Acoustic_attenuation - Wikiparency

# Acoustic attenuation **Acoustic attenuation** is the loss of energy from a [[Sound|sound]] wave as it travels through a medium, over and above the thinning that comes from simply spreading out. Part of the loss is [[Absorption_(acoustics)|absorption]], in which the wave's energy is turned into heat by [[Viscosity|viscosity]], heat conduction and molecular relaxation; part is scattering and reflection, in which energy is redirected out of the beam. Over a path of length x the amplitude falls as `A(x) = A0·e^(−αx)`, so the level in [[Decibel|decibels]] falls on a straight line, and the attenuation coefficient α is quoted in nepers or decibels per unit length. In almost every real medium α grows with [[Frequency|frequency]], and over wide frequency bands it follows a power law, `α(f) = α0·f^η`, with the exponent η between 0 and 2. A viscous liquid such as [[Water|water]] has η = 2, the result of [[Stokes's_law_of_sound_attenuation|Stokes's law of sound attenuation]]; soft tissue has η close to 1. That single fact sets the working limits of [[Medical_ultrasound|medical ultrasound]], [[Sonar|sonar]] and outdoor noise: high frequencies see finely but do not reach far. The article's microsim is *Acoustic attenuation: pulses fading with depth*, in which the reader sends ultrasound pulses into water, blood, fat, soft tissue, liver, muscle, bone or lung and watches the envelope shrink exponentially while the level falls in a straight line on a decibel chart. On the Minnesota side, Mayo Clinic in Rochester has run a study scoring an attenuation measurement against liver biopsy as a way to grade fat in the liver. ## Power-law frequency-dependent acoustic attenuation Measured over one or two decades of frequency, the attenuation of most media fits `α(ω) = α0·ω^η` closely enough that the two numbers α0 and η are the standard way to describe a material. Thomas Szabo showed in 1994 that such a law, with η anywhere from 0 to 2, can be written into a time-domain [[Wave_equation|wave equation]] through a convolution term, and that the law implies a matching frequency dependence of the [[Speed_of_sound|speed of sound]].[^szabo1994] The exponent is a fingerprint of the loss mechanism: η = 2 for classical viscous and thermal losses in a simple fluid, η near 1 for biological tissue and many polymers, and intermediate values for porous and granular media. The subsections below follow the law from the equation the microsim computes, through the mechanisms that produce it, to the trade-offs it forces. ### Exponential decay and the decibel per centimeter Because each centimeter removes the same fraction of what arrives, attenuation compounds. An amplitude coefficient α in nepers per meter converts to decibels by the factor 8.686, and the decibel form is what instruments use: the level after a distance x is `L = L0 − α·x`, with α in dB/cm. The decibel itself is the textbook ratio `β = 10·log10(I/I0)`.[^up17-3] The intensity form of the same law, `I(x) = I0·e^(−μx)`, is called Beer's law in the ultrasound literature, with μ the intensity attenuation coefficient.[^us-185] For soft tissue a coefficient of about 0.5 dB per centimeter per megahertz is the working value. At the microsim's default of 3.5 MHz that is 1.75 dB/cm, so a pulse loses 14 dB on its way to a depth of 8 cm and another 14 dB on the way back. A scanner that can recover echoes about 100 dB below the transmitted pulse can therefore image to a depth of `100/(2α)` ≈ 29 cm at 3.5 MHz, 1 m at 1 MHz, and only 10 cm at 10 MHz. The same budget in bone, attenuating roughly twenty times more strongly per megahertz, runs out within a centimeter or two. *Try: set frequency (0.5–15 MHz, log scale), choose a medium and move depth d — the pulses in the slab shrink under their exponential envelope, the orange and blue lines on the dB chart give the one-way and round-trip level, and the readout reports α in dB/cm, the loss to depth d, and the maximum depth before the echo falls 100 dB below the pulse; switch to bone or lung to watch the beam die within millimeters.* ### Classical absorption: viscosity and heat conduction In a simple fluid a sound wave squeezes and stretches the medium, and neighboring layers move at slightly different speeds. Shear [[Viscosity|viscosity]] resists that motion and turns wave energy into heat. George Gabriel Stokes worked out the result in 1845: the amplitude coefficient is `α = 2μω²/(3ρc³)`, growing as the square of the angular frequency ω.