The inhomogeneity is represented by the sound speed profile, which depends on one coordinate, namely. We show that our mathematical model of the speed of sound in the ocean actually predicts some of the behavior of the observed physical phenomena in the underwater sound channel. The inhomogeneity is represented by the sound speed profile, which depends on one coordinate, namely the depth.
Below the sound speed minimum, the primary effect is due to the increase of the hydrostatic pressure with depth; thus the speed of sound increases with depth. Rays emanating from a source located at the depth of minimum sound speed (called the channel axis) are refracted toward the axis as illustrated in Figure 2-1. The path of these rays is determined only by the structure of the sound speed profile in the region of the minimum.
We will study the situation in an inhomogeneous medium 1n by which the sound speed c (z) is given. So we have an axisymmetric situation in which the sound speed changes only with depth.
We will show that for the Epstein profile (2.1) off-axis arrivals precede axial arrivals. We will also show that at small starting angles, the first rays that deviate the most from the axis arrive. For obvious reasons, the travel time is for "direct arrival". a ray traveling exclusively along the axis) is R/c0.
Now the duration of the signal is the time interval between the arrival of the beam going through one cycle (N = l) and the direct arrival, thus. More precisely, we will show that paraxial (ie small angle) rays converge periodically along the channel axis. Later in Part II we will show ~ using normal mode theory that we obtain a resonant effect along the axis in exactly the same positions.
In general, the equation for ray paths is specified by. therefore, those values of r are expected to be focused on, so that. where the envelope of the ray family crosses the axis). To gain a better qualitative understanding of our sound speed profile, we will now look at the full beam image. Finally, we will show which class of profiles leads to wave propagation in the absence of shadow zones.
The result of the above analysis is illustrated in Figure 5-1, and we can see that there is no geometric shading zone.
PART II
In particular, we verify some of the previous results derived from beam theory in Part I, this time of course by giving them using a more quantitative formulation. In principle, the solution for a general time dependence is obtained simply by taking a Fourier transform of these previously derived elutions. However, such an integral representation does not provide even a superficial understanding of the properties of the solution, and indeed the forms generally given for the time-harmonic solution are incomplete in the sense that not enough is known for an investigation of a Fourier integral of the representation .
We present a complete derivation of the solution, carefully highlighting certain relevant details needed in our subsequent analysis. Therefore, using (2.14) and inverting the Hankel and Fourier transforms, we conclude that this is the solution to our boundary value problem. Although (2.15) provides an exact answer to the problem, in its current form it does not lend itself to any form of analysis for extracting properties of the solution.
First, it is advantageous to formally exchange the orders of integration, so we first consider the integral Q given by. This, together with our earlier requirement that Re{f.t} > 0 implies that we cut branches as illustrated in Figure 2-1. I am{£} ~ 0; and as shown in Appendix C, the poles of the Gamma functions on the real axis lie in the £-plane between -v and v and are in fact given by the solutions £ of.
At this point it is convenient to restrict ourselves to the case where z = 0, i.e. both source and receiver on the channel axis)'. There will also be branch cuts from the requirement that Re{f-l} > 0, where f-1 is defined in (2.17), and clearly, here the position of the cuts depends on the values. For ~ on plL the requirement that Re{f-l} > 0 means that we take branch sections as illustrated in figure 2-3.
Moreover, for ~ on plL' the only poles in the v-plane are at that v for which r(f-1+v) is singular. So by deforming our contour to the upper half of the complex v-plane, as shown in Figure 2.3, we proceed in the usual way to obtain this. It is well known (see [ 4] , pp,466) that in the limit like r-oo the contribution of the integrals in (2.23) is of a smaller order of magnitude than that from the series.
In order to determine the properties of the far field, we will asymptotically evaluate the integrals I (r,t) given by (2.32). In this section, we will be primarily interested in the time history of the acoustic pressure p (r,O,t) at any fixed point in the far field. Over time (still satisfying t < T), more and more sequence modes in (3.7) come into play.
We found that for t < T the far field is adequately described by (3.7). An interesting alternative derivation is given in Appendix H.) It is now necessary to examine I (r,t) for t in the range t:::::T. This analysis originated in the study of pulse propagation in fluid layers and is used where the stationary phase method becomes invalid. In fact, from purely physical reasoning, we would expect some kind of change in the behavior of I (r ,t) for t close to T, because the wave tail.
To begin the analysis, we mention that the phase function in the integrand (3.1) can be written as It is well known that the high-frequency components of any function play a very important role in the Fourier synthesis of that function, and so because our analysis in the t - T range corresponds to a stationary phase point that moves to infinity in frequency. Since the quantity (Ius I r) is large, we can use the asymptotic form of the Hankel function; so for.
It will be shown that the poly gamma functions r (fJ.+l-v) and r (fJ.+v) lie on the real axis. To perform this analysis, we need to know where the real and imaginary parts of fJ are in the £-plane. We will first show that there are no poles in the first quadrant (Re£>0, Im£>0).
Similarly, it can be shown that there are no gamma function poles in the third or fourth quadrant. For the part of the imaginary axis 0 < Im s ~ 1 we have to consider two cases. Similarly, it can be shown that there are no poles on part of the real axis.
For the Res> 0 case shown in Figure 2-4, we apply the same analysis; but now the poles of the integrand in (E.l) are at those v = v such that. We will show that the poles r(tJ.+V) lie in the 'second quadrant of the v-plane for those s on the path plL in the s-plane; and as e-0 the poles approach the negative real axis. For those s on the plR path, we will show that the poles r(tJ.+l-v) lie in the first quadrant of the v-plane; and as e-O these poles approach the positive real axis.
We will show that the wave equation is a good approximation of the equation that is satisfied by the sound pressure in the ocean (see P. Bergmann [22].