3.1 Speed of sound

Speed of sound basics

The speed of sound describes how fast pressure disturbances (sound waves) propagate through a medium. In compressible aerodynamics, it serves as the reference speed that defines flow regimes and determines when compressibility effects become significant.

Definition of speed of sound

The speed of sound is the rate at which small pressure disturbances travel through a medium such as air, water, or a solid. It depends on the medium's properties: density, temperature, and compressibility (how easily the medium changes volume under pressure).

At standard sea-level conditions (15°C, 1 atm), the speed of sound in air is approximately 343 m/s (1,235 km/h or 761 mph).

Factors affecting speed of sound

• Temperature: Higher temperature means molecules have more kinetic energy and collide more vigorously, so sound propagates faster. This is the dominant factor in the atmosphere.
• Medium density and stiffness: What matters is the ratio of a medium's stiffness (bulk modulus) to its density. A stiffer medium transmits sound faster; a denser medium (all else equal) slows it down. In gases specifically, higher molecular weight lowers the speed of sound.
• Humidity: A minor effect in air. Water vapor (molecular weight ~18) is lighter than the nitrogen (28) and oxygen (32) it displaces, so humid air has a slightly lower average molecular weight and a slightly higher speed of sound.
• Altitude: In the standard atmosphere, temperature generally decreases with altitude (in the troposphere), which reduces the speed of sound. The lower air density alone does not directly change the speed of sound in an ideal gas; it's the temperature drop that matters.

Speed of sound in different mediums

The speed of sound varies widely across gases, liquids, and solids because of differences in stiffness and density.

• Gases: Speed depends on the square root of the ratio of bulk modulus to density. Air at 20°C: ~343 m/s. Helium at 20°C: ~1,007 m/s (helium is much lighter, so sound travels faster).
• Liquids: Molecules are more closely packed and the bulk modulus is much higher, so sound travels faster than in gases. Fresh water at 20°C: ~1,482 m/s. Seawater at 20°C: ~1,522 m/s.
• Solids: Strong intermolecular bonds make solids very stiff, giving the highest speeds of sound. Steel: ~5,960 m/s. Aluminum: ~6,320 m/s.

Mach number

The Mach number is the single most important dimensionless parameter in compressible aerodynamics. It tells you the ratio of a flow speed (or object speed) to the local speed of sound, and it determines which physical effects dominate the flow.

Definition of Mach number

Mach number is defined as:

$M = \frac{v}{a}$

where $v$ is the object's (or flow's) speed and $a$ is the local speed of sound.

• $M < 1$: subsonic
• $M = 1$: sonic
• $M > 1$: supersonic

Because $a$ depends on local temperature, the Mach number can vary from point to point in a flow field even if the object's speed is constant. The quantity is named after Austrian physicist Ernst Mach, a pioneer in the study of supersonic flow.

Subsonic vs supersonic speeds

Subsonic flow ($M < 1$): Pressure disturbances can travel upstream (ahead of the object) because the sound waves move faster than the flow. This means the flow "knows" the object is coming and adjusts smoothly around it.

Supersonic flow ($M > 1$): The object outruns its own pressure disturbances. The flow ahead has no warning, so the air is forced to adjust abruptly through shock waves. This fundamental difference in information propagation is why supersonic aerodynamics behaves so differently from subsonic.

Transonic speed range

The transonic regime spans roughly Mach 0.8 to Mach 1.2. Even though the freestream may be subsonic, the air accelerating over curved surfaces (like the top of a wing) can locally exceed Mach 1. This creates a mixed flow field with both subsonic and supersonic regions.

Transonic flow is characterized by:

• Local shock waves forming on the wing surface
• A sharp rise in drag called drag divergence
• Reduced control effectiveness and potential buffeting

Commercial airliners cruise near Mach 0.8–0.85, right at the edge of this regime, which is why transonic aerodynamics is so important to their design.

Hypersonic speeds

Hypersonic flow is generally defined as Mach 5 and above. At these speeds, several new phenomena appear:

• Thin shock layers: The shock wave wraps tightly around the body, and the region between the shock and the body surface becomes very thin.
• High-temperature effects: Kinetic energy conversion to heat produces temperatures high enough to cause molecular dissociation and ionization of air.
• Viscous interaction: The thick, hot boundary layer interacts strongly with the outer inviscid flow.