[^stokes1845] Heat conduction between compressed (warm) and rarefied (cool) regions adds a second term of the same ω² form, found by Gustav Kirchhoff. For water at 20 °C, with μ = 1.0 mPa·s, ρ = 998 kg/m³ and c = 1,482 m/s, Stokes's formula gives about 0.0007 dB/cm at 1 MHz and 0.07 dB/cm at 10 MHz. Measured water attenuates about 0.0022 dB/cm at 1 MHz, also rising as f², roughly three times the Stokes value at every frequency.[^szabo-water] The excess comes from bulk (volume) viscosity, a resistance to pure compression that Stokes's derivation set to zero. The slope of 2 survives; only the prefactor changes. The See-also variant for [[Stokes's_law_of_sound_attenuation|Stokes's law]] draws both lines on the microsim's log-log chart. ### Molecular relaxation in air and seawater Many media carry a slower, stronger loss: molecular relaxation. When a compression arrives, energy is shared into an internal degree of freedom (a molecular vibration in [[Oxygen|oxygen]] or [[Nitrogen|nitrogen]], a chemical equilibrium between dissolved ions) that takes a finite time to respond. At frequencies near the inverse of that time the exchange falls out of step with the wave and drains it. Each relaxation contributes an f² rise below its relaxation frequency and a plateau above it, so a medium with several relaxations shows a local exponent that changes from band to band. In air the relaxation frequencies depend strongly on humidity, which is why [[Atmosphere_of_Earth|atmospheric]] absorption is specified by an international standard rather than a single number. The ISO 9613-1 formulae give, at 20 °C and 50% relative humidity, about 0.4 dB/km at 125 Hz, 4.7 dB/km at 1 kHz, 30 dB/km at 4 kHz and 105 dB/km at 8 kHz.[^iso9613] Over a kilometer, then, thunder and distant traffic lose their treble but keep their rumble. Seawater adds two chemical relaxations, of boric acid near 1 kHz and magnesium sulfate near 100 kHz. The simplified formula of Michael Ainslie and James McColm gives, for water at 10 °C, salinity 35 and pH 8 near the surface, about 0.06 dB/km at 1 kHz, 1 dB/km at 10 kHz and 34 dB/km at 100 kHz.[^ainslie1998] Low-frequency sound can cross ocean basins; the high-frequency sonar that resolves small targets reaches only a few kilometers, the central trade-off of [[Underwater_acoustics|underwater acoustics]]. ### Soft tissue, bone and the near-linear law Biological tissue is a heterogeneous, hierarchical material with a broad spread of relaxation processes, and the sum of many overlapping relaxations produces an attenuation that rises almost linearly with frequency over the diagnostic band of 1–15 MHz. The ultrasound text used on the portal puts it simply: attenuation "generally increases linearly with increasing frequency among different body tissues," fluid-filled structures attenuate much less than solid ones, and absorption into heat accounts for about 60–80% of the total, the rest being scattering and reflection.[^us-185] | Medium | α0 at 1 MHz (dB/cm) | η | Note | |---|---|---|---| | Water | 0.0022 | 2 | nearly lossless; used as an acoustic window | | Blood | about 0.18–0.20 | about 1.2 | the ultrasound text gives 0.20 dB/MHz·cm | | Soft tissue (average) | 0.5 | 1 | the working value for scanner design | | Liver | 0.5–1.1 | about 1 | the range covering 95% of patients in clinical liver measurements | | Bone | 10–20 | about 1 | the microsim uses 10; the ultrasound text gives about 20 | | Lung (air-filled) | about 40 | about 1 | gas pockets scatter and absorb | The table combines the portal's microsim values with the sources cited in its notes;[^us-185][^qiba2022] published values for any one tissue vary with species, temperature and measurement method, and the rows are ILLUSTRATIVE rather than reference data. ### Causality, dispersion and wave equations An attenuation that depends on frequency cannot leave the wave speed alone. Causality links the two through the Kramers–Kronig relations: a medium that absorbs more at high frequency must also carry high frequencies at a slightly different speed, a form of [[Acoustic_dispersion|acoustic dispersion]]. For a power law the relation takes a compact form, and the dispersion depends on η. For η = 2 the first-order dispersion vanishes, for η = 1 the phase speed rises logarithmically with frequency, and for other exponents it rises as a power of frequency.