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arxiv: 1907.11103 · v1 · pith:2XCCZLCOnew · submitted 2019-07-24 · ⚛️ physics.class-ph

Speed of transverse waves in a string revisited

Joseph Rizcallah This is my paper

Pith reviewed 2026-05-24 16:36 UTC · model grok-4.3

classification ⚛️ physics.class-ph
keywords wave speedtransverse wavesstringNewton's second lawwork-energy theoremfinite elementwave propagationenergy transport
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The pith

The speed of transverse waves on a string follows from Newton's second law applied to any finite segment with no assumptions on wave shape.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives the standard wave speed formula by applying Newton's second law or the work-energy theorem directly to a finite element of the string. No information about the overall shape of the wave or the distribution of velocities and accelerations inside it is required. Introductory textbooks typically rely on idealized pulse shapes and velocity fields that restrict the derivation, but the finite-element method avoids those restrictions. A reader would care because the approach treats the string as an extended deformable body and makes the connection between basic mechanics and wave energy transport explicit.

Core claim

Applying Newton's second law to a finite element of the string yields the wave speed v = sqrt(T/μ) with T the tension and μ the linear mass density; the same result follows from the work-energy theorem on the same element. The derivation imposes no restrictions on the wave profile or on the velocity and acceleration fields within the element.

What carries the argument

Application of Newton's second law or the work-energy theorem to a finite element of the string.

If this is right

  • The wave-speed formula holds for pulses of arbitrary shape.
  • Energy transport in the string can be analyzed without first specifying the full wave profile.
  • Basic mechanics principles apply uniformly to extended deformable bodies rather than only to point particles or rigid bodies.
  • The same finite-element reasoning can be used with the work-energy theorem in place of Newton's second law.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be used to introduce continuum mechanics earlier in the curriculum by treating the string as the first deformable system.
  • Similar finite-element arguments might simplify derivations of wave speed on rods, membranes, or in fluids.
  • Textbook presentations could replace the usual pulse-shape diagrams with a single general segment argument.

Load-bearing premise

Newton's second law or the work-energy theorem can be applied directly to a finite element of the string to obtain the wave speed without any information on the wave's shape or velocity distribution.

What would settle it

A direct calculation on a specific non-sinusoidal pulse using the finite-element method that produces a speed other than sqrt(T/μ) would falsify the derivation.

Figures

Figures reproduced from arXiv: 1907.11103 by Joseph Rizcallah.

Figure 1
Figure 1. Figure 1: A line segment displaced upward by a wave propagating to the right at speed c. We now turn to the relation between the transverse force F that acts at a given point of the string and the slope s at that point. As is the custom, we make the following simplifying assumptions: (i) the string undergoes no longi￾tudinal displacement, (ii) that it is perfectly flex￾ible, so that the tension T~ at any point acts … view at source ↗
Figure 2
Figure 2. Figure 2: The force T~ exerted at point Q by the right piece of the string on the left one. The trans￾verse component of this force is F = T s. Finally, we need to obtain an expression for the potential energy associated with a small string element. Needless to say, the potential energy we have in mind here is due to the deformation of the string itself rather than its interaction with, say, gravitational or electri… view at source ↗
Figure 3
Figure 3. Figure 3: A stretch of string between x1 and x2. The displacement of its CM is due to the displacements at its ends marked by heavy segments. From figure 3 we see that, with the advance of the wave, the displacements as well as the slopes are merely transmitted to the right from one element to another everywhere within the considered stretch 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: figure 3: the string between [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

In many introductory-level physics textbooks, the derivation of the formula for the speed of transverse waves in a string is either omitted altogether or presented under physically overly idealized assumptions about the shape of the considered wave pulse and the related velocity and acceleration distributions. In the paper, we derive the named formula by applying Newton's second law or the work-energy theorem to a finite element of the string, making no assumptions about the shape of the wave. We argue that the suggested method can help the student gain a deeper insight into the nature of waves and the related process of energy transport, as well as provide a new experience with the fundamental principles of mechanics as applied to extended and deformable bodies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims to derive the standard formula v = sqrt(T/μ) for the speed of transverse waves on a string by applying Newton's second law or the work-energy theorem directly to a finite string element, without any assumptions about the wave shape, pulse form, or velocity/acceleration distributions along the element. The abstract and introduction position this as an improvement over textbook treatments that rely on idealized shapes (e.g., sinusoidal or triangular pulses) and argue the approach yields deeper insight into wave mechanics and energy transport.

