International audienceA method is proposed for analyzing the steady-state response of nonlinear dynamic systems. The method iterates to obtain the discrete Fourier transform of the system response, returning to the time domain at each iteration to take advantage of the ease in evaluating nonlinearities there-rather than analytically describing the nonlinear terms in the frequency domain. The updated estimates of the nonlinear terms are transformed back into the frequency domain in order to continue iterating on the frequency spectrum of the steady-state response. The method is demonstrated by solving a problem with friction damping in which the excitation has multiple discrete frequencies

Although sound-producing (sonic) muscles attached to fish swimbladders are the fastest known vertebrate muscles, the functional requirement for such extreme speed has never been addressed. We measured movement of the swimbladder caused by sonic muscle stimulation in the oyster toadfish Opsanus tau and related it to major features of the sound waveform. The movement pattern is complex and produces sound inefficiently because the sides and bottom of the bladder move in opposite in and out directions, and both movement and sound decay rapidly. Sound amplitude is related to speed of swimbladder movement, and slow movements do not produce perceptible sound. Peak sound amplitude overlaps fundamental frequencies of the male's mating call because of muscle mechanics and not the natural frequency of the bladder. These findings suggest that rapid muscle speed evolved to generate sound from an inefficient highly damped system.

Recent reports of high frequency sound production by cusk-eels cannot be explained adequately by known mechanisms, i.e., a forced response driven by fast sonic muscles on the swimbladder. Time to complete a contraction-relaxation cycle places a ceiling on frequency and is unlikely to explain sounds with dominant frequencies above 1 kHz. We investigated sonic morphology in the fawn cusk-eel Lepophidium profundorum to determine morphology potentially associated with high frequency sound production and quantified development and sexual dimorphism of sonic structures. Unlike other sonic systems in fishes in which muscle relaxation is caused by internal pressure or swimbladder elasticity, this system utilizes antagonistic pairs of muscles: ventral and intermediate muscles pull the winglike process and swimbladder forward and pivot the neural arch (neural rocker) above the first vertebra backward. This action stretches a fenestra in the swimbladder wall and imparts strain energy to epineural ribs, tendons and ligaments connected to the anterior swimbladder. Relatively short antagonistic dorsal and dorsomedial muscles pull on the neural rocker, releasing strain energy, and use a lever advantage to restore the winglike process and swimbladder to their resting position. Sonic components grow isometrically and are typically larger in males although the tiny intermediate muscles are larger in females. Although external morphology is relatively conservative in ophidiids, sonic morphology is extremely variable within the family.

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Physiological work on fish sound production may require exposure of the swimbladder to air, which will change its loading (radiation mass and resistance) and could affect parameters of emitted sounds. This issue was examined in Atlantic croaker Micropogonius chromis by recording sounds from the same individuals in air and water. Although sonograms appear relatively similar in both cases, pulse duration is longer because of decreased damping, and sharpness of tuning (Q factor) is higher in water. However, pulse repetition rate and dominant frequency are unaffected. With appropriate caution it is suggested that sounds recorded in air can provide a useful tool in understanding the function of various swimbladder adaptations and provide reasonable approximation of natural sounds. Further, they provide an avenue for experimentally manipulating the sonic system, which can reveal details of its function not available from intact fish underwater.