[^szabo1994] In soft tissue the effect is small: with α0 = 0.5 dB/cm/MHz and η = 1, the relation gives a rise of about 3 m/s in phase speed between 1 and 10 MHz, some 0.2%, but it matters in precise speed-of-sound imaging. Simulating such media is harder than simulating a lossless one, because the ordinary wave equation has no memory. Szabo's convolution form gave one route.[^szabo1994] Bradley Treeby and Ben Cox showed in 2010 that the same power-law absorption and its matching dispersion can be produced by two terms built on the fractional Laplacian, a spatial operator that is cheap to evaluate with Fourier methods; the approach is the absorption model of the k-Wave simulation toolbox.[^treeby2010][^kwave] The [[Acoustic_wave_equation|acoustic wave equation]] article shows the lossless starting point. ### The depth–resolution trade in ultrasound imaging The engineering consequence of the near-linear tissue law is that every scanner design is a compromise between depth and detail. Axial resolution improves as the wavelength shrinks, which means raising the frequency; attenuation in decibels per centimeter rises in proportion. Abdominal, cardiac and brain scanning therefore uses 2–5 MHz, while the eyes and peripheral vessels are scanned at 5–15 MHz; the higher band sees only a few centimeters, but sees them sharply.[^us-freq] The receiver makes up for the round-trip loss with gain that the operator sets by depth, so that deep echoes are displayed as brightly as shallow ones.[^us-gain] Attenuation also writes itself into the image. Behind a strongly attenuating or reflecting structure such as a gallstone or bone, the beam is too weak to return echoes, and an [[Acoustic_shadow|acoustic shadow]] appears; behind a fluid-filled structure such as the gallbladder, the bladder or a cyst, the beam arrives stronger than the gain curve expects, and the tissue below looks too bright, an enhancement artifact.[^us-shadow] Radiologists read both as clues to what the structure is. Measured deliberately, attenuation becomes a biomarker. Fat droplets in liver cells raise the attenuation coefficient, so an estimate of α from the echo signal can grade hepatic steatosis without a needle. A 2022 review for the joint AIUM–RSNA Quantitative Imaging Biomarkers Alliance reported that liver attenuation coefficients in 95% of patients fall within 0.5–1.1 dB/cm/MHz and that thresholds for grading steatosis were not yet agreed.[^qiba2022] The earliest widely deployed version, the controlled attenuation parameter (CAP), measures attenuation at 3.5 MHz with the probe of a transient-elastography device and reports it in dB/m.[^sasso2010][^lee2022] ## Minnesota *This section is specific to Wikitube.* Mayo Clinic in Rochester ran a clinical study to evaluate "the diagnosis accuracy of the Controlled Attenuation Parameter (CAP) measured by FibroScan (both with M and XL probes) in all patients who are undergoing liver biopsy for any liver disease," with Kymberly Watt as principal investigator; the study is listed as closed for enrollment.[^mayo-cap] The design is the standard one for a new biomarker: every patient already scheduled for a biopsy also gets the attenuation measurement, so the histology serves as the reference against which the acoustic number is scored. Offering two probes, M and XL, is itself the depth budget of the previous section at the bedside: the thickness of the body wall above the liver is attenuation the beam must pay before any measurement begins. ## See also - [[Stokes's_law_of_sound_attenuation]] - [[Absorption_(acoustics)]] - [[Sound_pressure]] (section 10) - [[Acoustic_impedance]] (section 15) - [[Ultrasound]] (section 25) - [[Underwater_acoustics]] (section 26) ## References [^szabo1994]: Szabo, Thomas L. (1994). "Time domain wave equations for lossy media obeying a frequency power law." *Journal of the Acoustical Society of America* 96 (1): 491–500. https://ui.adsabs.harvard.edu/abs/1994ASAJ...96..491S/abstract [^up17-3]: OpenStax (2016). *University Physics Volume 1*, §17.3 "Sound Intensity," Eq. 17.12 (sound intensity level in decibels), p. 822. Portal Book 077. https://openstax.org/details/books/university-physics-volume-1 [^us-185]: Thapaliya, Arbin; Sithole, Alec; Welsh, Michael; Dana, Gaston (2024). *Ultrasound Physics and its Application in Medicine*. PALNI