to -ioj times the particle velocity, is p~^ d(>/px)/dz. Note that in Figure 18 layer 2 is a 

 constant K layer. The ability to mix linear and constant layers is necessary in a general pro- 

 gram because, as the gradient, |3, goes to zero, the argument of the Airy functions increases 

 without limit. Thus, depending on frequency, layer thickness and computer word length, 

 there is a minimum gradient that can be used. Layers with gradients smaller than this must 

 be represented by constant K layers. 



COMPARISON OF THE TWO MODELS 



It is instructive to see how these two models, the Hquid multilayer and the linear 

 gradient multilayer, compare. To do this, consider Figure 19. On the left hand side our 

 linear model has a sound speed that increases from 1500 m/s at the water/sediment inter- 

 face to 1 800 m/s at a sediment depth of 300 meters, which corresponds to an average 

 sound speed gradient equal to (1800 - 1500) m/s ^ 300 m, or 1 s~ . The constant 

 K model is shown for two layers. The layers have the same thickness and the sound speed 

 at the center of the layers (i.e., at 75 and 225 meters) is set equal to the sound speed of 

 the hnear layer at that depth. 



On the right hand side of Figure 19 is a diagram that indicates the main physical 

 events. Most of the energy either reflects at the surface or is refracted in the sediment 

 because of the gradient. Morris (1973) has used a ray description to calculate the energy in 

 each path and compare the ray description with the wave model. Of course there are second 

 and higher order effects as indicated by the dashed arrows that are impHcit in the wave 

 model. 



In Figure 20, the first comparison of the two models is shown. For the calculations 

 we used a frequency of 100 Hz, a density ratio (p in sediment)/(p in water) equal to 2.0 

 and zero attenuation. The reflection coefficient was calculated for grazing angles from 

 degree to 20 degrees which are of interest in sound propagation. With zero attenuation 

 both models return all sound to the water for these grazing angles so the modulus of R is 

 1 .0 or the bottom loss is zero. Figure 20 shows plots of phase, i.e., the argument of R, for 

 different cases. The curve marked L is for the linear K- model, while the curves labeled 1, 

 3 or 10 correspond to 1, 3 or 10 constant K layers. The 10 layer case has a layer thickness 

 of 30 meters, which is equal to 2 wavelengths in the water. For 30 layers (or a thickness of 

 0.67 X^) there is a maximum phase difference of 2.2 degrees at a grazing angle of 3.5 

 degrees which cannot be plotted on this scale. For 100 layers there is a maximum phase 

 difference of 0.2 degrees. 



In Figure 21 the same models are used except that there is an attenuation of 0.05 

 dB/m in both models. As in the previous case, the 10 layer model (thickness - 2X^) has a 

 maximum difference of ~10 degrees and the 30 layer model has essentially the phase as 

 the linear model. It is interesting to note that the attenuation has slowed the phase change 

 considerably. This will have a noticeable effect on the wave theory propagation models 

 where a shift in phase of 360 degrees will add a new mode to the sound field (Bucker, 

 1964). 



To complete the comparison of the linear and constant layers, the bottom loss 

 curves are shown in Figure 22. The one layer case has much less bottom loss because the 

 sound speed is equal to the sound speed of the linear model at 150 meter depth, which is 

 1629.6 m/s and corresponds to a critical angle greater than 20 degrees. 



51 



