experience upon reflection. Figure 3 is an example of the use of this computation. It shows 

 phase and amplitude of the reflection coefficient in shallow water over a sandy-silt sediment 

 lying over rock. The frequency is 1 500 Hz. Reflections are given only at discrete points 

 determined by the individual modes. 



The model in figure 3 is for a liquid bottom. That is, no rigidity is supplied in this 

 program and the sound speed, density, and attenuation determine the reflection coefficients. 



The reflection coefficients computed by eq (45) can be closely approximated by 

 dividing the mode attenuation by the loop length of the corresponding ray. The loop length 

 must be determined from ray theory for the ray of the same phase velocity or vertexing veloc- 

 ity. However, an interesting analog of the ray loop length is the intermode interference length. 

 This is discussed by Guthrie (ref 10). Specifically, if the difference between eigenvalues, ReXj, 

 for two adjacent modes is AX, the interference length 1 = 2it/AX. This distance will usually 

 equal the ray loop length for some ray with phase velocity between that of the two modes. 



As each mode after the first is computed, the length, 1, is computed and printed out. 

 Also routinely printed out for each mode is the mode damping or mode attenuation coeffi- 

 cient, in units of dB per km. This attenuation, aj, is computed from the relationship 



Oj = -1000 Im Xj logio® 

 = -8686ImXj. 

 This quantity multiplied by range gives the damping of mode i, in dB. 



10. The Connection Between Normal Modes and Rays in Underwater Acoustics, by KM Guthrie; J of Sound 

 and Vibration, vol 32, no 2, p 289-293, 1974. 



SOUND SPEED, m/s 

 1520 1530 1540 1550 



T 



400 L FINAL LAYER 



2.5 









03 









T3 



O 2.0 



_ 



#-^ 



'***_ 



H 





J' 





O 



x 







HI 



X 



/ 





-■ 1.5 



- \ 



/ 





LU 



tr 









m 1.0 



- / 



^^■"N^,.^^^^ 



— 



Q. 



/ 



^^**^ , 





« 



/ 





^1^~ 



M 



/ 







yo.5 



- 





- 







1 



1 1 1 



1 



150 f- 



2 4 6 8 10 



GRAZING ANGLE, DEGREES 



Figure 3. A shallow-water profile with resulting phase and amplitude of the reflection coefficient at 1 .5 kHz. 

 Parameters at the top of the sediment layers are as follows: 1st layer - c = 1606.45 m/s, y = 1.5s"l, a = 

 0.18dB/m,p= 1.68; 2nd layer -c= 1684.0 m/s, 7= 1.5s-l,a= 1.10dB/m,p= 1.91; final layer - 7 = -0.1. 



21 



