Undulation is the propagation of a wave along a flexible body to produce movement, and it is arguably the most widespread locomotion strategy in the animal kingdom. From microscopic sperm cells to ten-meter-long constrictor snakes, organisms across an enormous range of sizes rely on traveling waves to push themselves through water, over sand, across flat ground, and even through granular material. The physics changes dramatically depending on the scale and the medium, but the core idea stays the same: a wave moves down the body in one direction, and the animal moves the other way.
How Fish Undulate, and Why Not All of Them Do It the Same Way
Fish swimming looks deceptively simple from the outside, but biomechanists distinguish several distinct modes based on which parts of the body participate in the traveling wave. Slender-bodied fish like eels, lampreys, and many sharks swim in what is called the anguilliform mode, where much of the body undulates at high amplitude. Fish with broad tails and a narrow stalk just ahead of the tail, like tuna and mackerel, swim in the carangiform mode, where only the tail region undulates with large amplitude while the front of the body stays relatively rigid.1PubMed Central. Disentangling the Functional Roles of Morphology and Motion in the Swimming of Fish
These different swimming styles produce measurably different fluid signatures. Carangiform swimmers shed two staggered vortices per tail beat and drive a strong jet of water backward, which is efficient for sustained cruising. Anguilliform swimmers produce a more complex wake with at least two pairs of vortices per tail beat and relatively little downstream flow.2Integrative and Comparative Biology. Disentangling the Functional Roles of Morphology and Motion in the Swimming of Fish That might sound less efficient, but anguilliform swimming excels at maneuverability and acceleration from a standstill, which matters more than top-end cruising speed for an eel hunting through reef crevices or a lamprey navigating a rocky stream.
Between these two poles are subcarangiform swimmers (like trout) that use a moderate portion of the body, and thunniform swimmers (like tuna) that confine high-amplitude motion almost entirely to a crescent-shaped tail. The spectrum is continuous, not a set of rigid categories, and plenty of species shift their style depending on speed or context.
How Muscles Coordinate Along the Body
A traveling wave requires exquisite timing. If all the muscles along a fish’s body fired at once, the body would simply bend into a static curve and go nowhere. Instead, muscle activation rolls from head to tail with a slight delay between each segment, so that each section of the body reaches peak contraction just after the section in front of it. This head-to-tail sequence is what makes the wave “travel.”
One consistent finding across many fish species is that muscle activation begins earlier in the local bending cycle at more posterior positions along the body. In plainer terms, tail muscles start firing before the tail has fully straightened from its previous stroke, while muscles near the head fire closer to the moment of peak bending.3Journal of Experimental Biology. Fish swimming: patterns in muscle function This timing shift means that posterior muscles are doing more work to stiffen the tail and transmit force to the water, while anterior muscles contribute more to bending the body into the correct wave shape.
Studies on trout have shown that muscle recruitment also varies along the body’s length during steady swimming, with anterior red (slow-twitch) muscle being activated for longer durations than posterior red muscle. The pattern suggests that different regions of the body play distinct mechanical roles even within a single swimming gait.4Comparative Biochemistry and Physiology Part A: Molecular & Integrative Physiology. Red muscle recruitment during steady swimming correlates with rostral–caudal patterns of power production in trout
The Spinal Circuits That Generate the Rhythm
Fish do not have to consciously decide to undulate any more than you consciously decide the timing of each leg during a walk. The basic swimming rhythm is produced by networks of spinal interneurons known as central pattern generators, or CPGs. Each spinal segment contains its own local oscillator, and these oscillators are coupled so that a wave of activation propagates smoothly from segment to segment.5PubMed. Central mechanisms underlying fish swimming Command signals from the brain initiate swimming and set its speed, but the detailed timing of muscle contractions is handled locally in the spinal cord.
