УДК 534.134:565.14

Ivan S. Maksymov, Andrey Pototsky

Introduction

Vibrations and fluid-structure interactions are essential for efficient communication between living beings and they also underpin human-made imaging, spectroscopy and sensing techniques such as medical ultrasound and photoacoustic imaging modalities [1], Brillouin Light Scattering spectroscopy [2], and laser vibrometry [3] to name a few. Sound and vibrations are also likely to play an important role in the propagation of nerve impulses [4, 5] as well as they can be used to develop new methods of bacteria and virus killing [6, 7, 8]. Furthermore, using vibrations one could monitor, understand and control the behaviour of some animals, such as earthworms [9], and exploit them to sense and modify soil structure as well as to increase crop yields [10, 11, 12].

Earthworms – tube-shaped, segmented worms that have a world-wide distribution and are commonly found in soil – have become a subject of intensive research focused on their response to vibrations and sound. Some of these studies aim to explain the response of these animals to natural vibrations produced by predators, rain or plants [9]. Furthermore, the glial cell wrapping of the giant axons of earthworms resembles the myelin sheath of vertebrate nerve fibers [13]. Therefore, earthworms serve as a platform for neurobiological studies [14]. Earthworms are also cheap and using them does not require ethics approval. Hence, we choose these animals to demonstrate the onset of Faraday-like subharmonic body waves in a living organism subjected to external mechanical vibration.

Classical nonlinear standing Faraday waves appear on the horizontal surface of an infinitely extended liquid supported by a vertically vibrating container [15]. For any given vibration frequency ω, when the vibration amplitude exceeds a certain critical value, the flat surface of the fluid becomes unstable and subharmonic surface waves oscillating at the frequency are formed. These oscillations are due to a parametric resonance between the forcing at the frequency ω and gravity-capillary surface waves with the dispersion relation Ω(k), being k a certain wave vector selected as Ω(k) = .

Faraday waves have become a paradigmatic example of nonlinear wave systems exhibiting complex periodic [16] and quasi-periodic [17, 18, 19] dynamics as well as chaotic behaviour [20, 21, 22, 23]. Recently, a number of applications of Faraday waves in the fields outside the area fluid dynamics have been suggested, including novel photonic devices [24, 25], metamaterials [26, 27], alternative sources of energy [28], and applications in biology [29].

Parametrically excited vibrations and surface waves have also been observed in isolated liquid drops subjected to external mechanical forcing [30, 31, 32, 33, 34, 35, 36, 37, 38]. In response to vibration, the drop can either adopt a regular star shape [30, 31, 32, 33] or exhibit a more dramatic transformation by spontaneously elongating in horizontal direction to form a worm-like structure of gradually increasing length [34, 35, 36, 37].

In contrast to the classical Faraday instability in infinitely extended systems, in isolated liquid drops the boundary conditions at the drop edge dictate the existence of a discrete set of vibrational modes [39, 40, 41, 42]. The eigenfrequency Ω of each mode depends on the boundary conditions at the contact line [40, 41]. When a drop is vibrated at the frequency ω, the fundamental subharmonic resonance occurs when the resonance condition Ω = is fulfilled [30].

In inviscid fluids, the subharmonic response sets in at a vanishingly small vibration amplitude at frequencies that satisfy the resonance condition. For frequencies that do not satisfy the resonance condition, the critical amplitude is nonzero. In experiments with viscous isolated drops, the dependence of the subharmonic critical amplitude on the vibration frequency ω was shown to exhibit periodic variations [30, 32, 33]. This feature is in stark contrast with the Faraday instability in infinitely extended fluids, where the critical amplitude monotonically increases with the driving frequency ω [43].

In this work, we observe experimentally the subharmonic oscillations of the body of living earthworms lying horizontally on a flat solid surface subjected to vertical vibration. We measure the critical amplitude of the onset of subharmonic response as a function of the vibration frequency f, and we reveal that the obtained dependence exhibits signature characteristics of parametrically excited capillary surface waves in vibrated liquid drops [30, 31, 32, 33]. In particular, we show that the critical amplitude varies periodically with f. We explain the observed results by modelling the body of the worm as a horizontally-extended, liquid-filled elastic cylinder subjected to vertical vibration.

Because the excitation of Faraday-like waves in living organisms has thus far received little attention [44], our findings promise to push the frontiers of our knowledge of fundamental nonlinear phenomena and chaotic behaviour in biological systems. For instance, our results should be qualitatively reproducible in other living systems such as bacteria, biological cells or individual organs in the body including the brain and blood vessels.

Experimental results

We tested four different earthworm species encountered in the south eastern regions of Australia [45]. To correctly identify the earthworm species, we used an earthworm identification guide [46]. Eisenia fetida earthworms were purchased from a local fishing goods store, and on average they were 100 – 120 mm long and 5 – 6 mm wide. Lumbricus terrestris earthworms were harvested in the field and closely related to them Lumbricus rubellus earthworms were obtained from a local compost worm supplier. In this group, we selected the worms that measured approximately 120 – 150 mm in length and 8 – 10 mm in width. Several smaller 6 – 8-mm-long and 2 – 3-mm-wide Aporrectodea caliginosa earthworms were also harvested in the field and outcomes of their test were qualitatively similar.

Earthworms are non-regulated animals, and therefore this research did not require the approval of our Institutional Animal Ethics Committee. However, the worms were treated as humane as practical and afterwards they were placed into a worm farm where they fully recovered.

In preparation for experiments, earthworms were first placed in 20% ethanol for approximately 2 minutes, which immobilised them to simplify handling. Then, the entire body of an immobilised worm was placed on top of a thin Teflon plate that was vertically vibrated with the harmonic frequency f. The vibrations were detected by using an in-house laser vibrometry setup [38] consisting of a red laser diode and a photodetector. The intensity of light reflected from the worm is modulated due to the vertical vibration as well as the onset of parametrically excited body waves. We recorded these signals with Audacity software and Fourier-transformed them with Octave software to obtain frequency spectra. When required, the skin of the worm was moistened with water to avoid drying. However, in those cases special care was taken to remove all liquid drops from the Teflon plate. This ensured that Faraday waves are not excited on the liquid drop surface [38] and also dramatically simplified the analysis of the results.