equation in 3 dimensions, and 2) the Helmholtz equation in either 2 or 3 dimensions. Since the Helmholtz equation is a prototypical equation for modeling the spatial variation of classical waves, Mathieu functions can be used to describe a variety of wave phenomena. For instance, in computational electromagnetics they can be used to analyze the scattering of electromagnetic waves off elliptic cylinders, and wave propagation in elliptic waveguides. In general relativity, an exact plane wave solution to the Einstein field equation can be given in terms of Mathieu functions. More recently, Mathieu functions have been used to solve a special case of the Smoluchowski equation, describing the steady-state statistics of self-propelled particles. The remainder of this section details the analysis for the two-dimensional Helmholtz equation. In rectangular coordinates, the Helmholtz equation is Elliptic coordinates are defined by where , , and is a positive constant. The Helmholtz equation in these coordinates is The constant curves are confocal ellipses with focal length ; hence, these coordinates are convenient for solving the Helmholtz equation on domains with elliptic boundaries. Separation of variables via yields the Mathieu equations where is a separation constant. As a specific physical example, the Helmholtz equation can be interpreted as describing normal modes of an elastic membrane under uniform tension. In this case, the following physical conditions are imposed: Periodicity with respect to , i.e. Continuity of displacement across the interfocal line: Continuity of derivative across the interfocal line: For given , this restricts the solutions to those of the form and , where . This is the same as restricting allowable values of , for given . Restrictions on then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by . For instance, clamping the membrane at imposes , which in turn requires These conditions define the normal modes of the system. Dynamical problems In dynamical problems with periodically varying forces, the equation of motion sometimes takes the form of Mathieu's equation. In such cases, knowledge of the general properties of Mathieu's equation— particularly with regard to stability of the solutions—can be essential for understanding qualitative features of the physical dynamics. A classic example along these lines is the inverted pendulum. Other examples are vibrations of a string with periodically varying tension stability of railroad rails as trains drive over them seasonally forced population dynamics the phenomenon of parametric