Speed of sound

The speed of sound is defined as the distance traversed by a sound wave per unit of time as it propagates through an elastic medium. Fundamentally, it represents the velocity at which vibrations transmit. At a temperature of 20 °C (68 °F), the approximate speed of sound in air is 343 m/s (1,125 ft/s; 1,235 km/h; 767 mph; 667 kn), which translates to 1 km in 2.92 s or one mile in 4.69 s. This velocity is significantly influenced by both temperature and the specific medium of propagation.

The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. More simply, the speed of sound is how fast vibrations travel. At 20 °C (68 °F), the speed of sound in air is about 343 m/s (1,125 ft/s; 1,235 km/h; 767 mph; 667 kn), or 1 km in 2.92 s or one mile in 4.69 s. It depends strongly on temperature as well as the medium through which a sound wave is propagating.

Under conditions of 0 °C (32 °F) in dry air at sea level (14.7 psi), the speed of sound is approximately 331 m/s (1,086 ft/s; 1,192 km/h; 740 mph; 643 kn).

For an ideal gas, the speed of sound is solely determined by its temperature and chemical composition. In dry air, the speed exhibits a minor dependence on frequency and pressure, indicating a slight deviation from ideal gas behavior.

While the term speed of sound commonly denotes the velocity of sound waves in air, its actual value is substance-dependent. Generally, sound propagates slowest in gases, at an intermediate rate in liquids, and most rapidly in solids.

For instance, sound propagates at approximately 343 m/s in air, whereas in fresh water at 20 °C (68 °F), its velocity is 1481 m/s (exceeding the speed in air by over 4.3 times). In iron, the speed reaches 5120 m/s (nearly 15 times faster than in air). Furthermore, in exceptionally rigid materials like diamond, sound can achieve velocities of 12,000 m/s (39,000 ft/s), which is approximately 35 times its speed in air and represents close to the maximum attainable speed under typical conditions.

Conceptually, the speed of sound corresponds to the velocity of vibrations. In solid media, sound waves comprise both compression waves (analogous to those in gases and liquids) and shear waves, a distinct type of wave unique to solids. Shear waves typically propagate at velocities different from compression waves within solids, a phenomenon observed in seismology. The velocity of compression waves in solids is governed by the medium's compressibility, shear modulus, and density. Conversely, the velocity of shear waves is solely dependent on the solid material's shear modulus and density.

Within the field of fluid dynamics, the speed of sound in a fluid medium (either gas or liquid) serves as a comparative metric for the velocity of an object traversing that medium. The Mach number of an object is defined as the ratio of its speed to the speed of sound within the identical medium. Objects that exceed the speed of sound (Mach§23§) are characterized as traveling at supersonic velocities.

Earth

Within the Earth's atmosphere, the speed of sound exhibits significant variability, ranging from approximately 295 m/s (1,060 km/h; 660 mph) at elevated altitudes to about 355 m/s (1,280 km/h; 790 mph) under conditions of high temperature.

History

Archytas, a Pythagorean philosopher, posited that sounds of higher pitch propagate more rapidly. This perspective gained acceptance among several later philosophers, including members of the Academy and the Peripatos, and potentially Aristotle.

Sir Isaac Newton's 1687 work, Principia, presented a calculation for the speed of sound in air, yielding 979 feet per second (298 m/s). This value was approximately 15% lower than observed. The primary reason for this discrepancy was the omission of the then-unrecognized effect of rapid temperature fluctuations within a sound wave; in contemporary terminology, the compression and expansion of air during sound wave propagation constitute an adiabatic process, not an isothermal one. Newton subsequently introduced several ad hoc adjustments, such as the "crassitude of the solid particles of the air," to align his calculated value with experimental measurements. Both Lagrange and Euler attempted, unsuccessfully, to account for this disparity. The discrepancy was ultimately resolved by Pierre-Simon Laplace. In his publication, Traité de mécanique céleste, Laplace incorporated the findings from the 1819 Clément-Desormes experiment, which determined the heat capacity ratio of air to be 1.35. This integration led to a close concordance between theoretical predictions and experimental results for the speed of sound. The contemporary value of 1.40 was established several years later, achieving full agreement.

The 17th century witnessed multiple endeavors to precisely quantify the speed of sound. In 1630, Marin Mersenne reported two distinct values. His initial measurement, derived from timing the interval (using a seconds pendulum) between observing a gun's flash and hearing its sound over a predetermined distance, yielded 1,380 Parisian feet per second (448 m/s). Conversely, when he measured the delay between firing a gun and perceiving its echo from a known reflective surface, he obtained 970 Parisian feet per second. This discrepancy prompted speculation that echoed sound might propagate more slowly than direct sound. Subsequent researchers predominantly adopted Mersenne's first experimental approach.

Subsequent measurements included Pierre Gassendi's 1635 finding of 1,473 Parisian feet per second, and Robert Boyle's determination of 1,125 Parisian feet per second. By 1650, G. A. Borelli and V. Viviani, affiliated with the Accademia del Cimento, calculated the speed at 350 m/s. A more precise measurement was published in 1709 by the Reverend William Derham, Rector of Upminster, who reported 1,072 Parisian feet per second. (The Parisian foot measured 325 mm, exceeding the modern "international foot" officially established in 1959 at 304.8 mm. This difference implies that the speed of sound at 20 °C (68 °F) is equivalent to 1,055 Parisian feet per second).