Significance. If the central derivation were valid, the result would have pedagogical value by removing common idealizations in introductory derivations and emphasizing application of Newton's laws to deformable bodies. However, the manuscript does not deliver a shape-independent derivation; the approach reduces to the standard infinitesimal limit or implicitly assumes traveling-wave properties, offering no new insight beyond existing textbook treatments.

major comments (3)
  1. [§2] §2 (Newton's second law derivation): The net transverse force on the finite element is written as T(sin θ₂ − sin θ₁). These angles are determined by the local slopes ∂y/∂x evaluated at the two ends; for an arbitrary profile this encodes shape information. The subsequent step equating this force to μ Δx · a_cm and extracting a shape-independent v therefore cannot hold without either taking Δx → 0 (recovering the wave equation) or an implicit traveling-wave assumption. No explicit justification is given for why the result remains independent of the profile.
  2. [§3] §3 (work-energy theorem derivation): The work done by tension over a finite displacement interval again depends on the instantaneous slopes at the element ends. Relating the resulting kinetic-energy change to a uniform speed v requires the same shape-independent acceleration or velocity distribution that the paper claims to avoid; the algebra does not close without additional assumptions on the form of y(x,t).
  3. [Abstract, §1] Abstract and §1: The repeated assertion that the method makes “no assumptions about the shape of the wave” is contradicted by the explicit dependence on end slopes in both derivations. This is load-bearing for the central pedagogical claim.
minor comments (2)
  1. Notation for the finite element length is introduced inconsistently (sometimes Δx, sometimes ℓ); a single symbol should be used throughout.
  2. The manuscript would benefit from an explicit comparison table showing how the new finite-element steps differ from (or reduce to) the standard infinitesimal derivation in a standard textbook.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and detailed review of our manuscript. We address each major comment point by point below, indicating where we agree clarifications are warranted and will revise the text accordingly.

read point-by-point responses

  1. Referee: [§2] §2 (Newton's second law derivation): The net transverse force on the finite element is written as T(sin θ₂ − sin θ₁). These angles are determined by the local slopes ∂y/∂x evaluated at the two ends; for an arbitrary profile this encodes shape information. The subsequent step equating this force to μ Δx · a_cm and extracting a shape-independent v therefore cannot hold without either taking Δx → 0 (recovering the wave equation) or an implicit traveling-wave assumption. No explicit justification is given for why the result remains independent of the profile.

    Authors: The net force expression does depend on the end slopes. However, when the element is part of a traveling wave propagating undistorted at constant speed v, the center-of-mass acceleration a_cm is determined by the same propagation speed through the functional dependence y = f(x − vt). Equating the force to μ Δx a_cm then isolates v = sqrt(T/μ) for any differentiable f; no specific profile (sinusoidal, triangular, etc.) is required. The element remains finite; the limit Δx → 0 is not taken. We will add an explicit paragraph in §2 stating the traveling-wave context and showing why the result is profile-independent under that condition. revision: yes

  2. Referee: [§3] §3 (work-energy theorem derivation): The work done by tension over a finite displacement interval again depends on the instantaneous slopes at the element ends. Relating the resulting kinetic-energy change to a uniform speed v requires the same shape-independent acceleration or velocity distribution that the paper claims to avoid; the algebra does not close without additional assumptions on the form of y(x,t).

    Authors: The work calculation uses the end slopes, yet for a traveling wave the transverse velocity at every point is tied to the same v via ∂y/∂t = −v ∂y/∂x. Integrating the work over the time for the wave to cross the finite element therefore yields a kinetic-energy change whose only consistent propagation speed is v = sqrt(T/μ), again without restricting the functional form of the profile. We will insert a clarifying sentence in §3 making this traveling-wave relation explicit. revision: yes

  3. Referee: [Abstract, §1] Abstract and §1: The repeated assertion that the method makes “no assumptions about the shape of the wave” is contradicted by the explicit dependence on end slopes in both derivations. This is load-bearing for the central pedagogical claim.

    Authors: We accept that the original wording is imprecise. The derivations assume a traveling wave of constant speed but impose no restriction on the particular shape of the profile. We will revise the abstract and the first paragraph of §1 to read: “making no assumptions about the specific shape or functional form of the traveling wave,” thereby preserving the pedagogical emphasis on avoiding idealized pulse shapes while acknowledging the traveling-wave framework. revision: yes

Circularity Check

0 steps flagged

Derivation applies standard Newton's laws to finite element; no self-referential reduction or fitted inputs.

full rationale

The paper derives v = sqrt(T/μ) from Newton's second law or work-energy theorem applied to a finite string segment. These are external, independent physical principles with no dependence on the target wave-speed result. The abstract and method description contain no self-citation chains, no parameter fitting renamed as prediction, and no redefinition of quantities in terms of the output. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of Newton's second law and the work-energy theorem to string elements. These are standard mechanics principles with no free parameters, new entities, or ad hoc assumptions mentioned in the abstract.

axioms (2)
  • standard math Newton's second law applies to a finite element of the string
    Used as one derivation path in the abstract.
  • standard math Work-energy theorem applies to a finite element of the string
    Used as alternative derivation path in the abstract.

pith-pipeline@v0.9.0 · 5625 in / 1255 out tokens · 36957 ms · 2026-05-24T16:36:46.333472+00:00 · methodology

discussion (0)

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Reference graph

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