STICKLER: NORMAL MODES IN OCEAN ACOUSTICS 

 NORMAL-MODE THEORY 



Integral Representation 



A typical sound-speed profile in the ocean, shown in Figure 1, 

 has the following special features: it shows the presence of shear 

 in the bottom, and both the shear and sound-speed profiles in the 

 bottom are terminated in isovelocity, constant-density half spaces. 

 Some consequences of this choice for the termination of the sound 

 profile will be discussed later. 



The Hankel transform, chosen for the initial representation of 

 the pressure field, is useful for two reasons: (1) This approach was 

 used by Pekeris (1948), Ewing, Jardetsky and Press (1957), and 

 Brekhovskikh (1960) and, hence, should be familiar to most workers 

 in underwater acoustics. The alternative representation, based on 

 Titchmarsch (1946) and described by Labianca in his paper on surface 

 ducts (1973) is another possibility and, indeed, many of the subtle 

 analytical properties are best discussed by that method; and (2) 

 several points about proper or trapped modes , improper or leaky modes 

 and branch cut integrals, and the physical interpretation of these 

 terms, seem to fit best in the context of the Hankel transform. 



The Hankel representation for the pressure field p at an observa- 

 )n 

 by 



tion point (r, z) due to a point harmonic source at (O, z ) is given 



/ 



p (r,z,z ) = / P (z,z ,k) J (kr) kdk, (D 



J 

 k=o 



where J is the zero order Bessel function of the first kind, and 

 o 



P (z,z ,k) is the transform of p (r,z,z ) with respect to r. This 

 o o 



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