6.1 Speed of sound

Speed of sound in fluids

The speed of sound tells you how fast small pressure disturbances (acoustic waves) travel through a fluid. It's one of the most important reference quantities in compressible flow analysis because it sets the dividing line between subsonic and supersonic behavior. Everything in this unit builds on it: Mach number, shock waves, and compressible flow relations all depend on knowing the local speed of sound.

Sound speed varies widely between gases and liquids because of differences in density and compressibility. In gases, you can derive it from the ideal gas law combined with an isentropic (adiabatic, reversible) assumption. In liquids, the bulk modulus dominates. The sections below cover the physics, the key equations, and how sound speed connects to compressibility through the Mach number.

Factors affecting sound speed

Fluid density effects

The speed of sound in a fluid is inversely proportional to the square root of its density. All else being equal, a denser fluid means a slower sound speed.

• Density variations within a fluid change the local speed of sound. Temperature gradients and pressure changes both cause density fluctuations, so the speed of sound can vary from point to point in a flow field.

Be careful with a common misconception here: sound actually travels faster in water (~1480 m/s) than in air (~343 m/s) at standard conditions, even though water is far denser (998 kg/m³ vs. 1.225 kg/m³). That's because water's bulk modulus is enormously higher than air's, and that effect more than compensates for the higher density. Density alone doesn't determine sound speed; you always need to consider compressibility too.

Fluid compressibility impact

Compressibility measures how easily a fluid changes volume under pressure. The less compressible a fluid is (the "stiffer" it is), the faster sound travels through it.

• Gases are far more compressible than liquids, which is why sound speeds in gases are generally much lower than in liquids.
• The relevant stiffness measure is the bulk modulus $K$, which is the inverse of compressibility.

The general equation for the speed of sound is:

$c = \sqrt{\frac{K}{\rho}}$

where $c$ is the speed of sound, $K$ is the bulk modulus, and $\rho$ is the fluid density. This single equation explains why water beats air in sound speed: water's bulk modulus (~2.2 GPa) is roughly 20,000 times that of air, while its density is only about 800 times greater.

Speed of sound in gases

Ideal gas approximation

For an ideal gas undergoing isentropic (adiabatic + reversible) perturbations, the speed of sound reduces to:

$c = \sqrt{\frac{\gamma R T}{M}}$

• $\gamma$ = ratio of specific heats (adiabatic index), e.g., 1.4 for air
• $R$ = universal gas constant (8.314 J/(mol·K))
• $T$ = absolute temperature (in Kelvin)
• $M$ = molar mass of the gas

You can also write this using the specific gas constant $R_s = R/M$:

$c = \sqrt{\gamma R_s T}$

Two things to notice: sound speed in an ideal gas depends only on temperature (not on pressure directly), and lighter gases have higher sound speeds. For example, helium ($M$ ≈ 4 g/mol, $\gamma$ = 5/3) has a sound speed of about 1007 m/s at 20 °C, roughly three times that of air.

Adiabatic vs. isothermal processes

Why do we use the adiabatic (isentropic) assumption rather than isothermal?

• Acoustic compressions and rarefactions happen so quickly that there's essentially no time for heat to transfer between adjacent fluid parcels. That makes the process adiabatic.
• Newton originally derived the speed of sound assuming an isothermal process ($c = \sqrt{RT/M}$, without the $\gamma$), which underestimated the measured speed of sound in air by about 16%. Laplace corrected this by including $\gamma$, and the prediction matched experiments.
• An isothermal assumption would only be appropriate for extremely low-frequency disturbances where the gas has time to equilibrate thermally with its surroundings.

Speed of sound in liquids

Bulk modulus of liquids

In liquids, you go back to the general relation $c = \sqrt{K/\rho}$ because the ideal gas law doesn't apply.

• The bulk modulus $K$ represents a liquid's resistance to uniform compression. A higher bulk modulus means the liquid is stiffer and transmits sound faster.
• Water has $K$ ≈ 2.2 GPa, giving a sound speed of about 1480 m/s at 20 °C. Most oils have lower bulk moduli, so sound travels more slowly through them.