Examples of hypersonic vehicles include the X-15 (Mach 6.7), the X-43A scramjet demonstrator (Mach 9.6), and reentry vehicles like the Space Shuttle.

Compressibility effects

At low Mach numbers (below about 0.3), density changes in the flow are negligible and the air behaves as if it's incompressible. As the Mach number increases toward and beyond 1, density variations become large and fundamentally alter the flow behavior.

Compressibility at high speeds

When flow speeds approach the speed of sound, the fluid can no longer simply "get out of the way." Instead, it compresses, and its density, pressure, and temperature all change significantly. The threshold where compressibility matters is roughly $M \approx 0.3$; above this, density changes exceed about 5% and can no longer be ignored.

At higher Mach numbers, these effects intensify and lead to shock wave formation, altered pressure distributions, and changes in the aerodynamic forces on a body.

Density changes in compressible flow

In compressible flow, density is no longer constant. It varies with local pressure and temperature according to the equation of state for the gas.

• Upstream of a shock wave, the flow is at its original (lower) density.
• Across a shock wave, density increases abruptly as the flow is compressed.
• These density variations directly affect lift, drag, and stability because the aerodynamic forces depend on the local flow conditions, not just the freestream values.

Pressure changes in compressible flow

Pressure distributions around a body change significantly in compressible flow. As the Mach number increases:

• Pressure rises on forward-facing surfaces increase beyond what incompressible theory predicts.
• Suction peaks on upper surfaces become more intense.
• Shock waves create sudden, discontinuous pressure jumps that contribute to wave drag, a form of drag that does not exist in purely subsonic flow.

Temperature changes in compressible flow

As flow decelerates (for example, at a stagnation point or across a shock), kinetic energy converts to internal energy, raising the temperature. The stagnation temperature is given by:

$T_0 = T \left(1 + \frac{\gamma - 1}{2} M^2\right)$

At Mach 2 in standard air, stagnation temperature reaches roughly 260°C. At Mach 5, it exceeds 1,000°C. These temperatures create serious challenges for structural materials and require thermal protection or active cooling systems.

Shock waves

Shock waves are extremely thin regions (on the order of a few mean free paths, roughly $10^{-7}$ m in air) across which flow properties change almost discontinuously. They form whenever a supersonic flow is forced to decelerate abruptly.

Formation of shock waves

When an object moves faster than the speed of sound, the pressure disturbances it creates cannot propagate upstream to warn the oncoming air. These disturbances pile up and coalesce into a shock wave.

Across the shock, the flow properties change nearly instantaneously:

• Pressure, density, and temperature all increase
• Flow velocity decreases
• The process is irreversible, meaning entropy increases across every shock (energy is dissipated as heat)

Normal vs oblique shock waves

Normal shocks are perpendicular to the flow direction. They produce the strongest changes in flow properties for a given upstream Mach number. The flow downstream of a normal shock is always subsonic.

Oblique shocks are inclined at an angle to the flow. They form around wedges, cones, and other angled surfaces in supersonic flow. Compared to normal shocks at the same upstream Mach number:

• The property changes (pressure, density, temperature jumps) are smaller
• The downstream flow can remain supersonic
• The shock angle depends on both the upstream Mach number and the deflection angle of the surface

Properties across a shock wave

Across any shock wave:

The stronger the shock (higher upstream Mach number), the larger these changes. The loss of total pressure across a shock represents an irreversible energy loss, which is why strong shocks are aerodynamically costly.

Shock wave interactions

Shock waves don't exist in isolation. They interact with each other and with boundary layers, creating complex flow patterns.

• Shock-shock interactions: When two shocks intersect, they can produce reflected shocks, slip lines (contact discontinuities), and regions of very high pressure and temperature.
• Shock-boundary layer interactions: A shock impinging on a boundary layer can cause the boundary layer to thicken or separate entirely, leading to increased drag, localized heating, and potential loss of control effectiveness.
• Unsteady effects: Shock interactions can produce oscillations (buffeting) that cause structural fatigue and vibration problems, particularly in the transonic regime.

Speed of sound measurement

Accurate measurement of the speed of sound matters both for fundamental fluid property characterization and for calibrating instruments used in wind tunnels and flight testing.

Direct measurement techniques

Direct methods measure the transit time of a sound pulse over a known distance.

