An acoustic eigenvalue problem and its application to electrochemistry

The fundamental process that lies at the foundation of batteries, capacitors, and solar cells is the electron transfer process. This takes place at an interface or boundary in each device and is governed by its corresponding chemical reaction. Making these devices more efficient can help decrease our negative impact on the environment. Recent experiments in the field of Electrochemistry demonstrate that sound waves act as a catalyst for these electron transfer reactions. A model is developed using an Euler equation (conservation of momentum), conservation of mass equation, boundary motion equation, and surface tension equation. Chemically, it is clear that the catalytic phenomenon is derived from the sound waves and how they are affected by the top boundary. When combining these four equations we arrive at a boundary condition involving the top boundary only. We place this condition and the other contributing boundary and initial conditions on the wave equation to understand the interaction that occurs between the waves and the cell. We establish a self-adjoint operator and further use its inverse. Overall, using the Variational form and the Galerkin Method an approximation converges to the solution of the wave equation. With the help of MATLAB these eigenfunctions can be articulated as standing waves.