Derham conducted his experiments from the tower of St. Laurence Church in Upminster, employing a telescope to observe the flash of a distant shotgun discharge. He then used a half-second pendulum to time the interval until the gunshot's sound was perceived. These measurements involved gunshots originating from various local landmarks, such as North Ockendon church. Distances were ascertained through triangulation, enabling the calculation of sound propagation speed. Derham meticulously repeated these measurements under diverse conditions to investigate the influence of wind, barometric pressure, temperature, and humidity on sound velocity. For instance, he observed that sound speed increased when the wind blew towards the observer and decreased when it blew away. He erroneously concluded that temperature had no effect, citing consistent speeds in both summer and winter. Furthermore, his assertion that rain and fog diminished sound speed was accepted until Tyndall's subsequent refutation.

Initial measurements of sound speed exhibited inconsistencies, leading to the hypothesis that wind velocity and temperature might influence its propagation. In 1740, G. L. Bianconi demonstrated that the speed of sound in air is directly proportional to temperature. The Academy of Sciences of Paris, in 1738, utilized cannon fire as a sound source and determined that, in the absence of wind, the speed of sound at 0 °C was 332 m/s, a value within 1% of the currently accepted standard.

Chladni determined the speed of sound in solid materials by comparing the pitch of sounds produced in an air-filled tube and a solid bar. He observed that sound propagated approximately 7.5 times faster in tin than in air, and about 12 times faster in copper. In 1808, Biot measured the speed of sound within an iron pipe approximately 1000 meters in length, finding it to be 10.5 times greater than in air. However, he considered this an order-of-magnitude estimate, given that his time measurement accuracy of 0.5 seconds exceeded the actual propagation time through the pipe.

The inaugural measurement of the speed of sound in water was conducted by Jean-Daniel Colladon and Charles Sturm at Lake Geneva in 1826. Positioned on two boats separated by 10 kilometers, Colladon operated a lever that simultaneously ignited gunpowder above the water and rang an underwater bell. Sturm, using an underwater tube, listened for the bell and recorded the time until the sound was perceived. Their experiment yielded a value of 1437.8 m/s in water at 8 °C, which deviates from the contemporary accepted value by 1 m/s. The findings were subsequently published in a monograph.

In 1860, Samuel Earnshaw recounted his observation from an 1822 experiment where the sound of cannon fire was perceived before the accompanying officer's shouted command of "fire." He theorized that sufficiently loud sounds could generate discontinuities in the air—termed shock waves in contemporary terminology—which propagate at a speed exceeding that of typical sound waves. To substantiate this hypothesis, Earnshaw demonstrated that an ideal fluid is incapable of uniform wave propagation, a concept subsequently recognized as the Earnshaw paradox.

Compression and shear waves

In gases or liquids, sound manifests as compression waves. In solid materials, however, waves propagate in two distinct forms. A longitudinal wave involves compression and decompression along its direction of travel, functioning identically in gases and liquids, and is analogous to a compression wave in solids. Only compression waves are transmitted through fluids (gases and liquids). An additional wave type, the transverse wave, also known as a shear wave, is exclusively observed in solids due to their capacity to sustain elastic deformations. This phenomenon arises from the elastic deformation of the medium perpendicular to the wave's propagation direction; the orientation of this shear deformation defines the wave's polarization. Typically, transverse waves manifest as a pair of orthogonal polarizations.

These distinct wave types—compression waves and the various polarizations of shear waves—can exhibit varying propagation velocities even at identical frequencies. Consequently, their arrival times at an observer differ. A prominent illustration is an earthquake, where rapid compression waves precede the arrival of oscillating transverse waves by several seconds.

The velocity of a compression wave in a fluid is governed by the medium's compressibility and density. In solids, compression waves, analogous to those in fluids, also depend on compressibility and density, but are additionally influenced by the shear modulus. This shear modulus impacts compression waves through off-axis elastic energies that modulate effective tension and relaxation during compression. Conversely, the velocity of shear waves, exclusively found in solids, is solely determined by the solid material's shear modulus and density.

Equations

Conventionally, the speed of sound is mathematically denoted by c, a symbol derived from the Latin term celeritas, signifying "swiftness."

For fluids generally, the speed of sound, c, is defined by the Newton–Laplace equation: $${\displaystyle c={\sqrt {\frac {K_{s}}{\rho }}},}$$ where

• ${\displaystyle K_{s}}$ represents the coefficient of stiffness, specifically the isentropic bulk modulus (or, for gases, the modulus of bulk elasticity);
• ${\displaystyle \rho }$ denotes the density.

${\displaystyle K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}}$, where ${\displaystyle P}$ signifies pressure, and the derivative is evaluated isentropically, implying constant entropy s. This approach is justified because the rapid propagation of a sound wave allows its process to be approximated as adiabatic, precluding substantial heat conduction and radiation within a single pressure cycle.

