Supersonic Motion, Mach Cones, and Shock Waves

Supersonic Motion, Mach Cones, and Shock Waves

M06-14 showed that the ordinary moving-source Doppler formula becomes singular as a source speed approaches the wave speed from below.

That divergence is not an infinite physical frequency. It signals a change in wavefront geometry.

The key dimensionless ratio is the Mach number,

(1)

Here V is the source or flow speed relative to the medium and c is the local small-disturbance sound speed.

The basic regimes are

(2)

For steady uniform supersonic motion, the small-disturbance Mach angle μ satisfies

(3)

This article derives that geometry and then develops the finite-amplitude physics of shock waves.

1 Mach number as a comparison of time scales

Suppose a body of characteristic length L moves at speed V .

The flow transit time is

(4)

A sound disturbance crosses the same distance in

(5)

Therefore

(6)

For M < 1, acoustic information can cross the characteristic distance before the body does.

For M > 1, the source can cross that distance faster than a small disturbance can propagate through the medium.

2 Wavefront construction for a moving source

Let a point source move at constant speed V through a stationary uniform medium.

Every small disturbance expands at speed c relative to the medium.

A disturbance emitted at time zero has radius

(7)

at later time t, while the source has moved

(8)

The ratio of those distances is the Mach number.

3 Subsonic motion

For

(9)

we have

(10)

The source remains inside each earlier disturbance sphere.

Signals can therefore propagate ahead of the source.

This is the geometric basis for upstream communication in subsonic flow.

4 Sonic motion

For

(11)

the source lies on the forward edge of each earlier disturbance sphere.

Forward wavefront spacing collapses.

This is the geometric limit behind the singular behavior of the moving-source Doppler formula.

5 Supersonic motion

For

(12)

the source lies outside each earlier disturbance sphere.

The envelope tangent to the family of spheres forms a cone in three dimensions.

That envelope is the Mach cone.

6 Derivation of the Mach angle

Suppose the source travels from point O to point S in time t.

Then

(13)

The disturbance emitted at O has radius

(14)

The tangent from S to that sphere produces a right triangle. Therefore

(15)

Hence

(16)

or

(17)

7 Example 1: Mach angle at Mach 2

For

(18)

(19)

Therefore

(20)

8 Example 2: Mach angle at Mach 3

For

(21)

(22)

Thus

(23)

9 High-Mach-number limit

At the sonic limit,

(24)

the Mach angle approaches

(25)

At large Mach number, the angle is small.

Using

(26)

for small μ in radians,

(27)

10 Mach wave versus shock wave

A Mach cone is first a statement about the geometry of infinitesimal disturbances.

For an infinitesimally weak disturbance, the cone is made of Mach waves.

A finite disturbance can instead produce a shock wave.

The distinction is:

• a Mach wave is the small-disturbance limit,
• a shock contains finite jumps in pressure, density, temperature, and velocity,
• a physical shock produces entropy.

For a weak shock, the shock angle approaches the Mach angle.

For stronger compression, the shock angle is generally larger than μ.

Thus the finite shock angle is not, in general, identical to the Mach angle.

11 Why finite-amplitude compression waves steepen

Linear acoustics assumes every part of a waveform propagates with the same speed c.

Finite-amplitude compression waves are nonlinear.

For a weak right-moving simple wave in an ideal gas, the local characteristic speed is approximately

(28)

where c₀ is the undisturbed sound speed, u is local particle velocity, and γ is the ratio of specific heats.

A stronger compression therefore propagates faster than a weaker compression ahead of it.

The rear of the wave catches the front.

The profile steepens.

When the gradient becomes sufficiently large, viscosity and thermal conduction become important in a thin layer.

That layer is a shock.

12 Shock-fixed reference frame

A shock is easiest to analyze in a frame where the shock surface is stationary.

Let region 1 be upstream and region 2 downstream.

Use:

• ρ₁, p₁, u₁, h₁ upstream,
• ρ₂, p₂, u₂, h₂ downstream.

