APPENDIX A 

 NUMERICAL EVALUATION OF l 1 AND I 2 



With the exception of special cases, such as the shallow water approximation, the 

 integrals /, and /„ must be evaluated numerically. A FORTRAN program was written for this 

 purpose, and a summary of the numerical analysis is presented here. For simplicity, and 

 where the meaning is clear, the integrands of the integrals that follow will be omitted. 



It was pointed out before that K Q (9) is the positive, real root of the transcendental 



equation KH - — sec 2 9 tanh KH = 0. This root is obtained for each required value of 9 



F 2 

 by means of the Newton-Raphson method. 2 



Although the analysis pertaining to the integrals is similar in both cases it is some- 

 what more convenient to treat subcritical and supercritical speeds separately. 



SUBCRITICAL SPEED {F < 1) 



This is the simpler of the two cases, primarily because 9 Q = 0. For this reason together 

 with the fact that the integrands of both I ^ and / 2 are functions of cos 9, the 9 integrations 

 lend themselves to the Tchebysheff quadrature 3 in the form, 



77/2 n 



d9 /(cos 9) = — y /(cos 9) 



J 



/ = 1 



2/-1 



where 9- = n. In addition to the good accuracy associated with the Gaussian quadra- 



7 4n 



ture, it is especially advantageous for computer usage in that both the abscissae and (con- 

 stant) weight functions can be readily derived within the program. It is used initially for / 

 and the 9 integral of /j by setting n equal to 10. Then the integrals are reapproximated with 

 n doubled. If the estimates are not sufficiently close as determined by a convergence 

 criterion, n is redoubled, and the process is continued until convergence is obtained. 



For each value Q. of the double integral I x the root K Q is found and the integration on 

 K is performed by first writing the infinite integral as the sum 



2 K Q 2 K Q + 2 2 K Q + 4 



dK + ■•• 



-i. dK + f< dK + J 



2 K 2K Q + 2 



and approximating each of the integrals in the series by a repeated application of the 10-point 

 Gauss-Legendre quadrature; 3 that is, the range of integration of each integral is repeatedly 



14 



