Taut String Model: Getting the Right Energy versus Getting the Energy the Right Way

The initial boundary value problem of the transverse vibration of a taut string is a classic that can be found in many vibration and acoustics textbooks. It is often used as the basis for derivations of elementary numerical models, for instance finite element or finite difference schemes. The model of axial vibration of a prismatic elastic bar also serves in this capacity, often times side-by-side with the first model. The stored (potential) energy for these two models is derived in the literature in two distinct ways. We find the potential energy in the taut string model to be derived from a second-order expression of the change of the length of the string. This is very different in nature from the corresponding expression for the elastic bar, which is predictably based on the work of the internal forces. The two models are mathematically equivalent in that the equations of one can be obtained from the equations of the other by substitution of symbols such as the primary variable, the resisting force and the coefficient of the stiffness. The solutions also have equivalent meanings, such as propagation of waves and standing waves of free vibration. Consequently, the analogy between the two models can and should be exploited, which the present paper successfully undertakes. The potential energy of deformation of the string was attributed to the seminal work of Morse and Feshbach of 1953. This book was also the source of a misunderstanding as to the correct expression for the density of the energy of deformation. The present paper strives to settle this question.