INTRODUCTION 



This report describes a normal-mode program that has been used successfully for 1 2 

 years to compute sound propagation in idealized underwater acoustic ducts. The theory and 

 considerations used in developing the program are discussed here, and a copy of the FOR- 

 TRAN statements are mcluded as appendix A. Appendix B consists of sample inputs and 

 outputs to assist users in gaining familiarity with the program. It is hoped that this report 

 contains sufficient information to allow a user to run the program and to modify it as desired. 



This program follows the methods developed by Furry and Freehoffer (ref 1) to com- 

 pute electromagnetic propagation in the 1 940s. Marsh adapted these methods to underwater 

 sound in his doctoral thesis (ref 2). Using this material, Pedersen, at NOSC in the late 1950s, 

 adapted the method to digital computers and developed the programs to compute the auxil- 

 iary functions. This original program used two layers to define the sound-speed profile (ref 3). 

 This program was expanded to three layers by DF Gordon and RF Hosmer and finally to the 

 multiple-layer program reported here. In this program the only constraints on the number of 

 layers are computer space and running time. The program is normally configured to permit 

 up to 12 layers. 



The earlier programs were used to study sound propagation in ocean surface ducts. 

 Programs that permit more layers have proven useful also for studying propagation in the 

 deep ocean, although the number of modes required generally limits computations to fre- 

 quencies below 300 Hz. The multiple-layer program has also proven useful in modeling sedi- 

 ment layers and thus in computing shallow-water propagation. 



The principal Umitation in the apphcation of this program to real-world situations is 

 the requirement of ideal conditions: boundaries must be smooth and horizontal, and no 

 variation of boundary conditions with range is permitted. Despite this limitation, the pro- 

 gram has proven useful in predicting and explaining acoustic propagation and has applications 

 in a number of related areas. These include checking other types of wave-theory models or 

 corrections such as caustic corrections; determining group velocities, dispersion curves, and 

 reflection coefficients; and determining acoustic coupling between ducts. 



The following paragraphs describe the specific topics covered by the sections in this 

 report. In GENERAL SOLUTION are the equations required to solve the wave equation 

 with the boundary conditions used here. DETERMINANT is part of the basic solution but 

 is concerned with the particular numerical method used in this program to evaluate the con- 

 ditions imposed by the boundaries. Other approaches could be used instead. A later section, 

 NUMERICAL BREAKDOWN, is also part of the basic solution, but deals with special numer- 

 ical problems that have arisen but are not apparent from the basic equations. 



1. The Bilinear Modified-Index Profile, by WH Furry, in Propagation of Short Radio Waves, DE Kerr, ed; 

 MIT Rad Lab series, vol 13, p 140-168, McGraw-Hill, New York, 1951. 



2. Navy Underwater Sound Laboratory Report 111, Theory of the Anomalous Propagation of Acoustic 

 Waves in the Ocean, by HW Marsh, 1950. 



3. Normal-Mode Theory Applied to Short-Range Propagation in an Underwater Acoustic Surface Duct, by 

 MA Pedersen and DF Gordon; J Acoust Soc Am, vol 37, p 105-118, January 1965. 