The velocities are Normal to the shock in the shock-fixed frame.

For a steady adiabatic shock with no shaft work, conservation of mass, momentum, and total energy connects the two states.

13 Mass conservation

For unit area,

(29)

The common mass flux is

(30)

A compressive shock has

(31)

Therefore

(32)

14 Momentum conservation

Normal momentum flux is

(33)

Thus

(34)

Since the flow slows through a compressive shock, static pressure rises:

(35)

15 Energy conservation

For steady adiabatic flow with no shaft work,

(36)

Bulk kinetic energy per unit mass decreases while thermal enthalpy increases.

16 Rankine-Hugoniot conditions

The three jump relations

(37)

(38)

and

(39)

are the one-dimensional Rankine-Hugoniot conditions.

They express conservation.

A separate entropy condition selects the physically admissible compressive shock.

17 Entropy condition

A physical adiabatic shock is irreversible.

Therefore

(40)

Across a normal compressive shock:

• static pressure increases,
• density increases,
• temperature increases,
• normal flow speed decreases,
• entropy increases.

18 Upstream and downstream Mach numbers

Define

(41)

For an ideal-gas normal shock,

(42)

A normal shock converts supersonic normal flow into subsonic normal flow.

19 Ideal-gas density ratio

For a calorically perfect ideal gas,

(43)

As

(44)

the ratio approaches one.

For a strong shock,

(45)

(46)

For air with

(47)

the strong-shock density ratio approaches

(48)

20 Ideal-gas pressure ratio

The normal-shock pressure ratio is

(49)

Equivalently,

(50)

21 Temperature ratio

Using the ideal-gas law,

(51)

(52)

22 Downstream Mach number

For a normal shock,

(53)

23 Example 3: normal shock at Mach 2

Take

(54)

The density ratio is

	=	(55)
	= 2.667.	(56)

The pressure ratio is

	= 1 + (4 − 1)	(57)
	= 4.50.	(58)

Therefore

(59)

For the downstream Mach number,

M22	=	(60)
	= .	(61)

Hence

(62)

24 Shock thickness

The jump equations model the shock as a discontinuity.

A real shock has small but finite thickness.

Within the layer:

• viscous stresses matter,
• heat conduction matters,
• strong gradients generate entropy.

These transport processes regularize the nonlinear steepening.

At macroscopic scales, the layer is often thin enough that the discontinuity approximation is excellent.

25 Stagnation quantities

Across an adiabatic shock with no shaft work, total enthalpy is conserved.

For a calorically perfect ideal gas,

(63)

But the shock is irreversible, so stagnation pressure decreases:

(64)

The stagnation-pressure loss measures mechanical irreversibility.

26 Sonic-boom geometry

A supersonic aircraft continuously carries a shock system.

Consider ideal horizontal flight at altitude h.

Let the Mach angle be μ.

For an observer directly below the flight path, the shock cone reaches the observer when the aircraft is horizontally ahead by distance x.

The right triangle gives

(65)

Therefore

(66)

The ideal delay after overhead passage is

(67)

27 Example 4: Mach 2 sonic-boom delay

Take

(68)

(69)

and

(70)

Then

(71)

and

(72)

The aircraft is ahead by

x	= h cot 30∘	(73)
	≈ 17.32 km.	(74)

The ideal delay is

Δt	=	(75)
	≈ 25.25 s.	(76)

28 A sonic boom is not only a Mach-one crossing event

A common misconception is that a sonic boom occurs only at the instant an aircraft first crosses Mach one.

A continuously supersonic aircraft continuously carries a shock system.

Different observers hear the boom when that shock system passes them.

The relevant event is shock passage, not merely the earlier acceleration through the sonic threshold.

29 N-wave signature

Far from a slender supersonic aircraft, the pressure history often resembles an N-shaped waveform.

The observer can experience:

• a rapid positive pressure jump,
• a slower pressure evolution,
• a later rapid pressure change from the rear shock system.

The exact signature depends on aircraft geometry, atmospheric structure, propagation distance, and nonlinear evolution.

