Free boundary wave problems are every solution of the corresponding parabolic equation on a variable unknown domain with free boundary conditions. The problem can have a unique solution providing the definition of the wave’s boundary conditions are meticulously selected. In order to derive a unique solution, the standing waves must satisfy the boundary conditions.
At a free boundary, the restoring force is zero and the reflected wave has the same polarity as the incident wave. A boundary equation generates many solutions depending on the input. In order to solve a free boundary wave equation, the boundary conditions must be specified correctly. Therefore, boundary problems should be well-posed in order to be useful in applications. Since the solution to the equation depends on the input, considerable theoretical work is dedicated to proving the input is well-posed.
Dirichlet’s Principle is used by mathematicians and physicists. Dirichlet’s Principle demonstrates that for every bounded, connected domain there are infinitely many functions where at least one function will reduce to a given value. Dirichlet’s Principle considers open sets for which the boundary consists of analytic curves and the boundary values are analytic. As long as restrictive assumptions about the given boundary conditions are satisfied, every regular problem reduces to a single solution.