Consequently, the speed of sound rises with the material's stiffness (defined as an elastic body's resistance to deformation under an applied force) and diminishes with increasing density. In the case of ideal gases, the bulk modulus K is equivalent to the gas pressure multiplied by the dimensionless adiabatic index, which approximates 1.4 for air under standard pressure and temperature conditions.

For general equations of state, the speed of sound, denoted as c, can be derived using classical mechanics, as outlined below:

A sound wave propagates at a speed of ${\displaystyle v}$ through a pipe, which is aligned with the ${\displaystyle x}$ axis and possesses a cross-sectional area of ${\displaystyle A}$.

When relativistic effects become significant, the speed of sound is determined using the relativistic Euler equations.

Within a non-dispersive medium, the speed of sound remains constant irrespective of its frequency, implying that the rates of energy transport and sound propagation are identical across all frequencies. Air, composed primarily of oxygen and nitrogen, exemplifies a non-dispersive medium. However, air also contains a minor proportion of CO₂, which is inherently dispersive and induces dispersion in air at ultrasonic frequencies (exceeding 28 kHz).

Conversely, in a dispersive medium, the speed of sound varies with frequency, as defined by the dispersion relation. Individual frequency components travel at their distinct phase velocities, whereas the energy of the disturbance advances at the group velocity. An analogous phenomenon is observed with light waves.

Influence of Medium Properties

The speed of sound is not constant; it fluctuates based on the intrinsic properties of the substance transmitting the wave. For solids, the velocity of transverse (or shear) waves is determined by the material's resistance to shear deformation under stress (known as the shear modulus) and its density. Longitudinal (or compression) waves in solids are influenced by these two factors, in addition to the material's compressibility.

Within fluids, only the medium's compressibility and density are significant determinants, given that fluids cannot sustain shear stresses. In heterogeneous fluids, like a liquid containing gas bubbles, the liquid's density and the gas's compressibility cumulatively influence the speed of sound, a phenomenon exemplified by the hot chocolate effect.

For gases, adiabatic compressibility correlates directly with pressure via the heat capacity ratio (or adiabatic index). Concurrently, pressure and density exhibit an inverse relationship with temperature and molecular weight. Consequently, only the truly independent properties of temperature and molecular structure are critical, as the heat capacity ratio can be derived from these, though molecular weight alone is insufficient for its determination.

Sound travels more rapidly in gases with lower molecular weights, such as helium, compared to heavier gases like xenon. In the case of monatomic gases, the speed of sound approximates 75% of the average atomic velocity within that gas.

For a specific ideal gas with a fixed molecular composition, the speed of sound is solely temperature-dependent. At a constant temperature, gas pressure does not influence the speed of sound because an increase in density, which is proportional to pressure, produces effects on sound speed that are equal and opposite to those of pressure, resulting in a precise cancellation. Analogously, while compression waves in solids, like those in liquids, rely on both compressibility and density, in gases, density's contribution to compressibility simplifies the relationship. This leaves the speed of sound dependent only on temperature, molecular weight, and the heat capacity ratio, which can be independently determined from temperature and molecular composition. Consequently, for a particular gas (assuming constant molecular weight) and within a narrow temperature range (where heat capacity remains relatively stable), the speed of sound is exclusively a function of the gas's temperature.

Under conditions of non-ideal gas behavior, where the Van der Waals equation is applicable, the proportionality is inexact, leading to a minor dependence of sound velocity on gas pressure.

Humidity exerts a minor yet quantifiable influence on the speed of sound, typically increasing it by approximately 0.1% to 0.6%. This occurs because lighter water molecules displace heavier oxygen and nitrogen molecules in the air, representing a straightforward mixing phenomenon.

Altitude Variation and its Consequences for Atmospheric Acoustics

Within the Earth's atmosphere, temperature stands as the primary determinant of the speed of sound. For a specified ideal gas possessing constant heat capacity and composition, the speed of sound relies solely on temperature. In this idealized scenario, the impacts of reduced density and pressure at higher altitudes mutually negate, leaving only the residual influence of temperature.

As temperature, and consequently the speed of sound, diminishes with ascending altitude up to approximately 11 km, sound waves undergo upward refraction, diverting them from ground-level observers and thereby forming an acoustic shadow zone at a certain distance from the sound source. This reduction in the speed of sound as height increases is termed a negative sound speed gradient.

Beyond 11 km, this pattern exhibits notable variations. Specifically, within the stratosphere, at altitudes exceeding approximately 20 km, the speed of sound escalates with increasing height. This phenomenon is attributed to a temperature rise resulting from heating processes within the ozone layer. Consequently, this zone manifests a positive speed-of-sound gradient. Furthermore, an additional region exhibiting a positive gradient is observed at extremely high altitudes, specifically in the thermosphere above 90 km.

