Wave Equation via Laplace Transform

PDEs · Axiom Academy

LESSON Wave Equation via Laplace Transform Master the technique of solving the wave equation using Laplace transforms with boundary and initial conditions 1. The Wave Equation with Initial Conditions We begin with the one-dimensional wave equation on a semi-infinite domain: Where c is the wave speed. The initial conditions specify the initial displacement and velocity: For a semi-infinite domain (x ≥ 0), we also need a boundary condition at x = 0: 2. Applying the Laplace Transform in Time We take the Laplace transform with respect to time t , treating x as a parameter. Define: Using the transform properties for derivatives: The wave equation becomes an ODE in x : 3. Solving the Ordinary Differential Equation Rearranging the transformed equation: This is a second-order constant-coefficient ODE with characteristic equation: For boundedness as x → ∞, we require A(s) = 0, leaving: 4. Incorporating the Boundary Condition Consider the boundary condition u(0,t) = g(t). Taking the Laplace transform: From our solution U(x,s) = B(s)e -sx/c , evaluating at x = 0: Therefore, the complete solution in the transform domain is: 5. Inverse Transform: The Role of the Heaviside Function To find u(x,t), we use the shifting theorem for Laplace transforms: Where H(t) is the Heaviside step function. Applying this to our solution: The Heaviside function ensures causality: the solution at position x and time t depends only on the boundary value at the earlier time t - x/c.

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