[Paper Review] Strings attached: New light on an old problem

Jeanne N. Clelland, Peter J. Vassiliou|arXiv (Cornell University)|Feb 27, 2013

Advanced Mathematical Modeling in Engineering7 references3 citations

TL;DR

This paper demonstrates that the classical wave equation $u_{tt} = c^2 u_{xx}$ is an exact model for transversely vibrating elastic strings in the plane, not a small-amplitude approximation, by deriving it from physical principles without simplifying assumptions. It generalizes this result to elastic strings in arbitrary Riemannian surfaces via the wave map equation $\nabla_{\mathbf{u}_t}\mathbf{u}_t = c^2 \nabla_{\mathbf{u}_x}\mathbf{u}_x$, showing that perfect elasticity—replacing transverse vibration—leads to an exact, intrinsic nonlinear description in curved spaces.

ABSTRACT

The wave equation $u_{tt} = c^2 u_{xx}$ is generally regarded as a linear approximation to the equation describing the amplitude of a transversely vibrating elastic string in the plane. But, as is shown in \cite{BC96}, the assumption of transverse vibration in fact implies that the wave equation describes the vibration precisely, with no need for approximation. We give a simplified proof of this result, and we generalize to the case of an elastic string vibrating (transversely or not) in a Riemannian surface $M$. In the more general setting, the assumption of transverse vibration is replaced by the assumption of "perfect elasticity," and we show that the wave map equation $ abla_{\bu_t} \bu_t = c^2 abla_{\bu_x} \bu_x$ gives a precise description of the vibration of a perfectly elastic string in $M$, with no need for approximation. Finally, we give examples describing the motion of various vibrating strings in $\R^2$, $S^2$, and $\mathbb{H}^2$.

Motivation & Objective

• To resolve the longstanding misconception that the wave equation is only a linear approximation to the true dynamics of a vibrating elastic string.
• To show that the wave equation arises exactly from physical principles without small-amplitude or constant-tension assumptions.
• To generalize the wave equation to elastic strings vibrating in Riemannian surfaces, where transverse vibration is ill-defined.
• To establish the wave map equation as the exact governing equation for perfectly elastic strings in curved ambient spaces.
• To provide explicit solutions and numerical examples in $\mathbb{E}^2$, $S^2$, and $\mathbb{H}^2$ to illustrate the dynamics in non-flat geometries.

Proposed method

• Derive the wave equation from Newtonian mechanics and energy principles without approximations, using the full nonlinear expression for string length and tension.
• Introduce the concept of 'perfect elasticity' as a replacement for transverse vibration in curved spaces, ensuring energy conservation and consistent dynamics.
• Formulate the motion of the string as a map $\mathbf{u} : [0,L] \times \mathbb{R} \to M$ into a Riemannian surface $M$, governed by the wave map equation.
• Use the Levi-Civita connection $\nabla$ on $M$ to express the wave map equation $\nabla_{\mathbf{u}_t}\mathbf{u}_t = c^2 \nabla_{\mathbf{u}_x}\mathbf{u}_x$ as the exact equation of motion.
• Solve the wave map equation numerically in $\mathbb{E}^2$, $S^2$, and $\mathbb{H}^2$ using parametrized initial curves and time evolution.
• Analyze the behavior of solutions in non-flat spaces, particularly in $\mathbb{H}^2$, to demonstrate intrinsic nonlinearity and loss of periodicity at large amplitudes.

Experimental results

Research questions

• RQ1 Is the classical wave equation $u_{tt} = c^2 u_{xx}$ an exact description of transverse vibrations in an elastic string, or merely a small-amplitude approximation?
• RQ2 What physical principle replaces transverse vibration in curved Riemannian surfaces, and how does it lead to a consistent dynamical model?
• RQ3 Does the wave map equation $\nabla_{\mathbf{u}_t}\mathbf{u}_t = c^2 \nabla_{\mathbf{u}_x}\mathbf{u}_x$ arise exactly from physical laws without approximations in general Riemannian geometries?
• RQ4 How do solutions to the wave map equation behave in non-Euclidean spaces like $\mathbb{H}^2$, particularly at large amplitudes?
• RQ5 What geometric and dynamic properties emerge in wave map systems on $S^2$ and $\mathbb{H}^2$, and how do they differ from the flat case?

Key findings

• The wave equation $u_{tt} = c^2 u_{xx}$ is derived exactly from physical principles without assuming small amplitudes or constant tension, proving it is not an approximation.
• The assumption of transverse vibration in the plane is shown to be sufficient to derive the wave equation exactly, with all nonlinear terms canceling out identically.
• In Riemannian surfaces, the wave map equation $\nabla_{\mathbf{u}_t}\mathbf{u}_t = c^2 \nabla_{\mathbf{u}_x}\mathbf{u}_x$ governs the motion of a perfectly elastic string, with no approximations.
• Solutions in $\mathbb{H}^2$ are intrinsically nonlinear and non-periodic, with large-amplitude initial conditions leading to non-embedded configurations and asymmetric motion.
• In $\mathbb{H}^2$, a horizontal string of length 2 at $y=1$ shows nearly periodic behavior, but increasing amplitude causes significant distortion and loss of periodicity.
• A sinusoidal perturbation of a vertical geodesic in $\mathbb{H}^2$ exhibits asymmetric leftward motion during each cycle, highlighting the influence of curvature on dynamics.