Page 42 - Volume 9, Issue 3

Why don’t you see blobby interference patterns?
Why does the interference pattern in underwater acoustics always consist of parallel fringes (or striations) instead of just featureless blobs? The glib answer is, well, we’ve just proved it. But why? Think of other interference patterns such as directional gravity waves on the sea or nor- mal modes on a drumskin or a bell – these often have fea- tures that, though not necessarily randomly blobby, are cer- tainly not composed of parallel lines. In the ideal underwater case the source images are not just anywhere in the vertical plane, as they might be in room acoustics, but instead they form a relatively concentrated vertical array or grating sub- tending quite a small angle at the receiver. Therefore the streakiness of the fringes is just a manifestation of the dif- fraction pattern of a distant grating. On the other hand, sup- pose this were not an ideal case, for instance the sound speed might change spatially (or temporally) in an erratic manner, or there might be rough or wavy boundaries. Then the pat- tern is likely to break up, i.e. smooth out or tend to blobbi- ness. Perhaps this absence of a fringe pattern might also be a useful tool. The effects of internal waves on the stability of striation patterns has been studied by Rouseff (2001).
Some useful relationships
There are many other refracting sound speed profiles for which beta can be calculated analytically (see Harrison, 2011). To calculate them using this time domain approach we can make use of some rather surprising general relationships between modal quantities and ray quantities. In a sense these are all different interpretations of the same quantity T.
(5)
Going through this term by term, the quantity T was referred to by Weston (1959) as a “characteristic” time. The normal modes propagate with horizontal wavenumber K and the modes themselves are solutions to the vertical Helmholtz equation with a vertical wavelength [k(z)2 – k2]1/2, and the WKB solution is a good approximation to the mode shape.
The second term is known as the “WKB phase integral”, the phase across the waveguide (multiplied by a factor 2/ω), and the third term is the condition that these modes fit into the waveguide, the integer n being the mode number – essen- tially a vertical resonance phenomenon. The characteristic time T is then the time equivalent of this phase.
Since locally the ray angle is given by cosθ = k/K the sec- ond term can also be written as the fourth in terms of angle and sound speed. This quantity can be thought of as a ray invariant – it is useful for determining ray paths as they progress in range-dependent media, for instance if the water becomes shallower then the rays must become steeper in more or less the same proportion.
In a stratified or range-independent medium a ray start- ing at a particular angle cycles back and forth between upper
and lower extrema, which may be refraction turning points or surface or bottom reflections, and it obeys Snell’s law, which is equivalent to the horizontal wavenumber K being independent of depth. By considering the travel time along ray elements between extrema one arrives at the fifth term which relates T to the horizontal length of one complete ray cycle rc and the corresponding travel time tc. Noting that the group velocity U is just rc /tc since this is the speed that infor- mation travels along the waveguide, and the phase velocity V is by definition ω / K, we arrive at the last term (Harrison, 2012). This can be thought of as rc times the difference between the group slowness and the phase slowness.
The waveguide invariant in a refracting duct
We can extend the argument that led to Eq. (4) to cover refracting environments. Equation (3) is still valid but the time delay tn is more complicated and it may even decrease with n (since the steeper rays tend to propagate mainly through the faster layers). To see fringes at all (i.e. sets of par- allel lines rather than blobs) it must be possible to write tn as the product of a function of range only and a function of all the other parameters (e.g. n, H, zs etc.). In other words in the exponent of Eq. (3) the range dependence must be separable. This ensures that in going from one range to the next the fringes may shrink or stretch slightly but always retain their shape. That property forms the striation pattern, and without it there can be no fringes.
The image arrival delay can be written in terms of (tc – rcK/ω) (which by coincidence is the “characteristic” time T ) as
(6)
having dropped the source/receiver subscripts since we’re inter-
ested only in the separation of the groups of four images.
Although the right hand side is of the form n G(rc) this alone
(in combination with r = n×rc) does not ensure that delay is a
separable function of range r. The only function G that allows
separation is G(r ) = g (r )q (where g and q are constants), since cc
G(r)=G(r/n)=grq×n–q.Sotoseefringeswemusthave c
(7)
and the exponent in Eq. (3) must be iωrq × (n(1-q)g). By taking logs of the latter and differentiating we find behaviour exact- lyasinEq.(2)withb=–q.DoingthesametoEq.(7)wefind a relation for b in terms of ray cycle distance and travel time
(8)
It is now straightforward to plug in any sound speed pro- file, calculate rc, tc, K and calculate b, but straightaway we can see that the sign of b just depends on the dependence of cycle distance on angle. If cycle distance increases with angle b is negative, and vice versa.
Refracting examples
Using a standard propagation model (ORCA) run at many frequencies it is straightforward to see the interference
The Waveguide Invariant 41