Copyright © A. Filippone (1996-2001). All Rights Reserved.

High Speed Aerodynamics

Wave Propagation


Waves discussed in this section are limited to those aspects involving aeroacoustics and propagation of shocks associated with high-speed flows.

High speed flows include explosions and detonations (blast waves). In the physics of fluids other peculiar aspects can be traced in hydrodynamics (surface waves, waves in shallow waters, tidal waves, splashes, solitons, etc.).

Acoustic Waves and Shock Waves

Shock waves are equivalent to acoustic waves in the limit of vanishing strength. For this reason (and others) acoustic fields are generally regarded as small perturbations, whereas the shock waves, even when they are weak, are finite discontinuities in the flow domain. The figure below shows the progression of shock wave pattern around a biconvex airfoil.

Biconvex Airfoil at Transonic Speeds

Figure 1: progression of shocks on biconvex airfoil

Bow Shock on Supersonic Wedge Airfoil

Figure 2: Bow shock on supersonic wedge airfoil, M = 1.7


One-dimensional acoustic waves (plane or spheric) represent the simplest case of propagation. These waves are described by one hyperbolic equation, whose solution yields a forward and a regressive wave. The speed of both waves is the speed of sound. In acoustics the wave equation is always used. This equation, also hyperbolic, is based on the assumption of inviscid flow.

Characteristic Lines

Two-dimensional and axi-symmetric problems can be solved by searching characteristic invariants in the field, which is relatively easy when the equations are linearized (potential flow). In steady supersonic flow these invariants are two families of lines called characteristic lines (example shown in Fig. 1).

Speed of Sound

Radiation of waves from a source occur at a characteristic speed (speed of sound), that is a function of the thermodynamic properties of the air. The speed of sound in the air considered as an ideal gas is proportional to the square root of the absolute temperature. At very high temperatures and pressures, where the ideal gas approximation does not hold, corrections must be introduced.

Transport of Energy and Momentum

All modes of propagation transport momentum and energy at a characteristic speed (the speed of sound). Quantities defining the propagation characteristics are the energy flux, sound intensity and radiation pressure.

Of particular interest are the impulse of a shock wave on the ground (sonic boom) and the energy radiated by a detonation.

Propagation of Waves in a Duct

This case is typical of high speed vehicles (especially trains, subways) in a tunnel, but it is also important as a laboratory instrument, since it is used to produce supersonic flows in blowdown wind tunnels (shock tube), for internal ballistic problems (flows in cannons and pistons of internal combustion engines), and for special purposes in physics and chemistry.

Acoustic Waves

Wave propagation is limited by the walls, and is generally one-dimensional (except for those cases in which the walls interact elastically with the waves).

Acoustic waves propagate at a constant speed, and the flow of acoustic energy is independent of the duct section (contrary to the progagation of a blast). Propagation in a duct of variable cross-section is somewhat different.

Shock Waves

Physics and classification of the shock waves is discussed in the compressible flow chapter. Waves can travel with a pulse velocity v in a constant duct at subsonic, sonic and supersonic speed, depending on the value of v = u ± a (u = speed of the perturbation; a=speed of sound). When the propagation is supersonic, disturbances can propagate only downstream (law of the forbidden signals).

Propagation of Waves in Unbounded Field

Unbounded fields are a more general medium where the propagation front can assume spherical, planar or conical shape. The occurrence of either of these events is shown schematically in Fig. 2 and Fig. 3.

Propagation Front

An impulsive source of sound at rest travels the distance T = 2 f (f = frequency) in the unit of time. The volume involved by the sound is therefore spheric and of the order of T^3. The same source travelling at subsonic speed u < a after the time T will be close to its front end, Fig. 1.

At sonic conditions the source will be coincident with the front end of the wave. The volume involved is the infinite space behind the source.

At supersonic speed the source will always be ahead of its sound. The waves radiate from the source along characteristic lines making an angle=acos (1/M). In three dimensions these lines make the so called Mach cone (Mach, 1887). The region outside the cone is the zone of silence.

Acoustic Waves

Acoustic waves that are radiated at low subsonic speeds have a nearly spherical wave front. The deviation from spherical propagation is a Doppler effect. The energy radiated in the space and attenuate as 1/r (the inverse of the radius).

Shock Waves

Shock waves propagating from a supersonic point source are shown in Fig. 3. Shock waves created by supersonic bodies with a blunt nose travel ahead of the body and have a characteristic shape resembling a hyperboloid of revolution (bow shock).

Detonation Waves

Fast chemical reactions of the detonation-type are characterized by a highly exothermic process in a stationary medium (air, in our case). The propagation front is spheric, the speed can be subsonic, sonic, or supersonic. Detonations with front advancing at sonic speeds are the most common (Chapman-Jouguet detonation). Slower detonations are called deflagrations.

Subsonic Propagation

Figure 2: Subsonic source radiation

Mach Cone

Figure 3: Supersonic source radiation (Mach cone)

Wave Effects in Bounded Mediums

The most notable effects on wave propagation in bounded mediums is the refraction, reflection and interaction of the waves. A fairly large amount of cases can occur:

Shock interaction

Reflection from a wall or closed end; intersection of shocks of opposite families; intersection of shocks is the same family; shock incident on contact surface, etc.

Shock reflection from rigid boundary is particularly interesting for transonic wind tunnel testing, as well as for the analysis of integrated systems at high speeds (cascades, many components of high speed turbomachinery, etc.)

Related Material

Selected References

  • Shapiro AH. The Dynamics and Thermodynamics of Compressible Fluid Flow, Pergamon Press, 1953.

  • Thompson PA, Compressible Fluid Dynamics, McGraw-Hill, 1972.

  • Whitham GB. Linear and Non Linear Waves, John Wiley, 1974.

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Copyright © A. Filippone (1996-2001). All Rights Reserved.