In lampreys, the CPG generates waves of muscle activity that travel from head to tail, bending the body into wave shapes that also propagate rearward.6PubMed. Rostral versus caudal differences in mechanical entrainment of the lamprey central pattern generator for locomotion Zebrafish spinal networks use a combination of electrical and chemical connections between neurons that allows the correct activation sequence to be established quickly from random initial conditions, which is exactly what you need for a sudden swimming burst. That same wiring also keeps the body waveform relatively stable across different swimming speeds, so the shape of the undulation does not fall apart when the fish speeds up or slows down.7PubMed Central. Intersegmental coordination of the central pattern generator via interleaved electrical and chemical synapses in zebrafish spinal cord
Sensory Feedback and What Happens When the Spine Is Cut
CPGs can produce rhythmic output on their own, even in an isolated spinal cord. But in a living animal, sensory feedback refines and stabilizes the pattern. Stretch receptors in the body wall detect how much each segment is bending, and pressure sensors register the fluid forces acting on the skin. In computational models of elongate fish like eels, stretch and pressure feedback together contribute to rapid pattern generation and can, in principle, replace the direct neural couplings between oscillators.8PubMed Central. Multisensory feedback makes swimming circuits robust against spinal transection and enables terrestrial crawling in elongate fish
That finding has a dramatic implication: eels can keep swimming shortly after a complete spinal cord transection. Because the segments below the cut still have functional local oscillators and still receive stretch feedback from the body, they can maintain a coordinated wave even without commands from the brain. Models suggest that the combination of stretch feedback and the oscillators’ ability to fire spontaneously explains this resilience.9PubMed Central. Multisensory feedback makes swimming circuits robust against spinal transection and enables terrestrial crawling in elongate fish
An even more surprising result comes from theoretical work showing that a simple form of proprioceptive feedback, where local muscle activation depends on how curved the body is at that point, can drive swimming gaits spontaneously without any CPG at all.10PubMed Central. Gait and speed selection in slender inertial swimmers This suggests that undulatory locomotion may be partly a mechanical inevitability for a flexible body in fluid, not purely a product of elaborate neural circuitry.
Fish also use sensory feedback to exploit their environment. When swimming behind a cylinder that sheds vortices into the current, fish shift their muscle timing so that body movement occasionally precedes muscle activation rather than the other way around. In laminar flow, muscle activity leads body motion almost all the time, with only about three percent of timing intervals showing the reverse order. In a vortex street, that fraction jumps to nearly half, suggesting the fish is passively riding the vortices and only intermittently adding active muscle effort.11PubMed Central. Beyond propulsion: muscle proprioception enables hydrodynamic sensing in fish body
Snake Undulation on Flat Ground
Undulation is not restricted to water. Lateral undulation is the most common mode of snake locomotion, and it works on the same wave-propagation principle as fish swimming. During lateral undulation, all points along the snake’s body move simultaneously, with regions bending left and right propagating along the entire length of the animal in sliding contact with the ground.12Integrative and Comparative Biology. What Defines Different Modes of Snake Locomotion?
Outdoors, snakes typically push off rocks, branches, and uneven ground to generate forward thrust. But they can also slither across perfectly flat, featureless surfaces. The key turns out to be frictional anisotropy: snake scales are oriented so that friction is lower in the forward direction than sideways or backward. Experimental measurements of snakeskin friction coefficients, combined with a theoretical model, show that this directional difference in friction is enough to propel the animal forward without any push-points at all.13PubMed Central. The mechanics of slithering locomotion The same study highlighted that how a snake distributes its weight during lateral undulation matters more than was previously assumed.
How Vertebral Count Shapes the Wave
A snake’s body is essentially a long chain of vertebrae, and the number of links in that chain influences how tightly the animal can bend. In garter snakes, individuals with relatively more vertebrae achieved greater lateral bending during locomotion than those with fewer.14Functional Ecology. The effects of substrate and vertebral number on locomotion in the garter snake Thamnophis elegans This makes intuitive sense: more joints per unit length means each joint contributes a small angular change, and the cumulative effect is a smoother, tighter curve.
