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-= cn + v. (1 ) dt
This defines a sound cone in spacetime given by the condition n2 = 1, i.e.,
− c2dt2 + (dx − vdt)2 = 0. (2 )
That is
2 2 2 − [c − v ]dt − 2v ⋅ dxdt + 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 [607Jump To The Next Citation Point, 622Jump To The Next Citation Point, 626Jump To The Next Citation Point, 624Jump To The Next Citation Point, 470Jump To The Next Citation Point]

[ | ] 2 −-(c2 −-v2) −-vT g = Ω − v | I , (4 )
where Ω 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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