Page 43 - Volume 9, Issue 3

Fig. 3. An interference pattern with a uniform upward refracting profile in water of 100m depth. The waveguide invariant b = -3, and the striations bend in the oppo- site direction from those in Fig. 1.
fringes regardless of their shape. Figure 3 shows an example for the uniform sound speed gradient case where a and co are constants
(9)
Using standard analytical techniques the cycle time and dis- tance formulas are
izontal dashed line shows the depth of the receiver used in the main plot.
Finally there is an interesting case of a parametrised pro- file that can display more or less all the properties of the oth- ers by altering a single parameter p.
(15) (Actually, a very similar solution is also available for c2(z) =
c 2/(1-(az)p).) This can be imagined as a one-sided function o
with 0 < z < ∞, in which case z = 0 represents a reflecting sur- face, or alternatively it can be imagined as a fully refracting, two-sided function with – ∞ < z < ∞. Now the analytical solution for b is
(16)
Fig. 4. Vertical fringes for the cosh profile where there is perfect focusing (as shown by the superimposed ray trace) which results in the ray cycle distance being inde- pendent of angle so that b = ∞. A raytrace is superimposed (blue).
and
(10)
(11)
where θo is measured at the low sound speed boundary. Here b = -3 and the fringes can be seen to tilt in the opposite direc- tion to those in Fig. 1.
Another important case is the “cosh” profile
which results in perfect, repeated focusing with
and
(12)
(13)
(14)
By definition these perfect focuses mean that the cycle dis- tance is independent of angle. In the context of interference fringes it sits on the dividing line between positive and nega- tive values of b, and it is pathological since b is infinite, and the fringes, as shown in Fig. 4, are vertical, i.e. independent of frequency. Superimposed on this plot is a ray diagram showing the repeated upper and lower focus points. The hor-
42 Acoustics Today, July 2013
Fig. 5. Normalised sound speed profiles c2 = co2(1+(az)p) for various values of the parameter p for which solutions are available. These can be regarded as one-sided functions (with zero depth representing a reflecting surface), or alternatively as fully refracting two-sided functions.