The same logic applies in reverse for animals that need a stiffer spine. Ground-dwelling chameleons tend to have fewer presacral vertebrae (roughly 15 to 19) than arboreal species (roughly 18 to 23), producing a less flexible vertebral column that suits a walking rather than a climbing lifestyle.15PubMed Central. Morphological and functional regionalization of trunk vertebrae as an adaptation for arboreal locomotion in chameleons Even among limbless lizards and snakes, which look superficially similar, the musculoskeletal differences and the way axial kinematic traits are decoupled allow different species to form distinct undulatory waves and adopt different locomotor modes despite broadly similar body plans.16PubMed. Angles and waves: intervertebral joint angles and axial kinematics of limbed lizards, limbless lizards, and snakes
Swimming Through Sand
One of the more striking examples of undulation occurs in the sandfish lizard, a skink that dives beneath loose desert sand and “swims” through it without using its legs. High-speed X-ray imaging reveals that once submerged, the lizard tucks its limbs against its body and propagates a single-period sinusoidal wave down its length, exactly like a swimming fish but in a granular medium.17PubMed. Undulatory swimming in sand: subsurface locomotion of the sandfish lizard
The physics of sand-swimming differs from water swimming because sand grains interact through friction rather than viscosity, making the medium behave like what researchers describe as a frictional fluid. Modeling work that balanced granular thrust and drag forces along the sandfish’s body predicted a wave efficiency of about 0.5, meaning the animal’s forward speed is roughly half the speed of the wave traveling down its body. By varying the ratio of undulation amplitude to wavelength, the models found that maximal swimming speed occurs at a specific ratio of about 0.2, which is the same kinematics the sandfish naturally uses.18PubMed Central. Mechanical models of sandfish locomotion reveal principles of high performance subsurface sand-swimming In other words, evolution has tuned the sandfish’s undulation to the optimum predicted by physics.
Undulation at Microscopic Scales
Scale the problem down by several orders of magnitude and undulation still works, though the physics is completely different. At the size of a sperm cell or a single-celled organism, inertia is negligible and viscous drag dominates everything. A human swimmer can glide after a stroke; a sperm cell stops the instant it stops beating its flagellum.
The undulatory beat of a eukaryotic flagellum is driven by molecular motors called dyneins that cause microtubules inside the flagellum’s core structure to slide past one another. This sliding is converted into bending by constraints that prevent the tubules from simply slipping apart. The oscillation is self-regulatory: mechanical feedback within the structure controls when and where the motors activate, producing a sustained traveling wave without constant input from the cell body.19PubMed. Mechanical induction of oscillatory movement in demembranated, immotile flagella of sea urchin sperm at very low ATP concentrations
The nematode worm Caenorhabditis elegans, at about a millimeter long, lives in a physical regime where both viscosity and external load matter. When placed in fluids of increasing resistance, the worm smoothly adjusts its undulatory gait: both the wavelength and the frequency of undulation decrease as the load gets heavier.20PubMed Central. Biomechanical analysis of gait adaptation in the nematode Caenorhabditis elegans This continuous tunability has made C. elegans a popular model for studying how nervous systems and bodies interact to produce locomotion, since the worm’s entire neural wiring diagram is known.
Counter-Propagating Waves and Hovering
Most undulatory swimmers send a wave in one direction to move in the other. But the electric ghost knifefish has a trick that lets it hover in place: it sends two waves along its ribbon-like ventral fin simultaneously, one traveling forward and one traveling backward. The two waves meet at a nodal point slightly behind the fin’s midpoint, and their opposing thrusts cancel out, keeping the fish stationary.21Journal of Experimental Biology. Kinematics of the ribbon fin in hovering and swimming of the electric ghost knifefish
When the knifefish wants to swim forward, it shifts the nodal point toward the tail and reduces the number of undulations, letting the rearward-traveling wave dominate. At higher speeds, both wave frequencies increase, accompanied by changes in the amplitude of fin-ray motion.22Journal of Experimental Biology. Kinematics of the ribbon fin in hovering and swimming of the electric ghost knifefish This system gives the fish extraordinarily precise positional control, useful for an animal that navigates murky water using electric fields rather than vision.
