VALIDITY OF RAY ACOrSTICS 



63 



the point to whirh the range is measured, and on re- 

 fraction conditions, or more specifically on the tem- 

 perature-depth variation indicated by the bathy- 

 thorniosraph. 



The intensity contour diagram is a set of lines 

 drawn on a ray diagram indicating the intensity loss. 

 On each contour the intensity loss has a constant 

 value, in a fashion similar to the curves of constant 

 barometric pressure on a weather map. The contours 

 are obtained from a ray diagram by using one of the 

 methods discussed in Sections 3.4 and 3.5. On each 

 ray, or for each pair of adjacent rays, the intensity, 

 or transmission anomaly, is computed at suitably 

 chosen intervals. Then one finds, by interpolation, 

 the points where the intensity loss is 55 db, 60 db, 

 65 db, and so on . After this process is carried through 

 for all the rays, intensity contours can be drawn by 

 joining the points of equal transmission loss on all the 

 rays. 



Sample intensity contour diagrams for the oceano- 

 graphic situations treated in Section 3.5.2 are given in 

 Figure 25. The contour diagram for isothermal water 

 is shown for comparison since it indicates optimum 

 sound-ranging conditions, that is, the intensity losses 

 which would be observed if the water had no tem- 

 perature gradients, and if there were no attenuation 

 losses; for this situation, the intensity loss out to the 

 range R is given by the inverse square law and 

 amounts to 20 log R. The contour diagrams for the 

 split-beam cases are identical with that for the iso- 

 thermal case at depths near the sea surface and at 

 short to moderate ranges; at depths below the ther- 

 mocline, however, the predicted spreading loss is 

 much increased; the amount of increase depends on 

 the depth to the thermocline and the sharpness of the 

 thermocline gradient. In the case of downward refrac- 

 tion, the intensity contours which denote large values 

 of the intensity loss are piled together in the vicinity 

 of the predicted shadow boundary. 



A more detailed discussion of intensity contours 

 with a derivation of some of the basic equations de- 

 rived at the beginning of this chapter is given in a 

 report by UCDWR.* Sample theoretical intensity 

 contours for different temperature patterns are also 

 discussed in this reference. A comparison of these pre- 

 dicted intensities with sound intensities found from 

 explosive pulses is given in Chapter 9. The encyclo- 

 pedia of ray diagrams in reference 5 includes intensity 

 contours on most of the diagrams and thus may be 

 used to find the type of predicted sound field for many 

 different varieties of temperature-depth patterns. 



It will be seen in Chapter 5 that the intensity pre- 

 dictions of the contour diagram are not, in general, 

 sufficiently accurate to be trusted for the prediction of 

 maximum echo ranges. However, they are useful for 

 various special purposes, such as indicating howsound 

 intensities should vary with depth at a fixed range. 



3.6 VALIDITY OF RAY ACOUSTICS 



In Sections 3.1 to 3.5 of this chapter the method of 

 ray acoustics has been presented as an independent 

 theory without much connection with the rigorous 

 treatment of wave propagation presented in Chap- 

 ter 2. We first noted in Section 3.1.1 that the im- 

 portant features of the propagation of spherical waves 

 could be derived equally well by using the concept of 

 wave fronts connecting points which have equal 

 phase of condensation, and the concept of energy 

 transported by rays perpendicular to these wave 

 fronts. Then we generalized the definition of wave 

 fronts and rays, derived differential equations for the 

 ray paths from these definitions, and solved these 

 differential equations for the ray paths and the re- 

 sulting sound intensity. 



It is important to remember, however, that the 

 method of wave fronts for the general case placed no 

 requirement on the wave front, except for stipulating 

 that it be of the form (7) for some function W{x,y,z). 

 To make the idea of wave fronts intuitively signifi- 

 cant, it was implied that the wave front should always 

 join points of constant phase of condensation; but this 

 implication was never used. The ray paths depended 

 only on the form of the function W and the variation 

 of c; the intensity calculations depended, in addition, 

 on the assumption that energy is transported out 

 along the rays. In this section, where we try to find 

 a connection between ray acoustics and wave acous- 

 tics, we must assume a physical significance for the 

 wave fronts. Accordingly, we shall make the explicit 

 assumption that the wave fronts join points of equal 

 phase of condensation since we already know that 

 the assumption brings ray acoustics and wave acous- 

 tics into agreement for the case of spherical waves. 



In this section, we shall examine whether wave 

 acoustics and ray acoustics with this definition of 

 wave fronts are equivalent in general or only under 

 some special conditions. Since soimd field calcula- 

 tions are much simpler by the ray method than by a 

 rigorous solution of the wave equation, it will be ex- 

 tremely vahiable to know when the ray theory can 

 be applied without much error and when it will lead 

 to definitely wrong results. 



