Introduction to Acoustics

along rays. These can be regarded as lines or paths in
space.
3.16.1 Wavefront Propagation
A wavefront is a hypothetical surface in space over
which distinct waveform features are simultaneously received.
The theory of plane wave propagation predicts
that wavefronts move locally with speed c when viewed
in a coordinate system in which the ambient medium
appears at rest. If the ambient medium is moving with
velocity v, the wave velocity cn seen by someone moving
with the fluid becomes v +cn in a coordinate system
at rest. Here n is the unit vector normal to the wavefront;
it coincides with the direction of propagation if the coordinate
system is moving with the local ambient fluid
velocity v. A ray can be defined [3.94] as the time trajectory
of a point that is always on the same wavefront,
and for which the velocity is
vray = v + nc . (3.608)
To determine ray paths without explicit knowledge of
wavefronts, it is appropriate to consider a function τ(x),
which gives the time at which the wavefront of interest
passes the point x. Its gradient s is termed the wave
slowness vector and is related to the local wavefront
x
n(x)
τ(x) = t1
τ(x) = t1 + ∆t
Fig. 3.51 Sketch illustrating the concept of a wavefront as
a surface along which characteristic waveform features are
simultaneously received. The time a given waveform passes
a point x is τ(x) and the unit normal in the direction of
propagation is n(x)
Basic Linear Acoustics 3.16 Ray Acoustics 95
Listener
Shadow
zone
Fig. 3.52 Multipaths and shadow zones. The situation depicted
is when the sound speed decreases with distance
above the ground so that rays are refracted upward
normal by the equations
s = n
, (3.609)
c + v·n
n = cs
, (3.610)
Ω
where the quantity Ω is defined by
Ω = 1 − v·s . (3.611)
Given the concepts of ray position and the slowness
vector, a purely kinematical derivation leads to the raytracing
equations
dx
dt = c2s + v , (3.612)
Ω
ds
dt =−Ω∇c
− (s·∇)v − s×(∇×v) . (3.613)
c
Simpler versions result when there is no ambient flow,
or when the ambient variables vary with only one coordinate.
If the ambient variables are independent of
position, then the ray paths are straight lines.
The above equations are often used for analysis of
propagation through inhomogeneous media (moving or
nonmoving) when the ambient variables do not vary
significantly over distances of the order of a wavelength,
even though they may do so over the total propagation
distance. The rays that connect the source and listener
locations are termed the eigenrays for that situation.
If there is more than one eigenray, one has multipath
reception. If there is no eigenray, then the listener is in
a shadow zone.
3.16.2 Reflected and Diffracted Rays
This theory is readily extended to take into account solid
boundaries and interfaces. When an incident ray strikes
a solid boundary, a reflected ray is generated whose
Part A 3.16