CHAPTER II 

 A NEW MODEL 



The Model 



The purpose of this report is to improve the equations which are presently utilized to predict the 

 resonant frequency and acoustic cross section of a swimbladder fish, in order to facilitate the correlation of 

 acoustic and biological volume reverberation data. A new model for a swimbladder fish is proposed and the 

 appropriate equatbns developed for it. The model consists of a small spherical shell, enclosing an air cavity, 

 in water. The shell is chosen to be a viscous, heat-conducting Newtonian fluid, with the physical properties of 

 fish flesh, and the interface between the shell and the cavity supports a surface tension. 



The shell is insonified by a harmonic plane compressional wave whose wavelength is large compared to 

 the shell diameter. For convenience, the center of the shell will be taken as the origin of the coordinate system 

 and the wave will travel in the positive direction along the z-axis. Spherical coordinates (r, 6, <p) will be used 

 to describe the field, where r is the distance from the origin, 6 the polar angle and cp the azimuthal angle. 

 Since the problem is axisymmetric, 9/3cp = 0. 



Acoustically, fish flesh may be considered to resemble soft rubber. In such a substance, shear waves are 

 much less important than transverse waves and the substance can be closely approximated by a viscous fluid. 

 Modelling fish flesh as a viscous liquid has advantages over modelling it as an elastic solid with a complex shear 

 modulus. Specifically, exact equations (Navier-Stokes) exist for the motion of viscous fluids, whereas a 

 phenomenological approach is required if a complex shear modulus is utilized. 



The body of a fish which surrounds its swimbladder will cause an increase in stiffness over that of a free 

 bubble. It is assumed that most of that increased stiffness in concentrated in the swimbladder wall and may be 

 modelled by a surface tension at that interface. Unlike an elastic modulus, swimbladder tension may be under 

 the fish's control and could account for otherwise unexpected variations in resonant frequency. 



Limits and Physical Properties 



Several limits will be placed on this model. These include limits on applicable depth, frequency, and size 

 ranges. The depth range of interest is from the surface to 1 ,000 m, which corresponds to an ambient 

 pressure of 10 6 to 10 8 dynes/cm 2 . The frequency range is 100 Hz to 40 kHz, which corresponds to a circular 

 frequency, a), of 2n x 10 2 to 8n x 10 4 rad/sec. The fish size range is 1 cm to 1 m. This roughly corresponds 

 to an inner shell radius, a, of 1 0- < to 5 cm. (Appendix A contains a discussion of swimbladder volumes.) The 

 ratio of outer shell radius, b, to a is 2.5 < b/a < 6 for small fish and 2.5 < b/a < 3.2 for large fish. Two further 

 limitations will be that 6 x 10 2 cm/sec < coa< 2.4 x 10" cm/sec and that cob< 7 x 10 4 cm/sec. 



In addition to swimbladder size, the surface tension at the swimbladder wall must be specified. For small 



fish, 



10 2 dyne/cm < s < 10 6 dyne/cm 



and for large fish, 



10 2 dyne/cm < s < 10 9 dyne/cm. 



The rationale for this choice of ranges is discussed in Appendix A. 



Several other parameters must also be specified. These include the velocity of a compressional wave 

 (sound velocity), c; density, p ; specific heat at constant pressure, c p ; ratio of specific heats, y; thermal 

 conductivity, k; shear viscosity, n s ; and bulk viscosity q b ; for air, sea water, and fish flesh. Subscripts a, w, 

 and f will be utilized to indicate the properties of air, sea water, and fish flesh, respectively. For convenience, 

 a viscosity parameter, E„ is defined as 



5 = fins, + n bf ■ OH) 



The parameters are listed in Table 1, for a temperature of 10°C and, unless specified, a pressure of one 

 atmosphere (10 6 dyne/cm 2 ). The properties of air were obtained from references 42 and 43 and those of 

 sea water from reference 44. The properties of fish flesh are discussed in Appendix A. 



