The Resolution of Boundary Value Problems by Means of the Finite Fourier Transformation: General Vibration of a String

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Abstract
The finite sine transformation and the finite cosine transformation are defined as the linear functional operations S{G}=∫0πG(x)sinnxdx=gs(n) and C{G}=∫0πG(x)cosnxdx=gc(n), respectively. The inversion of the product of two transforms can be made by means of four Faltung theorems. The finite sine transformation was applied to a boundary value problem of a general vibrating string, in which the partial differential equation has coefficients which may be functions of the time. A resolution was made of the transformed boundary value problem by the introduction of a fundamental set of solutions of the homogeneous transformed problem; an inversion in closed form was accomplished by the use of the Faltung theorems. A formal verification of this solution was made as well as a short survey of the applicability of the finite Fourier transformation to problems in engineering and physics.

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