Liquid density considerations

Density still matters through the $\sqrt{K/\rho}$ relation, but liquid densities are far less sensitive to temperature and pressure changes than gas densities. This means the speed of sound in a liquid stays relatively stable across a wide range of conditions, unlike in gases where a temperature swing can shift $c$ noticeably.

Mach number and compressibility

Definition of Mach number

The Mach number is the ratio of flow velocity to the local speed of sound:

$Ma = \frac{v}{c}$

where $v$ is the flow velocity and $c$ is the local speed of sound.

The flow regime classification:

Because $c$ depends on local temperature (in a gas), the Mach number can vary throughout a flow field even if the actual velocity is constant.

Compressible vs. incompressible flow

A practical rule of thumb: compressibility effects become significant when $Ma \gtrsim 0.3$. Below that threshold, density changes in the flow are small (less than about 5%), and you can treat the fluid as incompressible, which greatly simplifies the governing equations.

Above $Ma$ ≈ 0.3, you need the full compressible flow equations. Density, pressure, and temperature all couple together, and new phenomena appear:

• Shock waves (abrupt, nearly discontinuous jumps in flow properties) at supersonic speeds
• Choked flow in converging nozzles when the throat reaches $Ma = 1$
• Expansion fans in supersonic flow turning around convex corners

Acoustic wave propagation in fluids

Longitudinal wave characteristics

Sound waves in fluids are longitudinal waves: fluid particles oscillate back and forth parallel to the direction the wave travels. This creates alternating regions of compression (higher pressure/density) and rarefaction (lower pressure/density). The speed of sound determines how fast these regions move through the fluid.

Transverse wave absence

Fluids cannot sustain shear stress at rest, so they don't support transverse (shear) waves. Only longitudinal modes propagate. This is a key difference from solids, where both longitudinal and transverse waves exist, and it simplifies acoustic analysis in fluids considerably.

Measurement techniques for sound speed

Direct measurement methods

Time-of-flight is the most straightforward approach:

1. Emit a short sound pulse from a source.
2. Record the time it takes to reach a receiver at a known distance $d$.
3. Calculate $c = d / t$.

Acoustic interferometry uses interference patterns of sound waves to determine $c$. By adjusting the path length and finding constructive/destructive interference fringes, you can extract the wavelength and, combined with the known frequency, compute the speed of sound.

Indirect calculation approaches

You can also calculate $c$ from known fluid properties rather than measuring it directly:

• For gases, use $c = \sqrt{\gamma R T / M}$ if you know the gas composition, temperature, and $\gamma$.
• For liquids, use $c = \sqrt{K/\rho}$ with measured or tabulated values of bulk modulus and density. Empirical correlations exist for common liquids (e.g., seawater equations that account for salinity, temperature, and depth), though these are specific to particular fluid types.

Applications of sound speed in fluids

Sonic and ultrasonic flow meters

These devices use the speed of sound to measure fluid velocity. Two common approaches:

• Transit-time method: Sound pulses are sent upstream and downstream between two transducers. The pulse traveling with the flow arrives slightly sooner than the one traveling against it. The time difference is proportional to the flow velocity.
• Doppler method: Sound reflects off particles or bubbles in the fluid. The frequency shift between the emitted and reflected waves gives the fluid velocity.

Both methods require accurate knowledge of the speed of sound in the working fluid.

Acoustic levitation and manipulation

Standing acoustic waves can create pressure nodes where small objects or droplets are trapped and held in place without physical contact. The spacing of these nodes depends on the wavelength, which is set by the frequency and the speed of sound in the medium ($\lambda = c/f$). Controlling $c$ (by changing the gas or temperature) lets you tune the levitation positions.

Limitations and assumptions

Homogeneous fluid assumption

Most of the equations above assume the fluid has uniform properties everywhere. Real fluids often have spatial variations in temperature, density, or composition (think of the ocean, where salinity and temperature change with depth). In those cases, the local speed of sound varies, and you may need numerical methods or layered models to get accurate predictions.

Non-dispersive medium simplification

The standard equations also assume the speed of sound doesn't depend on wave frequency (a non-dispersive medium). This holds well for most common fluids under normal conditions. However, in viscoelastic fluids or when molecular relaxation processes are present (e.g., in $CO_2$ at certain frequencies), sound speed becomes frequency-dependent. This dispersion distorts waveforms and requires more complex models to describe accurately.