Analytical Solution of One-Dimensional Wave Equations Using the Laplace Transform Method

In this paper, the Laplace Transform Method has been applied to one-dimensional wave equations with different boundary conditions. The transformation was performed with respect to the time variable, converting the governing partial differential equation into an ordinary differential equation in the spatial variable. Two examples were considered. The first example involved homogeneous boundary conditions and sinusoidal initial displacement. The resulting solution was obtained in closed form as u(x,t) = sin(πx) cos(πct) and was verified by direct substitution into the governing equation and by checking the initial and boundary conditions. The second example involved a non-homogeneous sinusoidal boundary condition. The Laplace transform was used to obtain the transformed solution, and the inverse solution was represented through the natural modes of the finite interval. The analysis also identified the resonance condition associated with the natural frequencies of the system. The results demonstrate that the Laplace Transform Method provides a direct and systematic analytical framework for solving one-dimensional wave equations. The method is particularly useful for linear initial-boundary value problems and provides clear mathematical and physical interpretation of wave propagation and resonance.

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