Detailed Analysis

Acoustic Velocity in Ideal Gases and Atmospheric Air

In the context of an ideal gas, the bulk modulus, denoted as K (which corresponds to C, the coefficient of stiffness in solid materials, as referenced in preceding equations), is defined by the following expression: $${\displaystyle K=\gamma \cdot p.}$$ Consequently, deriving from the aforementioned Newton–Laplace equation, the velocity of sound propagation within an ideal gas is articulated as: $${\displaystyle c={\sqrt {\gamma \cdot {p \over \rho }}},}$$ where:

• γ represents the adiabatic index, alternatively termed the isentropic expansion factor. This parameter is defined as the ratio of the specific heat capacity of a gas at constant pressure to its specific heat capacity at constant volume (${\displaystyle C_{p}/C_{v}}$). Its significance stems from the adiabatic compression induced by a classical sound wave, where the heat generated during compression lacks sufficient time to dissipate from the pressure pulse, thereby augmenting the pressure produced by the compression;
• p denotes the absolute pressure;
• ρ signifies the mass density of the gas.

By applying the ideal gas law, substituting p with nRT/V, and replacing ρ with nM/V, the resulting equation for an ideal gas is formulated as: 121§ − §126127§ ⋅ γ T m ≈ 8.314 γ T M { = 1.9329086 ⋅ §191192§ − §197198§ ⋅ T m ≈ 11.640 T M , γ = §242243§ §244245§ ≈ 2.301 ⋅ §267268§ − §273274§ ⋅ T m ≈ 13.857 T M , γ = §318319§ §320321§ ≈ 1.840 ⋅ §343344§ − §349350§ ⋅ T m ≈ 11.086 T M , γ = §394395§ §396397§ c d r y   a i r ≈ 20.04687087513010149970678963 T {\displaystyle {\begin{aligned}c_{\mathrm {ideal} }&={\sqrt {\frac {\gamma p}{\rho }}}={\sqrt {\frac {\gamma RT}{M}}}={\sqrt {\frac {\gamma kT}{m}}}\\&={\sqrt {\frac {1.380649\cdot 10^{-23}\cdot \gamma T}{m}}}\approx {\sqrt {\frac {8.314\gamma T}{M}}}\\&{\begin{cases}&={\sqrt {\frac {1.9329086\cdot 10^{-23}\cdot T}{m}}}\approx {\sqrt {\frac {11.640T}{M}}},&&\gamma ={\frac {7}{5}}\\&\approx {\sqrt {\frac {2.301\cdot 10^{-23}\cdot T}{m}}}\approx {\sqrt {\frac {13.857T}{M}}},&&\gamma ={\frac {5}{3}}\\&\approx {\sqrt {\frac {1.840\cdot 10^{-23}\cdot T}{m}}}\approx {\sqrt {\frac {11.086T}{M}}},&&\gamma ={\frac {4}{3}}\\\end{cases}}\\c_{\mathrm {dry\ air} }&\approx 20.04687087513010149970678963{\sqrt {T}}\end{aligned}}} where

• c_ideal represents the speed of sound within an ideal gas.
• p denotes the pressure.
• ρ signifies the density.
• γ (gamma) represents the adiabatic index. According to kinetic theory, at room temperature, where thermal energy is fully distributed into rotational modes (indicating full rotational excitation) but quantum effects inhibit the excitation of vibrational modes, the value for diatomic gases (e.g., oxygen and nitrogen) is 7/5 = 1.400. Experimentally, for air at 0 °C, gamma has been measured within a range of 1.3991 to 1.403. For monatomic gases (e.g., argon), gamma is precisely 5/3 = 1.667, while for non-collinear triatomic gases, such as H
  §1819§O, it is 4/3 = 1.333 (a collinear triatomic gas like CO
  §2930§ is considered equivalent to a diatomic gas for these calculations). Furthermore,
  • γ exhibits temperature dependence, increasing at lower temperatures and decreasing at higher temperatures. For instance, in dry air, its value is approximately 1.404 at 258.15 K, 1.400 at 293.15 K, and 1.398 at 473.15 K.
• R represents the molar gas constant, which is 8.31446261815324 J⋅mol⁻¹⋅K⁻¹‍.
• k denotes the Boltzmann constant, with a value of 1.380649×10⁻²³ J⋅K⁻¹‍.
• T signifies the absolute temperature.
• M refers to the molar mass of the gas. For dry air, the average molar mass is approximately 28.9647 g/mol (equivalent to 0.0289647 kg/mol).
• n indicates the number of moles.
• m represents the mass of an individual molecule.

This equation is applicable exclusively when the sound wave constitutes a minor perturbation to the ambient conditions, and when other specified criteria, detailed subsequently, are satisfied. Discrepancies have been observed between calculated values for c_air and those determined experimentally.

Isaac Newton, predating significant advancements in thermodynamics, famously analyzed the speed of sound, but erroneously employed isothermal rather than adiabatic calculations. Consequently, his derived result lacked the factor of γ, though it was otherwise accurate.

The numerical substitution of the aforementioned values yields the ideal gas approximation for the velocity of sound in gases, which maintains accuracy at relatively low gas pressures and densities; for air, this encompasses standard Earth sea-level conditions. Furthermore, for diatomic gases, the application of γ = 1.4000 necessitates that the gas resides within a temperature range sufficiently high to ensure full excitation of rotational heat capacity (meaning molecular rotation functions entirely as a heat energy reservoir). Concurrently, the temperature must be low enough to prevent molecular vibrational modes from contributing to heat capacity (implying negligible heat transfer into vibration, as vibrational quantum modes beyond the minimum-energy state possess energies too high to be significantly populated by molecules at this temperature). For air, these conditions are satisfied at room temperature and at temperatures considerably below it.

