## PXYZT J fiioiJZgHtiuj RptwZe1

The outer expansion of this inner solution is obtained by setting 4.4l R r e , r (x2 + y2)2 , T t e and evaluating asyn totically for small e . Thus f (w)y w,Zg)y .z 2' (- -)eia,t e du Using the leading term of the asymptotic series for IT with large argument ( l7jl we obtain J f (w) ip(u),Zg)fp(u),z)(- pj-) e * e 5 da But this integral is suitable for an application of the stationary phase approximation. Let ojo uo(r,t) be a value of for which Then, making the usual approximations near when...

## Info

This agrees with (4.6) if we divide (4.59) by e , multiply by 2 ir e 1 1 1 ana identify a of (4.60) with a of (4.6), (4.10). o o We see froin (4.60) that there is a phase shift of in the signal corresponding to the ray which has touched the caustic relative to the signal for the direct ray even after allowance has been made for the differences in path length. Provided this phase shift is incorporated it is possible to use the naive ray theory of (4.6) even for rays which have grazed the...

## Velocity

A typical deep ocean velocity-depth profile. The surface duct and deep isothermal layer are separated by the main thermocline. 1 1 J 1 J * I i 1 I i i ill 1 I i i i i i i i i i i i j 1 1 J 1 J * I i 1 I i i ill 1 I i i i i i i i i i i i j Figure 5. Contour of the sound speed measured during the late summer of 1968 along the meridian 157 50'W. Bottom loss versus grazing angle at selected frequencies. The dashed lines indicate regions of little or no data. With permission of Urick 26 . vertical...

## Bvp

The boundary value problem (BVP) is solved by the method of normal modes, by the method of Hankel transforms and by the ray method. This leads to the three representations denoted by mode, Hankel and ray, respectively. Then the representations are transformed into one another by the method of residues, by the binomial expansion and by the Poisson summation formula, as indicated. Figure 1. The boundary value problem (BVP) is solved by the method of normal modes, by the method of Hankel...

## Hgf

4.53 Aq 0 at z 0, 0 at z Z (x,y,t) , similar equations hold for Bq . Thus as before we write The transport equations obtained from the V 1 coefficients are 4.5 5a 2 < *p4 > Sta0it - VS-Va + - V2S aQ + rt 'h Vptf (Vp)2 bo+ V Vpt -(Vp)2 bc ap < V 2 yptVt- ivp)2 o 0 k.55b 2 < iL t> S.b . - VS-VbJ + L. o > s*+ - K * fp> Ptt - (7P)2> 0 + VPtao,t Vp'7ao 0 Just as at (3.13-3.17) It is found that on defining that aQ satisfy the ordinary transport equation (4.24) and S the ordinary...

## Py

Which is independent of the integer p . As a consequence of (8.18) and (8.20), the only nonzero coefficient is a , a , . For brevitj , we define P.P-1 P-l.P S-W - b W + a (W .,-W ) - a (W -W , ), 2 < p < 1-1 dT p pp p+1 p+1 p ps p p-1 - * - We shall use A and 6 to denote froward and backward difference operators, respectively, i.e. Then, the general relation in (8.23) can be expressed as Notice that in (8.25), we have simply rewritten unity as 8.26 Ap (p+l)-p 1 , 6p p-(p-l) 1 Prom (8.14),...

## Table Of Contents

SURVEY OF WAVE PROPAGATION AND UNDERWATER ACOUSTICS (Joseph B. Keller) 2. Wave propagation in a deterministic 3. Wave propagation in a stochastic II. EXACT AND ASYMPTOTIC REPRESENTATIONS OF THE SOUND FIELD IN A STRATIFIED OCEAN (Daljit S. Ahluwalia and Joseph. . Keller) 1. Formulation and fundamental 2. Time harmonic 3. The homogeneous ocean of constant depth 3.2 Normal mode 3.3 Hankel transform 3.3A 3. Ray 3.5 Connections between the 3.5A U. The inhomogeneous stratified ocean of constant...

## X

One easily calculates that the distance to the focal plane is R(l+R kQB ), which is slightly less than the distance predicted by geometrical acoustics. Thus ray-trace predictions of the location of convergence zones are always slightly in error, the error becoming larger the greater the range and the lower the frequency. Parabolic approximations are therefore not unique in having errors that accumulate with range. The main features of the SOFAR sound channel that make possible long-range...

## IJH k0r 2 x 353ai IT iT

Eq. (3.53a) is solved in the usual way for the outgoing wave by starting from r 0 and marching out to the largest desired range. This stored solution is then put into the right hand side of (3.53b) and the solution of this equation is obtained by marching inward from large r backward toward the source at r 0. In this manner, the acoustic energy scattered back from the environment to the source can be computed within the parabolic approximation. In principle, this procedure could be iterated by...

## Cd

So*N Ss'N *40'N SS*N feo*H SB*N feTl Figure 8. Propagation loss versus range for a receiver at a depth of 2500 ft., a source at 500 ft., and a frequency of 31 hz. The receiver is fixed at 27 30* N, 157 50' W while the source moves northward. The top, central, and lower curves represent measured data, computer predictions, and measurements superimposed on predictions, respectively. FREQUENCY 31 Hi S8URCE 800 FEET RECEIVER 10800 FEET Figure 9 Same as Figure 8 except that the receiver is at a...

## The Parabolic Approximation Method

Courant Institute of Mathematical Sciences New York University 251 Mercer Street Mew York, NY 10012 The propagation of acoustic signals in the ocean to long ranges is made possible by the existence of the SOFAR sound channel which acts like a waveguide that confines the acoustic waves within the water column and prevents their interaction with the ocean bottom,which is generally quite lossy compared to the water itself. The parabolic approximation methods discussed in this article are based on...

## Survey Op Wave Propagation And Underwater Acoustics

Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY 10012 Underwater acoustics, the science of sound propagation in the ocean, has been developed extensively during the last forty years in response to practical needs. By now the theory is so well developed that it provides a general understanding and a detailed des-cription of how sound travels in the ocean, and of the Mechanisms affecting it. The theory can also be used to make quantitative...