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2.2 Geometrical acoustics

At the level of geometrical acoustics we need only assume that:

Then, relative to the laboratory, the velocity of a sound ray propagating, with respect to the fluid, along the direction defined by the unit vector n is

dx- dt = c n + v. (1)
This defines a sound cone in spacetime given by the condition 2 n = 1, i.e.,
- c2 dt2 + (dx - vdt)2 = 0 . (2)
That is
2 2 2 - [c - v ] dt - 2v .dx dt + dx .dx = 0 . (3)
View Image

Figure 4: A moving fluid will tip the “sound cones” as it moves. Supersonic flow will tip the sound cones past the vertical.
Solving this quadratic equation for dx as a function of dt provides a double cone associated with each point in space and time. This is associated with a conformal class of Lorentzian metrics [376Jump To The Next Citation Point387Jump To The Next Citation Point391Jump To The Next Citation Point389Jump To The Next Citation Point284Jump To The Next Citation Point]
[ 2 2 | T ] g = _O_2 --(c----v-)|-v--- , (4) - v | I
where _O_ is an unspecified but non-vanishing function.

The virtues of the geometric approach are its extreme simplicity and the fact that the basic structure is dimension-independent. Moreover this logic rapidly (and relatively easily) generalises to more complicated physical situations.3


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