search can be made along the real line. In simple profiles this can often give a successful set 

 of modes on the first try. Usually, only the three initial eigenvalues need to be located by 

 this means because further eigenvalues can be located by extrapolation on the previous three. 

 This is the function of the third type of control card. 



The third type of card specifies the number of additional modes to be determined by 

 extrapolation. The starting value of each eigenvalue is determined by extrapolating from the 

 three most recently determined eigenvalues to find v. The step size, Av, is chosen as 0.0001 

 times the distance between the last two eigenvalues. The exact eigenvalue is then determined 

 by iteration. The extrapolation is the simple parabolic form for equal steps: 



V = 3vjj - 3vn_i + Vn_2. (27) 



This method of locating modes works well when the modes Ue along a smooth curve, 

 as usually occurs for single ducts. But this relationship does not always occur for profiles 

 with multiple ducts. 



The final control card is punched by the program when requested and contains the 

 correct eigenvalue to full precision. Upon encountering this card, the program does not 

 iterate, but instead evaluates G for this eigenvalue and stores this value of G as the next 

 eigenvalue. A deck of such completed eigenvalues can be stored, saving the expense of 

 recomputing the eigenvalues for a given profile and frequency. 



ITERATION TERMINATION 



A full description of the iteration of eq (1 5) should include the method of termina- 

 tion. The usual criterion for stopping is that G fails to become smaller. As G approaches 

 minimum size, however, round-off error can act as noise so that G is no longer a predictable 

 function of v. The denominator of eq (15) can then be very small by chance, resulting in a 

 large value for 6 j. If this happens, the next value of v, which was as near to the root as pos- 

 sible, will be far away. A much better convergence criterion is that 6j has reached a minimum 

 in absolute value. In the program, iteration is stopped when \5^\ exceeds the previous value 

 by a factor of 2. However, this criterion is not apphed until three iterative steps have been 

 completed, to permit the process to become well established. An upper limit of 1 5 iterative 

 steps is permitted. We have not found an improvement on the root after 1 5 steps. 



SOUND SPEED PROFILE 



The normal mode program requires as inputs the depth of each layer and the sound 

 speed and sound speed gradient at the top of each layer. These variables are mapped into the 

 dimensionless internal variables of the program by eq (7). The purpose of the sound speed 

 profile processing portion of the program is to accept the profile parameters in a form con- 

 venient for the user and to translate them into the required sound speeds and gradients. 



The first function of the processing program is to make the sound speed continuous 

 at interfaces. This is done simply by using the sound speed at the bottom of one layer as the 

 sound speed at the top of the next. It may be necessary to compute the sound speed at the 

 bottom of the layer. The necessary parameters wiU have been given. Occasionally a 



11 