For air, the following shorthand notation is introduced: $${\displaystyle R_{*}=R/M_{\mathrm {air} }.}$$

R

Ultimately, the binomial approximation of the remaining square root, predicated on the assumption that θ is negligibly small, yields the following expression:

At zero degrees Celsius, the binomial approximation introduces no inaccuracy; however, the parameter γ does, as perfectly dry air at this temperature exhibits a heat capacity ratio of approximately 1.403. The corrected formulation is presented below:

35§ ) c a i r ≈ 331.67 m s × ( §8081§ + θ 546.3 ) ≈ 331.67 m s + 0.607 θ m s {\displaystyle {\begin{aligned}\gamma &=1.403~({\text{dry air at }}\theta =0)\\c_{\mathrm {air} }&\approx 331.67{\frac {\text{m}}{\text{s}}}\times \left(1+{\frac {\theta }{546.3}}\right)\\&\approx 331.67{\frac {\text{m}}{\text{s}}}+0.607\theta {\frac {\mathrm {m} }{\text{s}}}\end{aligned}}}

A graphical representation illustrates the comparative results derived from the two equations, employing a refined value of 331.5 m/s (1,088 ft/s) for the speed of sound at 0 °C.

Impacts of Wind Shear

The velocity of sound is contingent upon temperature. As both temperature and sound velocity typically diminish with ascending altitude, sound waves undergo upward refraction, diverting them from ground-level observers and consequently forming an acoustic shadow zone at a certain distance from the emission source. A wind shear of 4 m/(s · km) is capable of inducing refraction equivalent to a standard temperature lapse rate of 7.5 °C/km. Conversely, elevated wind gradient values will cause sound to refract downwards towards the surface in the downwind direction, thereby mitigating the acoustic shadow on that side. This phenomenon enhances the audibility of sounds in the downwind region. This downwind refraction is primarily attributable to the presence of a wind gradient, rather than the direct advection of sound by the wind itself.

For the purpose of sound propagation analysis, the exponential relationship between wind speed and height can be mathematically expressed as follows: 29§ ) h ζ , d U d h ( h ) = ζ U ( h ) h , {\displaystyle {\begin{aligned}U(h)&=U(0)h^{\zeta },\\{\frac {\mathrm {d} U}{\mathrm {d} h}}(h)&=\zeta {\frac {U(h)}{h}},\end{aligned}}} where

• U(h) represents the wind speed at a specific height h;
• ζ denotes the exponential coefficient, which is determined by the ground surface roughness and generally falls within the range of 0.08 to 0.52;
• dU/dh(h) signifies the instantaneous rate of change of wind speed concerning height h, exhibiting proportionality to the anticipated wind gradient at a constant height h.

During the 1862 Battle of Iuka in the American Civil War, an acoustic shadow, purportedly intensified by a northeast wind, prevented two divisions of Union soldiers from engaging in combat. This occurred because they were unable to perceive the sounds of the battle, despite being positioned merely 10 km (six miles) downwind.

Tables

Under standard atmospheric conditions:

• At T§23§, which corresponds to 273.15 K (= 0 °C = 32 °F), the theoretical speed of sound is 331.3 m/s (= §2223§086.9 ft/s = 1193 km/h = 741.1 mph = 644.0 kn). However, reference literature may present values ranging from 331.3 to 331.6 m/s;
• At T₂₀, equivalent to 293.15 K (= 20 °C = 68 °F), the speed of sound is 343.2 m/s (= §2223§126.0 ft/s = 1236 km/h = 767.8 mph = 667.2 kn);
• At T₂₅, which is 298.15 K (= 25 °C = 77 °F), the speed of sound measures 346.1 m/s (= §2223§135.6 ft/s = 1246 km/h = 774.3 mph = 672.8 kn).

Fundamentally, for an ideal gas, the speed of sound, denoted as c, is solely contingent upon its temperature and composition, being independent of pressure or density (as these parameters covary at a constant temperature, effectively canceling each other out). Given that air approximates an ideal gas, its temperature fluctuations with altitude result in corresponding variations in the speed of sound within the standard atmosphere; however, actual environmental conditions may deviate.

Under typical atmospheric conditions, both temperature and, consequently, the speed of sound exhibit variation with increasing altitude:

Effect of Frequency and Gas Composition

General Physical Considerations

The medium through which a sound wave propagates does not consistently exhibit adiabatic behavior, which can lead to variations in the speed of sound as a function of frequency.

The concept of sound speed is significantly limited by extreme attenuation. High-frequency attenuation prevalent at sea level extends to progressively lower frequencies as atmospheric pressure decreases or the mean free path increases. Consequently, the applicability of the sound speed concept (except for frequencies approaching zero) diminishes considerably at higher altitudes. The standard equations for sound speed are accurately applicable only in scenarios where the sound wave's wavelength is substantially greater than the mean free path of the gas molecules.