1. Time-of-flight: Generate a sound pulse at one location and detect it at another. Divide the known distance by the measured travel time to get the speed of sound. This is conceptually simple but requires precise timing.
2. Resonance tube: Fill a tube with the fluid of interest and excite standing waves. The resonant frequencies depend on the tube length and the speed of sound, so measuring the resonance frequencies lets you back out $a$.

Indirect calculation methods

Instead of measuring sound propagation directly, you can calculate the speed of sound from other measured properties.

For any fluid:

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

where $K$ is the bulk modulus and $\rho$ is the density.

For an ideal gas, this simplifies to:

$a = \sqrt{\gamma R T}$

where $\gamma$ is the ratio of specific heats (1.4 for air), $R$ is the specific gas constant (287 J/(kg·K) for air), and $T$ is the absolute temperature in Kelvin. This equation shows directly why temperature is the controlling variable for the speed of sound in air.

Factors affecting measurement accuracy

• Temperature control: Since $a$ depends strongly on temperature, even small temperature gradients in the test section can introduce errors. Accurate thermometry is essential.
• Humidity: Needs to be measured and accounted for when working in air, though the effect is small (a few tenths of a percent).
• Frequency selection: Higher-frequency sound waves provide better spatial resolution but may experience more attenuation. The measurement frequency should be chosen to balance resolution and signal strength.
• Boundary effects: Walls, surfaces, and reflections can interfere with measurements. Proper acoustic treatment and careful transducer placement help minimize these effects.

Applications of speed of sound

The speed of sound and the Mach number framework underpin the design and analysis of any system where compressibility matters, from aircraft to medical devices.

Aircraft design considerations

The speed of sound directly determines what aerodynamic challenges a given aircraft will face:

• Subsonic aircraft (e.g., general aviation, turboprops) operate well below Mach 1. Design focuses on maximizing lift-to-drag ratio using conventional airfoil shapes.
• Transonic aircraft (e.g., commercial jets cruising at Mach 0.80–0.85) use supercritical airfoils with flattened upper surfaces to delay shock formation and reduce drag divergence. Swept wings also help by reducing the effective Mach number seen by the wing.
• Supersonic aircraft (e.g., fighter jets, the retired Concorde) use thin, highly swept or delta wings and sharp leading edges to minimize wave drag. Area ruling (shaping the fuselage to smooth out the cross-sectional area distribution) is another key technique.

Supersonic flight challenges

Flying faster than the speed of sound introduces problems that don't exist in subsonic flight:

• Wave drag: Shock waves extract energy from the flow, creating a drag component that increases with Mach number. Minimizing wave drag requires careful shaping of the entire vehicle.
• Sonic boom: The conical shock wave system trailing a supersonic aircraft reaches the ground as a sudden pressure pulse heard as a loud boom. This has historically restricted supersonic commercial flight over land.
• Aerodynamic heating: At Mach 2+, skin temperatures rise significantly. The Concorde's fuselage stretched by ~15 cm in flight due to thermal expansion. Higher Mach numbers demand specialized materials like titanium alloys or ceramic composites.

Sonic booms and their impact

A sonic boom is not a one-time event that happens when an aircraft "breaks the sound barrier." It's a continuous phenomenon: the shock cone trails behind the aircraft for as long as it flies supersonically, sweeping across the ground below.

The boom's intensity depends on the aircraft's size, shape, altitude, and Mach number. Sonic booms can rattle windows, startle people and animals, and in extreme cases cause minor structural damage. These effects led to bans on supersonic overland flight in many countries.

Current research (e.g., NASA's X-59 QueSST) focuses on shaping aircraft to produce a softer "thump" instead of a sharp boom, potentially reopening the door to supersonic commercial travel over land.

Speed of sound in medical imaging

Ultrasound-based medical imaging relies on knowing the speed of sound in biological tissues (approximately 1,540 m/s in soft tissue).

• Ultrasound imaging: A transducer emits sound pulses into the body and listens for echoes. The system uses the assumed speed of sound to convert echo return times into distances, building up a 2D or 3D image.
• Elastography: Measures how fast shear waves travel through tissue. Stiffer tissues (like tumors) transmit waves faster, so mapping the local wave speed reveals tissue stiffness, aiding diagnosis of conditions like liver fibrosis.
• Photoacoustic imaging: Short laser pulses are absorbed by tissue, causing rapid thermal expansion that generates ultrasound waves. Reconstructing images from these waves requires accurate knowledge of the speed of sound in the tissue.