Wave Propagation
Summary
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.).
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.
Figure 1: progression of shocks on biconvex 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.
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).
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.
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.
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.
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.
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).
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.
An impulsive source of sound at rest travels the distance T = 2
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 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 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).
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.
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:
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.)
Acoustic Waves and Shock Waves
Mathematics
Characteristic Lines
Speed of Sound
Transport of Energy and Momentum
Propagation of Waves in a Duct
Acoustic Waves
Shock Waves
Propagation of Waves in Unbounded Field
Propagation Front
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.
Acoustic Waves
Shock Waves
Detonation Waves
Wave Effects in Bounded Mediums
Shock interaction
Related Material
Selected References