The molecular composition of a gas, encompassing both the mass (M) of its molecules and their heat capacities, significantly influences the speed of sound. Generally, for gases with identical molecular masses, monatomic gases exhibit a slightly higher speed of sound (exceeding 9%) compared to diatomic gases. This difference arises because monatomic gases possess a higher γ (5/3 = 1.66...) than diatomic gases (7/5 = 1.4). Consequently, at a constant molecular mass, the speed of sound in a monatomic gas increases by a factor expressed as: c g a s , m o n a t o m i c c g a s , d i a t o m i c = §8889§ / §9495§ §9899§ / §104105§ = §115116§ §117118§ = 1.091 … {\displaystyle {c_{\mathrm {gas,monatomic} } \over c_{\mathrm {gas,diatomic} }}={\sqrt {{5/3} \over {7/5}}}={\sqrt {25 \over 21}}=1.091\ldots }

This calculation accounts for the approximately 9% difference, representing a characteristic ratio for sound speeds at room temperature, such as between helium and deuterium, both having a molecular weight of 4. Sound propagates more rapidly in helium than in deuterium because adiabatic compression generates more heat in helium. This occurs because helium molecules, being monatomic, can store heat energy from compression exclusively through translational motion, not rotational motion. Consequently, monatomic helium molecules exhibit faster movement within a sound wave, leading to more rapid sound transmission. Notably, sound typically travels at approximately 70% of the mean molecular speed in gases, specifically 75% in monatomic gases and 68% in diatomic gases.

This example assumes a sufficiently low temperature, where molecular vibration does not influence heat capacities. Vibrational modes typically cause the adiabatic index (gamma) to decrease towards 1. This is because vibrational modes in a polyatomic gas provide additional mechanisms for heat storage that do not impact temperature, and therefore do not affect molecular or sound velocity. Consequently, the combined effect of elevated temperatures and vibrational heat capacity accentuates the disparity in sound speed between monatomic and polyatomic molecules, with monatomic gases consistently exhibiting a higher sound speed.

Practical Applications in Air

Temperature constitutes the predominant factor influencing the speed of sound in air. This velocity exhibits proportionality to the square root of the absolute temperature, resulting in an approximate increase of 0.6 m/s for each degree Celsius. Consequently, the pitch of a musical wind instrument rises with an increase in its operational temperature.

Humidity also contributes to an elevation in the speed of sound. The disparity between 0% and 100% humidity levels measures approximately 1.5 m/s under standard pressure and temperature conditions; however, the magnitude of this humidity effect intensifies significantly with rising temperature.

In practical contexts, the influence of frequency and pressure on the speed of sound is typically negligible. Within dry air, the speed of sound experiences an increase of approximately 0.1 m/s as the frequency ascends from 10 Hz to 100 Hz. For audible frequencies exceeding 100 Hz, this speed remains comparatively constant. Standard values for the speed of sound are conventionally cited under conditions of low frequencies, where the wavelength substantially exceeds the mean free path.

The approximate value of 1000/3, equivalent to 333.33... m/s, precisely corresponds to the speed of sound slightly below 5 °C. This value serves as a reliable approximation for typical ambient temperatures, particularly in temperate climates. Consequently, a common heuristic for estimating the distance to a lightning strike involves counting the seconds between observing the flash and hearing the thunder, then dividing this duration by 3 to obtain the distance in kilometers. Alternatively, dividing the number of seconds by 5 yields an approximate distance in miles.

Mach Number

The Mach number, a critical parameter in aerodynamics, represents the ratio of an object's airspeed to the local speed of sound. At higher altitudes, the Mach number is intrinsically linked to temperature. Nevertheless, aircraft flight instruments determine the Mach number through pressure differentials rather than direct temperature measurements. This operational approach relies on the premise that a specific pressure corresponds to a particular altitude and, by extension, a standard temperature. Such a methodology is necessitated by the fact that the stagnation pressure detected by a Pitot tube is influenced by both altitude and airspeed.

Experimental Methodologies

Various methodologies are available for determining the speed of sound in an atmospheric medium.

William Derham conducted the earliest demonstrably accurate estimation of the speed of sound in air, a contribution recognized by Isaac Newton. Derham positioned a telescope atop the tower of the Church of St Laurence in Upminster, England. During calm weather, an assistant, equipped with a synchronized pocket watch, would discharge a shotgun at a pre-arranged time from a prominent location several miles distant across the landscape, an event verifiable via telescope. Derham subsequently measured the temporal interval between observing the gunsmoke and hearing the sound, utilizing a half-second pendulum. The distance to the gun's firing point was ascertained through triangulation, and the velocity was then calculated by simple division (distance/time). Through numerous observations across varying distances, the inherent inaccuracies of the half-second pendulum were mitigated by averaging, yielding his definitive estimate for the speed of sound. Contemporary stopwatches now permit the application of this method over shorter distances, ranging from 200 to 400 meters, without requiring a sound source as powerful as a shotgun.

Single-Shot Timing Methodologies

The most straightforward conceptual approach involves measurement conducted with two microphones and a high-speed recording apparatus, such as a digital storage oscilloscope. This methodology operates on the subsequent principle.

When a sound source and two microphones are linearly aligned, with the source positioned at one extremity, the following parameters become measurable:

1. The spatial separation between the microphones (x), designated as the microphone basis.
2. The temporal difference in signal arrival (t) at the respective microphones.

