[Paper Review] Scaling the tail beat frequency and swimming speed in underwater undulatory swimming

Jesus Sanchez Roriguez, Christophe Raufaste+1 more|arXiv (Cornell University)|Jan 25, 2023

Fish Ecology and Management StudiesEnvironmental Science3 citations

TL;DR

This study proposes scaling laws linking tail beat frequency and swimming speed in undulatory swimmers by integrating muscle physiology and fluid dynamics. It reveals a crossover at 0.5–1 m body length: below this, frequency is tuned biologically (2–20 Hz); above it, fluid resistance dominates, making frequency inversely proportional to length, predicting a maximum speed of 5–10 m·s⁻¹ consistent with cavitation limits.

ABSTRACT

Due to its great efficiency and maneuverability, undulatory swimming is the predominant form of locomotion in aquatic vertebrates. A myriad of animals of different species and sizes oscillate their bodies to propel themselves in aquatic environments with swimming speed scaling as the product of the animal length by the oscillation frequency. Although frequency tuning is the primary means by which a swimmer selects its speed, there is no consensus on the mechanisms involved. In this article, we propose scaling laws for undulatory swimmers that relate oscillation frequency to length by taking into account both the biological characteristics of the muscles and the interaction of the moving swimmer with its environment. Results are supported by an extensive literature review including approximately 1200 individuals of different species, sizes and swimming environments. We highlight a crossover in length around 0.5-1 m. Below this value, the frequency can be tuned between 2-20 Hz due to biological constraints and the interplay between slow and fast muscles. Above this value, the fluid-swimmer interaction must be taken into account and the frequency is inversely proportional to the length of the animal. This approach predicts a maximum swimming speed around 5-10 m.s$^{-1}$ for large swimmers, consistent with the threshold to prevent bubble cavitation.

Motivation & Objective

• To resolve the lack of consensus on how tail beat frequency scales with body length in aquatic vertebrates.
• To identify the dominant physical and biological mechanisms governing frequency selection across species of varying size.
• To unify disparate experimental data into a coherent scaling framework by analyzing 1,200 data points from diverse species and swimming conditions.
• To distinguish between biological constraints (muscle type, activity level) and hydrodynamic interactions as governing factors at different size scales.
• To predict the upper limit of swimming speed in large swimmers, linking it to physical constraints like cavitation.

Proposed method

• Collected and compiled approximately 1,200 experimental data points on tail beat frequency and body length from diverse aquatic vertebrates, without filtering by activity level or species.
• Used logarithmic binning of length to compute bounds for burst and sustained swimming frequencies, ensuring robust statistical representation.
• Applied least absolute deviations (LAD) fitting to model the frequency-length relationship, with parameters optimized across multiple interval counts (N = 10 to 50).
• Identified and excluded unreliable data points—non-peer-reviewed sources, estimates, and rod-mounted device measurements—using objective criteria from Hirt et al. [49].
• Proposed a theoretical framework balancing muscle force and fluid reactive forces to explain frequency selection across size scales.
• Derived allometric scaling laws for frequency and swimming speed, distinguishing between small swimmers (biologically tuned) and large swimmers (hydrodynamically constrained).

Figure 1: Tail beat frequency $f$ as a function of length $L$ for amphibians (yellow), fish (blue), reptiles (green), birds (red) and mammals (purple). Thick black and grey lines represent the burst and sustained activity levels, respectively, fitted with the model. Thin lines are the scaling laws i

Experimental results

Research questions

• RQ1 What physical and biological mechanisms govern the scaling of tail beat frequency with body length in undulatory swimmers?
• RQ2 How does the dominant control mechanism for frequency change across different size classes of aquatic vertebrates?
• RQ3 What is the relationship between swimming speed, frequency, and body length across multiple orders of magnitude in size?
• RQ4 Why is there a lack of consensus in existing scaling laws for frequency, and how can a unified model be established?
• RQ5 What limits the maximum swimming speed in large aquatic animals, and is this consistent with physical constraints like cavitation?

Key findings

• A crossover in scaling behavior occurs at a body length of 0.5–1 m, separating two distinct regimes of frequency control.
• For animals below 0.5–1 m, frequency is tunable between 2–20 Hz due to biological constraints involving slow and fast muscle fibers.
• For animals above 0.5–1 m, fluid-swimmer interaction dominates, and frequency scales inversely with body length (f ∝ L⁻¹).
• Swimming speed scales as U ≈ 0.7Lf, with a consistent proportionality factor between 0.4 and 1 across fish and cetaceans.
• The model predicts a maximum swimming speed of 5–10 m·s⁻¹ for large swimmers, consistent with the threshold to avoid bubble cavitation.
• The exclusion of non-peer-reviewed, estimated, and rod-mounted device data improved the reliability of the frequency-length relationship, reducing artificial outliers.

Figure 2: Swimming speed $U$ as a function of length $L$ . Following the law $U=0.7Lf$ , we show our estimates of swimming speed from the tail beat frequency measurements displayed in Fig. 1 (closed circles): amphibians (yellow), fish (blue), reptiles (green), birds (red), and mammals (purple). Brow