Nero et al.: Low-frequency acoustic measurements of Merluccius productus 
333 
i'=i 
where cr fes . is the backscattering cross section of the 
i th scatterer and 0 is the number of individuals within 
a 1-m 2 vertical column that extends over the depth 
of the layer. At some stations, usually at night, sev- 
eral layers were evident. In these cases, Equation 1 
was solved for each individual layer, and the subse- 
quent distributions of ESR were summed. 
Swimbladder model 
The matrix c of individual backscattering cross sec- 
tions was calculated with Love’s model (1978) 
which models a fish swimbladder as an air-filled vis- 
cous spherical shell. The acoustic cross section in the 
back-scattered direction, <j bs , for a single swim- 
bladder of radius r is given by 
= 
( f 2 
( f2 j 
2 N 
1 ° + 
Ar-1 
f 2 H 2 
V 
f 2 
\ 1 / 
) 
(3) 
where o bs is in m 2 ; 
r is in m; 
f = the insonifying frequency in Hz; 
f 0 = the swimbladder’s monopole resonance 
frequency in Hz; and 
H = a damping factor. 
The resonance frequency is 
ever, Love (1978) shows that these terms are negli- 
gible for the present case, therefore they were omit- 
ted for clarity. The physical properties of fish used 
here were taken from Love (1978): p = 1050 kg/m 3 
and % = 50 Pa sec. 
Swimbladders are not spherical but resemble pro- 
late spheroids or cylinders with major-to-minor axis 
(length-to-diameter) ratios up to 10. However, it has 
been shown that spheroidal and cylindrical air 
bubbles have only a slightly higher resonance fre- 
quency than spherical bubbles (Weston, 1967; 
Feuillade and Werby, 1994; Ye, 1997). Love’s model 
with Weston’s correction to the resonance frequency 
has recently been used successfully to examine scat- 
tering from commercial-size fish in several ocean 
regions (Love, 1993; Thompson and Love, 1996) and 
what are presumed to be deep-water grenadiers off 
Oregon (Nero et al., 1997). 
To model hake for this study, we assumed the 
swimbladder to be a prolate spheroid with a major- 
to-minor axis ratio of 8. This assumption is reason- 
able based on our at-sea measurements on Pacific 
hake (see below) and reported measurements on 13 
pollack, Pollachius pollachius, from the North At- 
lantic (Foote, 1985), both of which gave major-to- 
minor axis ratios near 8. The resonance frequency of 
such a spheroid is about 20% higher than for a sphere 
of equal volume ( Weston, 1967); this correction factor 
was taken into account by increasing f () given in Equa- 
tion 4 by a factor of 1.2. Recently, Feuillade and Werby 
(1994) have shown that the broadside target strength 
of such spheroids is about 0.5 dB less than that of a 
sphere. Because this difference is well within the ex- 
pected error of our measurements, it was not incorpo- 
rated into our model calculations. 
3 r a P 
4n 2 r 2 p 
(4) 
where y a - 
P = 
P = 
the ratio of specific heats of air (y ft =1.4); 
the ambient pressure in Pa; and 
the density of fish flesh in kg/m 3 . 
The damping factor is 
1 _ 2jrr f 2 ! S 
H f 0 c nr 2 f 0 p 
(5) 
where c = the speed of sound in water in m/sec; and 
£, = the viscosity of fish flesh in Pa sec. 
Love’s model includes a term in Equation 4 that ac- 
counts for the effects of swimbladder wall tension on 
f 0 and a thermal damping term in Equation 5. How- 
Fish length and bladder size 
We made direct comparisons of ESR derived from 
the NRL acoustic data with estimates of ESR from 
the lengths of fish obtained in NMFS trawls. How- 
ever, the process of deriving swimbladder size from 
fish length is problematic. Because one of the pri- 
mary functions of a swimbladder is to act as a hy- 
drostatic organ, making a fish neutrally buoyant, one 
can assume ratios of swimbladder volume to fish 
volume near 0.05 (Jones and Marshall, 1953). How- 
ever, many fish are frequently less than neutrally 
buoyant. In addition, food and gonadal products can 
crowd the abdominal cavity and reduce swimbladder 
volumes. Repeated measurements indicate values 
near 0.03 are common for gadoids (Foote, 1985; Ona, 
1990) but that they are highly variable (Sand and 
Hawkins, 1973; Jones and Scholes, 1985; Foote, 1985; 
and Ona, 1990). Our field measurements on five hake 