What Roboticists Have Learned from Undulation
Engineers have been building undulatory robots for decades, partly to test biological theories and partly because undulatory propulsion has practical advantages: no exposed rotating parts, low noise, and good maneuverability in cluttered environments. One recent platform is a reconfigurable robotic fish that can replicate all four major fish swimming gaits (anguilliform, subcarangiform, carangiform, and thunniform) by rapidly tuning its body stiffness. Vacuum-driven jamming muscles in four joints enable stiffness changes within a second, with a stiffness ratio of nearly 47 between the softest and stiffest settings.23PubMed Central. Adaptive multimodal swimming gaits in a reconfigurable modular soft robotic fish
In thunniform mode, where only the tail oscillates, the robot reaches about 1.24 body lengths per second and generates roughly 142 percent more thrust than in anguilliform mode at the same frequency. In anguilliform mode, it sacrifices speed for agility, achieving a turning radius of just 0.26 body lengths. During tasks the robot dynamically switches gaits, using thunniform for straight-line speed and anguilliform for weaving through obstacles.24PubMed Central. Adaptive multimodal swimming gaits in a reconfigurable modular soft robotic fish
Simpler soft-robotic platforms have confirmed that body stiffness is a key variable. In one study using pneumatic actuators attached to a flexible foil, the system produced more thrust at higher tail-beat frequencies up to a plateau above 0.8 Hz, and achieved a self-propelled speed of about 0.8 foil lengths per second.25PubMed. Undulatory Swimming Performance and Body Stiffness Modulation in a Soft Robotic Fish-Inspired Physical Model That performance is modest compared to a real fish, but it validated the principle that an elastic body actuated with simple inputs can reproduce the essential features of undulatory propulsion. Separate experiments on passive elastic swimmers have shown that the wave-like character of the motion depends heavily on how energy dissipates along the body, an insight that matters for designing robots that use flexibility rather than motors at every joint.26PubMed Central. Passive elastic mechanism to mimic fish-muscle action in anguilliform swimming
The Acoustic Footprint of Undulatory Swimming
One underappreciated aspect of undulation is how quiet it can be. Numerical simulations of the far-field sound generated by a biomimetic swimmer found that the sound distribution takes on a dipole-like pattern, and that anguilliform motion, which involves the whole body undulating, generates lower sound during propulsion than other swimming modes.27Ocean Engineering. Parametric analysis on the acoustic characteristics generated by a biomimetic swimmer A thinner body profile also helps: a small thickness-to-length ratio allows large thrust with little acoustic output. These findings are relevant both for understanding how predators and prey detect each other and for designing underwater vehicles that need to operate stealthily.
On the detection side, the hydrodynamic disturbances created by swimming fish are surprisingly low-frequency. Laser-based flow measurements around hovering and swimming fish, frogs, and crustaceans found that the velocity spectra of the water movements they produce peak below 10 Hz, and transient motions like startle responses also register below that threshold.28PubMed. The time course and frequency content of hydrodynamic events caused by moving fish, frogs, and crustaceans These signals sit well below the hearing range of most terrestrial animals but are readily detected by the lateral-line organs of other fish, creating a private channel of information that undulatory swimmers both broadcast and eavesdrop on.
Undulation in the Fossil Record
Trackways left by early amphibians offer a rare window into when and how undulation made the transition from water to land. Exceptionally preserved Early Permian trackways from the Southern Alps document the shift from terrestrial-underwater walking to swimming in a single sequence of tracks. The patterns match closely with the anatomy and movement mechanics of living salamanders, providing evidence for a deep conservatism of locomotor mechanics among amphibians stretching back hundreds of millions of years.29Palaeogeography, Palaeoclimatology, Palaeoecology. Transition between terrestrial-submerged walking and swimming revealed by Early Permian amphibian trackways and a new proposal for the nomenclature of compound trace fossils The transition from walking to swimming in these early tetrapods appears to have been a graded balance between limb-driven and body-wave-driven propulsion, not an abrupt switch. That graded transition is still visible today when you watch a salamander walk into a pond and seamlessly shift from stepping to undulating.
Aquatic mammals took a very different evolutionary path. Rather than retaining the lateral body undulation of their fish ancestors, mammals that returned to the sea evolved vertical (dorsoventral) oscillation of the tail, flippers, or both. These lift-based swimming modes include caudal oscillation in dolphins and whales, pectoral oscillation in sea lions, and pelvic oscillation in some seals.30Oxford Academic. Secondary Evolution of Aquatic Propulsion in Higher Vertebrates: Validation and Prospect The reason for the vertical plane of motion traces back to the mammalian spine, which already flexed up and down during galloping. When these lineages returned to water, they co-opted that existing dorsoventral flexibility rather than re-evolving the lateral bending their distant fish ancestors used. The result is that a dolphin’s tail strokes up and down while a shark’s sweeps side to side, even though both are producing thrust through undulatory waves.