Open Press. §1.8.5 "Attenuation," pp. 16–17 (Beer's law; absorption 60–80% of attenuation, citing ter Haar 2011; blood about 0.20 and bone about 20.0 dB/MHz·cm; attenuation roughly linear in frequency). Portal Book 091. CC BY 4.0. [^stokes1845]: Stokes, G. G. (1845). "On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids." *Transactions of the Cambridge Philosophical Society* 8: 287–319. Page range not re-checked for this article. [^szabo-water]: The 0.0022 dB/cm·MHz² value for water is the portal acoustic pack's coefficient (`acoustic.attenuation.tissues`), matching the value of about 0.00217 dB/(MHz²·cm) tabulated in Szabo, Thomas L. (2004), *Diagnostic Ultrasound Imaging: Inside Out*, Elsevier Academic Press, appendix of acoustic properties (table number not re-checked). The Stokes values are computed from the formula above. [^iso9613]: International Organization for Standardization (1993). ISO 9613-1:1993, *Acoustics — Attenuation of sound during propagation outdoors — Part 1: Calculation of the absorption of sound by the atmosphere*. Values here are computed from the standard's formulae at 101.325 kPa, 20 °C and 50% relative humidity. [^ainslie1998]: Ainslie, Michael A.; McColm, James G. (1998). "A simplified formula for viscous and chemical absorption in sea water." *Journal of the Acoustical Society of America* 103 (3): 1671–1672. https://ui.adsabs.harvard.edu/abs/1998ASAJ..103.1671A/abstract (values computed from the formula for 10 °C, S = 35, pH 8, depth 0). [^treeby2010]: Treeby, Bradley E.; Cox, B. T. (2010). "Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian." *Journal of the Acoustical Society of America* 127 (5): 2741–2748. https://doi.org/10.1121/1.3377056 [^us-freq]: Thapaliya et al. (2024). *Ultrasound Physics and its Application in Medicine*, §2.11.1 "A-Mode Display," p. 48 (2–5 MHz for abdominal, cardiac and brain scanning; 5–15 MHz for eyes and peripheral vessels). Portal Book 091. [^us-gain]: Thapaliya et al. (2024). *Ultrasound Physics and its Application in Medicine*, §2.3 "Ultrasound Machine Components," p. 23 (amplifier gain adjusted to the required depth). Portal Book 091. [^kwave]: k-Wave. "Publications." http://www.k-wave.org/publications.php (lists Treeby and Cox 2010 among the papers describing the toolbox's models). [^us-shadow]: Thapaliya et al. (2024). *Ultrasound Physics and its Application in Medicine*, §2.9.1 "Shadowing Artifact" and §2.9.14 "Enhancement Artifact." Portal Book 091. [^qiba2022]: Ferraioli, Giovanna; Kumar, Viksit; Ozturk, Arinc; Nam, Kibo; de Korte, Chris L.; Barr, Richard G. (2022). "US Attenuation for Liver Fat Quantification: An AIUM-RSNA QIBA Pulse-Echo Quantitative Ultrasound Initiative." *Radiology* 302 (3). https://doi.org/10.1148/radiol.210736 [^sasso2010]: Sasso, M.; Beaugrand, M.; de Ledinghen, V.; Douvin, C.; Marcellin, P.; Poupon, R.; et al. (2010). "Controlled attenuation parameter (CAP): a novel VCTE guided ultrasonic attenuation measurement for the evaluation of hepatic steatosis: preliminary study and validation in a cohort of patients with chronic liver disease from various causes." *Ultrasound in Medicine and Biology* 36: 1825–1835. https://www.sciencedirect.com/science/article/pii/S0301562910003546 [^lee2022]: Lee, S.; Kim, K. W.; Kim, S. Y.; Seo, N.; Song, G. W.; Lee, S. G. (2022). "Controlled attenuation parameter measured using transient elastography for the noninvasive assessment of macrovesicular steatosis in potential living liver donors." *Ultrasonography* 41 (1): 164–170 ("The CAP was calculated as the attenuation of the ultrasonic signal at 3.5 MHz"). https://www.e-ultrasonography.org/upload/usg-21071.pdf [^mayo-cap]: Mayo Clinic. "A Study to Evaluate Diagnostic Accuracy of the Controlled Attenuation Parameter (CAP) Measured by FibroScan in Patients Scheduled for a Liver Biopsy." Clinical trial CLS-20307017 (IRB 16-009735), Rochester, Minn. https://www.mayo.edu/research/clinical-trials/cls-20307017  **Microsim — three.js (Wikitube framework):** *Acoustic attenuation* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Acoustic_attenuation.html" data-title="Acoustic attenuation"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Acoustic_attenuation.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_attenuation) : [Wikitube](https://en.wikitube.io/wiki/Acoustic_attenuation) - skeleton pinned to revision 1370001032 (2026-09-11).