Consequently, the velocity v is determined by the equation x/t.

Alternative Methodologies

Within these alternative approaches, the direct measurement of time is superseded by the measurement of its inverse, namely frequency.

Kundt's tube exemplifies an experimental setup suitable for determining the speed of sound within a confined volume. A notable advantage of this technique is its applicability to measuring the speed of sound in any gaseous medium. This method employs a fine powder to render the acoustic nodes and antinodes visually discernible. It represents a compact and efficient experimental configuration.

When a tuning fork is positioned near the opening of a long pipe partially submerged in water, the pipe can achieve resonance. This occurs when the air column's length within the pipe corresponds to (1 + 2n)λ/4, where n represents an integer. Given that the antinodal point at the open end of the pipe extends slightly beyond its mouth, a more accurate measurement involves identifying two or more resonance points and subsequently determining half a wavelength between them.

The relationship between these variables is expressed by the equation v = fλ.

High-Precision Measurements in Air

Impurities significantly influence the accuracy of high-precision sound speed measurements. While chemical desiccants can dry air, they simultaneously introduce contamination. Cryogenic drying, another method, removes carbon dioxide alongside moisture, leading many high-precision studies to utilize carbon dioxide-free air instead of natural atmospheric air. A 2002 review identified a 1963 measurement by Smith and Harlow, employing a cylindrical resonator, as yielding "the most probable value of the standard speed of sound to date." This experiment, initially conducted with air devoid of carbon dioxide, had its results subsequently adjusted to be applicable to natural air. Although the experiments were performed at 30 °C, the reported values were temperature-corrected to 0 °C. The resulting speed of sound for dry air at standard temperature and pressure (STP) was 331.45 ± 0.01 m/s, valid for frequencies ranging from 93 Hz to 1,500 Hz.

Non-Gaseous Media

Speed of Sound in Solids

Three-Dimensional Solids

A solid material exhibits non-zero stiffness in response to both volumetric and shear deformations. Consequently, sound waves can propagate through solids at varying velocities, depending on the specific mode of deformation. Sound waves that induce volumetric changes (compression) are termed pressure waves (or longitudinal waves), while those causing shear deformations (shearing) are known as shear waves (or transverse waves). In the context of earthquakes, these corresponding seismic phenomena are referred to as P-waves (primary waves) and S-waves (secondary waves), respectively. The propagation velocities of these two wave types within a homogeneous, three-dimensional solid are mathematically expressed as follows: c s o l i d , p = K + §4041§ §4243§ G ρ<!-- ρ --> = E ( §6566§ − ν ) ρ ( §8283§ + ν ) ( §9394§ − §9899§ ν<!-- ν --> ) , {\displaystyle c_{\mathrm {solid,p} }={\sqrt {\frac {K+{\frac {4}{3}}G}{\rho }}}={\sqrt {\frac {E(1-\nu )}{\rho (1+\nu )(1-2\nu )}}},} $${\displaystyle c_{\mathrm {solid,s} }={\sqrt {\frac {G}{\rho }}},}$$ where:

• K represents the bulk modulus of the elastic material;
• G denotes the shear modulus of the elastic material;
• E signifies Young's modulus;
• ρ indicates the material's density;
• ν refers to Poisson's ratio.

It is important to note that the last quantity, Poisson's ratio, is not an independent variable, as demonstrated by the relationship E = 3K(1 − 2ν). The velocity of pressure waves is influenced by both the material's resistance to pressure and shear, whereas the velocity of shear waves is solely determined by its shear properties.

Generally, pressure waves propagate through materials at a higher velocity than shear waves. This phenomenon explains why the initial phase of an earthquake is frequently characterized by a rapid vertical shock, occurring prior to the arrival of waves that induce lateral motion. For instance, a representative steel alloy with a bulk modulus K of 170 GPa, a shear modulus G of 80 GPa, and a density p of 7700 kg/m§1415§, would exhibit a compressional wave speed c_solid,p of approximately 6,000 m/s. This calculated value aligns well with an experimentally determined c_solid,p of 5,930 m/s for a potentially distinct steel composition. Utilizing these same parameters, the shear wave speed c_solid,s is estimated to be 3,200 m/s.

The velocity of sound in semiconductor solids can exhibit significant sensitivity to the concentration of electronic dopants present within their structure.

One-dimensional Solids

For stiff materials like metals, the speed of sound for pressure waves is occasionally reported for "long rods" of the material, a configuration that facilitates measurement. When the diameter of such rods is less than a wavelength, the velocity of pure pressure waves can be simplified and is expressed as: $${\displaystyle c_{\mathrm {solid} }={\sqrt {\frac {E}{\rho }}},}$$ where E represents Young's modulus. This formulation parallels the expression for shear waves, with Young's modulus substituting for the shear modulus. The velocity of pressure waves in long rods will consistently be marginally lower than that in homogeneous three-dimensional solids, with the ratio between these velocities being contingent upon the material's Poisson's ratio.

Acoustic Velocity in Liquids

Within a fluid, the sole non-zero stiffness pertains to volumetric deformation, as fluids are incapable of sustaining shear forces.

