MODE FOLLOWER PROGRAM 



Appendix D lists the Mode Follower Program in FORTRAN. It is not a part of the 

 general normal mode program, but is related in that it uses some parts of the general pro- 

 gram. The purpose of the mode follower is to trace a given eigenvalue as some parameter is 

 varied. This parameter is usually frequency, but any profile parameter can also be varied. 

 The eigenvalues at a given set of parameters are discrete points. By permitting the parame- 

 ter to vary, the eigenvalues become a set of lines, and this often clarifies their behavior at the 

 fixed points. Figures 7-9 illustrate this. 



Figure 7 is a sound speed profile consisting of two ducts. Figures 8 and 9 show the 

 real and imaginary parts of some eigenvalues of the profile over a range of frequencies. The 

 imaginary parts are expressed as mode attenuations. The figures show a region where both 

 ducts are exerting an influence on the eigenvalues. The broken lines show the location of 

 eigenvalues for a profile that consists of only the upper duct of figure 7. Considerable time 

 could be spent studying the interaction between the two ducts, but since the purpose here is 

 to illustrate eigenvalues as functions of a parameter, only a brief description will be given. 



Modes are numbered by the real parts of their eigenvalues. This numbering is con- 

 sistent with the number of beats or changes of n in the phase of the depth functions. Thus 

 the eigenvalue of a mode numbered 1 in a profile consisting of only the upper duct lies 

 exactly over the eigenvalues of a mode in the double duct in figures 8 and 9, but this mode 

 in the double duct changes number each time it crosses the real part of another mode. The 

 depth function actually gains an additional beat each time this happens. The background of 

 modes that are being crossed consists of the higher order, untrapped modes associated with the 

 lower duct. 



Mode 2, of the upper duct only, does not have a single mode in the double duct that 

 overhes it exactly. Instead, a mode attempts to follow it at frequencies above 1350 Hz. 

 Below this frequency, successive modes follow its path for short intervals. This interplay 

 between modes occurs when mode 2 of the upper duct is in some sense equally as untrapped 

 as the modes associated with the lower duct. 



The imaginary parts of the modes follow similar patterns; but because the mode 

 numbering is not determined by the imaginary parts, the mode numbers sometime jump 

 from one line to another. An important feature of these two plots is that if the real parts of 

 the eigenvalues cross, the imaginary parts do not; and vice versa. Thus two eigenvalues do 

 not tend to become equal at a point which would make them degenerate. 



The mode follower program will tend to follow the continuous curves. Thus if start- 

 ed in the right direction on mode 59 at 1450 Hz, it will follow along the continuous mode 

 which becomes successively mode 58, 57, 56, and 55. 



Figures such as 8 and 9 can be drawn by computing the eigenvalues at a sufficient 

 number of frequencies to determine the lines. The mode follower does this for a given 

 eigenvalue while adjusting the step size so the mode will not be lost, or so the program will 

 correctly follow the mode. The step size is permitted to become large where the eigenvalue 

 can be approximated by a parabolic curve, but it shortens when extrapolation to the next 

 point becomes less accurate. 



36 



