An overview of variational methods

Abstract

Variational methods provide a powerful method for determining the behavior of complex physical systems without needing to solve the equations of state or satisfy the boundary conditions exactly. This technique has been widely applied in the areas of structural vibration and scattering, but has not been used much in the areas of transduction and array modeling. The advantage of this method over many others is that the answers obtained from variational principles are always significantly better than the inputs to them. A totally general method for deriving stationary variational principles from the equations of state and the boundary/initial conditions will be shown. Techiniques for choosing the inputs, or trial functions, judiciously will be discussed. The method will be illustrated using two examples: Determining the resonance frequencies of a clamped vibrating string and of a string with an attached point mass. The results of the variational treatment will be compared with exact analytical solutions of the problem for both accuracy and simplicity of solution. A brief historical overview of variational methods in acoustics will also be given.