Consequently, the speed of sound within a fluid is determined by the following equation: $${\displaystyle c_{\mathrm {fluid} }={\sqrt {\frac {K}{\rho }}},}$$ where K denotes the bulk modulus of the fluid.

Acoustic Propagation in Water

In freshwater, sound propagates at approximately 1481 m/s when the temperature is 20 °C. Underwater sound finds applications in sonar systems, acoustic communication, and acoustical oceanography.

Acoustic Propagation in Seawater

In saline water devoid of air bubbles and suspended sediment, sound propagates at approximately 1500 m/s (specifically, §67§500.235 m/s at 1000 kilopascals, 10 °C, and 3% salinity, according to one methodology). The velocity of sound in seawater is influenced by pressure (and consequently depth), temperature (where a 1 °C variation corresponds to approximately 4 m/s), and salinity (a 1‰ change correlating to about 1 m/s); empirical equations have been developed to precisely compute sound speed based on these parameters. Additional factors influencing sound velocity are considered negligible. Given that temperature generally diminishes with increasing depth across most oceanic regions, the sound speed profile exhibits a decrease, reaching a minimum at depths of several hundred meters. Beyond this minimum, sound velocity subsequently rises, as the impact of escalating pressure supersedes the influence of diminishing temperature. Dushaw et al. provide further details on this topic.

Mackenzie developed an empirical equation to determine the speed of sound in seawater, expressed as follows: c ( T , S , z ) = a §2829§ + a §3839§ T + a §5051§ T §5859§ + a §6869§ T §7677§ + a §8687§ ( S − §9798§ ) + a §107108§ z + a §119120§ z §127128§ + a §137138§ T ( S − §150151§ ) + a §160161§ T z §170171§ , {\displaystyle c(T,S,z)=a_{1}+a_{2}T+a_{3}T^{2}+a_{4}T^{3}+a_{5}(S-35)+a_{6}z+a_{7}z^{2}+a_{8}T(S-35)+a_{9}Tz^{3},} where:

• T represents the temperature, measured in degrees Celsius.
• S denotes the salinity, expressed in parts per thousand.
• z signifies the depth, given in meters.

The constants a§23§, a§6_{7§, ..., a§1011§ are defined as follows:}

The Sound Speed vs. Depth graph does not directly correlate with the MacKenzie formula because temperature and salinity fluctuate at varying depths. However, when T (temperature) and S (salinity) are maintained as constants, the formula consistently indicates an increase with depth.

Alternative equations for calculating the speed of sound in seawater, such as those developed by V. A. Del Grosso and the Chen-Millero-Li Equation, offer accuracy across diverse conditions but are considerably more intricate.

Speed of Sound in Plasma

For a plasma where electrons are hotter than ions (though not excessively so), the speed of sound is determined by the following formula: c s = ( γ<!-- γ --> Z k T e m i ) §5758§ / §6364§ = ( γ Z T e μ ) §99100§ / §105106§ × 90.85   m / s , {\displaystyle c_{s}=\left({\frac {\gamma ZkT_{\mathrm {e} }}{m_{\mathrm {i} }}}\right)^{1/2}=\left({\frac {\gamma ZT_{e}}{\mu }}\right)^{1/2}\times 90.85~\mathrm {m/s} ,} where:

• m_i represents the ion mass;
• μ denotes the ratio of ion mass to proton mass, expressed as μ = m_i/m_p;
• T_e signifies the electron temperature;
• Z indicates the charge state;
• k refers to the Boltzmann constant;
• γ represents the adiabatic index.

Unlike in a gas, the pressure and density in a plasma are attributed to distinct species: electrons contribute to pressure, while ions determine density. These two components are interconnected via a fluctuating electric field.

Mars

On Mars, the speed of sound exhibits frequency-dependent variation, with higher frequencies propagating more rapidly than lower frequencies. Specifically, high-frequency laser-generated sound travels at 250 m/s (820 ft/s), whereas low-frequency sound propagates at 240 m/s (790 ft/s).

Gradients

In a three-dimensional isotropic propagation, sound intensity diminishes inversely with the square of the distance. Conversely, within oceanic environments, a specific stratum known as the 'deep sound channel' or SOFAR channel can restrict sound waves to a particular depth.

Within the SOFAR channel, the speed of sound is reduced compared to the adjacent layers both above and below it. Analogous to light waves refracting towards areas of higher refractive index, sound waves refract towards regions where their propagation speed decreases. Consequently, sound becomes confined within this layer, similar to how light is contained within a glass sheet or an optical fiber. This confinement effectively limits sound propagation to two dimensions, where intensity decreases inversely with distance, enabling waves to traverse significantly greater distances before becoming imperceptible.

An analogous phenomenon manifests within the atmosphere. Project Mogul notably leveraged this effect to successfully identify nuclear detonations from substantial distances.

• Acoustoelastic effect
• Second sound
• Sound barrier
• Underwater acoustics
• Bell X-1

References

• Speed of Sound Calculator
• Speed of sound: Temperature Matters, Not Air Pressure
• The Speed of Sound
• Did Sound Once Travel at Light Speed?
• Discovery of Sound in the Sea (uses of sound by humans and other animals)