2013/02/27 by Jeanne N. Clelland, Clelland, Jeanne N., Peter J. Vassiliou +1
Computer Science · Physics and Astronomy · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1302.6672
The wave equation utt = c2 uxx 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 \citeBC96, 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 ∇\but \but = c2 ∇\bux \bux 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 \R2, S2, and ℍ2.