Page 41 - Volume 9, Issue 3

Fig. 1. A typical impulse response in a duct with initially bunched up arrivals (left) spreading out on the right. The groups of four, two-up-two-down, correspond to up/down paths at source and receiver with almost the same ray angle.
how. This will lead to the correct answer eventually but it may be much more heavy-going. Inevitably the solution is in the form of a Green’s function in the frequency domain because that makes the differential equations easier to solve. Then in a ducted medium there are various choices of solu- tion, for instance the Green’s function can be expressed as a sum of vertical normal modes. However, if you want the impulse response, or to be precise, the waveform as a func- tion of time, either you need to work out the group velocities for each mode then add the modes according to their arrivals, or you need to calculate the Green’s function and modes for all frequencies and then Fourier transform them to time. As pointed out by David Weston (1971) there is a duality between rays and modes – if you include a large enough number of either rays or modes you will get the cor- rect answer, but in any particular problem there may be a dis- tinct computational advantage in choosing one or the other since a single ray is equivalent to many modes, and a single mode is equivalent to many rays. For instance, at low fre- quency and very long range there may be only a small num- ber of modes but a very large number of rays [they may even be chaotic (Smith, et al, 1992)], so one would choose modes in that case.
The original work on the waveguide invariant (Chuprov, 1982) and most subsequent work (e.g. D’Spain and Kuperman, 1999; Brown, Beron-Vera, Rypina, and Udovydchenkov, 2005) took a frequency domain approach, using wave models of ducted sound propagation. The defini- tion of the waveguide invariant, known as beta, in terms of angular frequency ω and range r is
(2)
which is also the slope of d(logω)/d(logr). It is a property of, above all, the sound speed profile, and if this remains con- stant then b is a constant, i.e. invariant.
Lloyd’s mirror fringes have been the basis of passive rang- ing for some time, and more generally striations are of interest because they can be measured with a single hydrophone. In recent years b has been considered as part of the toolset in geoacoustic inversion (Heaney, 2004) and has been applied to the detection of targets and reverberation estimation (Goldhahn, et al., 2008) and active sonar (Quijano, et al., 2008). It has also been tied into such topics as time reversal focusing (Kim, et al., 2003) and beam processing (Yang, 2003).
The waveguide invariant in a constant velocity duct
We can try to sweep away some of the mystique by using the impulse response crudely derived above but written more generally in Harrison (2011) and taking Fourier transforms. We can assume that at each n in Eq. (1), essentially each angle, the impulse has a slowly varying strength an determined by reflec- tion coefficient, and so on, so the Fourier transform is
(3)
Although this is potentially a complicated function, by the time we have substituted for tn,μ,v using Eq. (1) we see that the exponent is explicitly a function of (ω / r), so no matter what its dependence on n is or its functional form it has only one shape as ω and r vary. At any given r there will be a fringe pat- tern in ω, but moving to a different value of r we find the same pattern but stretched in ω in proportion to the increase in r. This automatically constructs a fringe pattern where the modulation takes a constant value along lines where ω ∝ r. In other words the condition for a fringe is that
(4)
Taking logs and differentiating we find this obeys Eq. (2) with b = 1, as is well known for this isovelocity case. The result is an interference pattern that looks like that shown in Fig. 2.
40 Acoustics Today, July 2013
Fig. 2. An interference pattern (“striations”) in water of 100m depth with constant sound speed. The waveguide invariant b = 1.