so on screen. A layer between the two media can cancel the reflection at one frequency. If the layer is a quarter of its own wavelength thick and has impedance √(Z₁Z₂), the wave reflected from its back face returns half a cycle behind the wave reflected from its front face, and the two cancel.[^kinsler] For a PZT element of 30 MRayl facing tissue of 1.63 MRayl, the ideal layer is about 7 MRayl. That is the job of the matching layers on the face of an [[Ultrasonic_transducer|ultrasonic transducer]], and the principle of the [[Acoustic_transmission_line|acoustic transmission line]] treated as an impedance transformer. *Try: pick medium 1 and medium 2 from the menus and compare the R and T bars; set soft tissue to air to see nearly everything bounce; slide the incidence angle past about 26° with soft tissue to bone; tick the quarter-wave matching layer with PZT to soft tissue and watch the reflected wave disappear.* ### Effect of temperature Because ρ₀ and c₀ both depend on temperature, so does the characteristic impedance of a gas. For an ideal gas at fixed pressure p₀, density falls as 1/T and sound speed rises as √T, so their product falls as 1/√T: `z₀ = p₀ √(γM / RT)`, with γ the ratio of specific heats, M the molar mass and R the gas constant.[^kinsler] The dependence follows from the [[Kinetic_theory_of_gases|kinetic theory of gases]], and OpenStax gives the matching rise in sound speed from 331 m/s at 0 °C to 343 m/s at 20 °C.[^ost-172] At sea-level pressure the characteristic impedance of dry air is about 428 rayl at 0 °C and 413 rayl at 20 °C. Across a Minnesota year the swing is larger: about 454 rayl on a −30 °C January morning and about 403 rayl on a 35 °C July afternoon, a 12% change. Altitude matters in the same proportion as pressure, so higher in the [[Atmosphere_of_Earth|atmosphere]] the impedance of air is lower still. For liquids the variation is smaller and less regular, because the sound speed of water rises with temperature while its density barely changes. ### Characteristic acoustic impedance Dividing the characteristic specific impedance by the cross-section of a duct gives the characteristic acoustic impedance of that duct, `Z₀ = ρ₀c₀ / A`.[^kinsler] It plays the role that the characteristic impedance plays on an electrical transmission line. A wave traveling down a uniform pipe sees Z₀ at every point; where the pipe changes cross-section, opens into a room or ends in a closed cap, the impedance changes and part of the wave reflects. An open pipe end is nearly a pressure-release boundary, so a pressure wave reflects inverted; a closed end is rigid, so it reflects upright. Those two reflections set the resonances of organ pipes and wind instruments, and of the [[Standing_wave|standing waves]] of [[Acoustic_resonance|acoustic resonance]]. A horn is a duct whose area grows gradually, so its impedance changes slowly and little of the wave reflects on the way out: it matches the small, high-impedance throat at a [[Loudspeaker|loudspeaker]] driver to the large, low-impedance mouth open to the room, which was the problem Webster's 1919 paper set out to solve.[^webster] ## Minnesota *This section is specific to Wikitube.* Ice fishing in [[Minnesota]] puts the air–water mismatch to work every winter. Vexilar, founded in Minneapolis in 1960 by John Uldrich and Robert Knutson and later based in Bloomington, introduced its FL-8 flasher in 1989 and a dedicated ice-fishing transducer, the Ice-Ducer, in 1997; the transducer hangs level in the water of the drilled hole, held vertical like a plumb bob.[^vexilar] The reason it goes into the water is the table above: from air into water only about 0.1% of the sound energy crosses the boundary, 30 dB lost before the first fish, and the same loss applies again to the returning echo. With the transducer immersed, the [[Sonar|sonar]] element and the water share nearly the same impedance and the pulse enters the lake almost whole. ## See also - [[Reflection_(physics)]] - [[Acoustic_transmission_line]] - [[Acoustic_attenuation]] · [[Soundproofing]] · [[Ultrasound]] · [[Underwater_acoustics]] — the neighboring sections of the Acoustics spine - [[Medical_ultrasound]] - [[PORTAL_Acoustics]] ## References [^kinsler]: Kinsler, Lawrence E.; Frey, Austin R.; Coppens, Alan B.; Sanders, James V. (2000). *Fundamentals of Acoustics* (4th ed.). Wiley. Chapters on plane waves (specific acoustic impedance, characteristic impedance ρ₀c₀ and its temperature dependence), transmission between media (normal and oblique incidence, quarter-wave layer) and pipes (acoustic impedance, Z = z/A). Section numbers not checked in this run. [^webster]: 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 [^rayleigh]: Rayleigh, John William Strutt, Baron (1877–1878). *The Theory of Sound*, 2 vols. London: Macmillan. [^ost-173]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax. §17.3 "Sound Intensity," pp. 818ff. (intensity of a sound wave in terms of pressure amplitude, density and speed). https://openstax.org/details/books/university-physics-volume-1 — on the [[PORTAL_Acoustics]] book shelf (077). [^ost-172]: Ling, Sanny and Moebs (2016), *University Physics Volume 1*, §17.2 "Speed of Sound," pp. 812–813, Table 17.1 and Eq. 17.7. [^ost-165]: Ling, Sanny and Moebs (2016), *University Physics Volume 1*, §16.5 "Interference of Waves," subsection "Reflection and Transmission," pp. 774–775, Figure 16.17. [^us-181]: *Ultrasound Physics and its Application in Medicine* (2024). Ch. 1 "Basic Principles of Ultrasound," §1.8.1 "Reflection," pp. 9–12 (reflection and transmission coefficients from Z₁ and Z₂; Snell's law in §1.8.2). — on the [[PORTAL_Acoustics]] book shelf (091). [^us-187]: *Ultrasound Physics and its Application in Medicine* (2024), §1.8.7 "Acoustic Impedance," p. 18. The text prints the values as "0.0004 rayls for air, 1.65 rayls for the liver and blood … around 5 rayls for the bone"; the unit is evidently megarayl (Z = ρc for air is about 413 rayl), and Wikitube quotes it so. [^us-25]: *Ultrasound Physics and its Application in Medicine* (2024), Ch. 2 "Ultrasound Instrumentation," §2.5 "Transducer Characteristics," p. 25 (quoted). [^vexilar]: Vexilar, Inc. "History." Company page. https://vexilar.com/pages/history (founding in 1960 by John Uldrich and Robert Knutson in Minneapolis, the FL-8 in 1989, the Ice-Ducer in 1997).  **Microsim — three.js (Wikitube framework):** *Acoustic impedance* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Acoustic_impedance.html" data-title="Acoustic impedance"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Acoustic_impedance.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_impedance) : [Wikitube](https://en.wikitube.io/wiki/Acoustic_impedance) - skeleton pinned to revision 1361691012 (2026-09-11).