30 Oblique shocks

A normal shock is perpendicular to the incoming flow.

Many supersonic flows contain oblique shocks.

If β is the shock angle measured from the upstream flow direction, the upstream normal Mach number is

(77)

Normal-shock relations apply to the normal velocity component.

In the ideal inviscid model, the tangential velocity component is unchanged across the shock.

For a weak oblique shock, the shock angle approaches the Mach angle.

31 Flow deflection and shock angle

For a calorically perfect ideal gas, the flow deflection angle 𝜃 and shock angle β satisfy

(78)

For a given supersonic Mach number and modest compression angle, weak- and strong-shock branches can exist.

If the required turning is too large, an attached shock is impossible and the shock detaches.

32 Bow shocks

A blunt body generally produces a detached bow shock.

Near the stagnation line the shock is nearly normal.

Farther away it becomes increasingly oblique.

Examples include:

• blunt supersonic vehicles,
• atmospheric entry capsules,
• supersonic flow around spheres,
• astrophysical bow shocks.

33 Information propagation in supersonic flow

The Mach-cone picture has a causal interpretation.

In a uniform supersonic stream, ordinary small disturbances are swept downstream.

They cannot propagate directly upstream against the flow.

This gives supersonic flow a different information structure from subsonic flow.

The geometry of characteristics determines which regions can influence one another.

34 Common mistakes

1. Treating Mach number as a speed rather than the ratio V∕c.
2. Using the Mach-angle relation for M < 1.
3. Writing cos μ = 1∕M instead of sin μ = 1∕M.
4. Assuming a larger Mach number gives a wider Mach cone.
5. Calling every Mach wave a finite shock.
6. Assuming finite shock angle always equals Mach angle.
7. Extending linear acoustics through a finite shock without jump conditions.
8. Forgetting that density increases and normal velocity decreases through a compressive normal shock.
9. Forgetting that entropy increases through a physical shock.
10. Assuming stagnation pressure is conserved across an irreversible shock.
11. Confusing stagnation temperature with static temperature.
12. Applying normal-shock formulas to the total velocity of an oblique shock instead of its normal component.
13. Thinking a sonic boom occurs only once when the aircraft first crosses Mach one.
14. Ignoring altitude and Mach angle in sonic-boom timing.
15. Treating a detached bow shock as an attached Mach line.

35 Practice exercises

1. Derive the Mach-angle relation from the wavefront-envelope construction.
2. Calculate Mach angles for M = 1.2, 2, 3, and 5.
3. Show that μ ≈ 1∕M radians for M ≫ 1.
4. Explain geometrically why upstream acoustic influence is possible in subsonic flow but not in a uniform supersonic stream.
5. Explain the distinction between a Mach wave and a finite shock wave.
6. Use the nonlinear characteristic-speed relation to explain waveform steepening.
7. Derive the mass and momentum jump conditions from a shock-fixed control volume.
8. State the energy jump condition and explain its physical meaning.
9. Reproduce the Mach-2 normal-shock density, pressure, temperature, and downstream Mach-number ratios.
10. Evaluate the strong-shock density-ratio limit for air.
11. Derive x = h cot μ for the ideal sonic-boom geometry.
12. Reproduce the Mach-2, 10 km altitude sonic-boom delay example.
13. Explain why a steadily supersonic aircraft produces a continuing shock system.
14. For an oblique shock, identify the normal Mach number and explain which velocity component remains unchanged.
15. Explain why a sufficiently blunt body produces a detached bow shock.

36 Summary

The Mach number is

(79)

For supersonic small-disturbance motion,

(80)

A finite shock obeys mass conservation,

(81)

momentum conservation,

(82)

and total-enthalpy conservation,

(83)

For a physical adiabatic shock,

(84)

For an ideal gas,

(85)

(86)

and

(87)

Mach cones describe characteristic wave geometry.

Shock waves describe finite irreversible jumps.

Their connection is where linear waves meet nonlinear compressible flow.

References

References