Mathieu functions occur in practical applications in two main categories:
Initial-value problems, in which only one equation (28.2.1) or (28.20.1) is involved. See §28.33(iii).
Physical problems involving Mathieu functions include vibrational problems in elliptical coordinates; see (28.32.1). We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho $ per unit area, and radial tension $\tau $ per unit arc length. The wave equation
28.33.1 | $$\frac{{\partial}^{2}W}{{\partial x}^{2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}-\frac{\rho}{\tau}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0,$$ | ||
with $W\left(x,y,t\right)={\mathrm{e}}^{\mathrm{i}\omega t}V\left(x,y\right)$, reduces to (28.32.2) with ${k}^{2}={\omega}^{2}\rho /\tau $. In elliptical coordinates (28.32.2) becomes (28.32.3). The separated solutions ${V}_{n}\left(\xi ,\eta \right)$ must be $2\pi $-periodic in $\eta $, and have the form
28.33.2 | $${V}_{n}\left(\xi ,\eta \right)=\left({c}_{n}{\mathrm{M}}_{n}^{\left(1\right)}\left(\xi ,\sqrt{q}\right)+{d}_{n}{\mathrm{M}}_{n}^{\left(2\right)}\left(\xi ,\sqrt{q}\right)\right){\mathrm{me}}_{n}\left(\eta ,q\right),$$ | ||
where $q=\frac{1}{4}{c}^{2}{k}^{2}$ and ${a}_{n}\left(q\right)$ or ${b}_{n}\left(q\right)$ is the separation constant; compare (28.12.11), (28.20.11), and (28.20.12). Here ${c}_{n}$ and ${d}_{n}$ are constants. The boundary conditions for $\xi ={\xi}_{0}$ (outer clamp) and $\xi ={\xi}_{1}$ (inner clamp) yield the following equation for $q$:
28.33.3 | $${\mathrm{M}}_{n}^{\left(1\right)}\left({\xi}_{0},\sqrt{q}\right){\mathrm{M}}_{n}^{\left(2\right)}\left({\xi}_{1},\sqrt{q}\right)-{\mathrm{M}}_{n}^{\left(1\right)}\left({\xi}_{1},\sqrt{q}\right){\mathrm{M}}_{n}^{\left(2\right)}\left({\xi}_{0},\sqrt{q}\right)=0.$$ | ||
If we denote the positive solutions $q$ of (28.33.3) by ${q}_{n,m}$, then the vibration of the membrane is given by ${\omega}_{n,m}^{2}=4{q}_{n,m}\tau /\left({c}^{2}\rho \right)$. The general solution of the problem is a superposition of the separated solutions.
For a visualization see Gutiérrez-Vega et al. (2003), and for references to other boundary-value problems see:
McLachlan (1947, Chapters XVI–XIX) for applications of the wave equation to vibrational systems, electrical and thermal diffusion, electromagnetic wave guides, elliptical cylinders in viscous fluids, and diffraction of sound and electromagnetic waves.
Meixner and Schäfke (1954, §§4.3, 4.4) for elliptic membranes and electromagnetic waves.
Daymond (1955) for vibrating systems.
Troesch and Troesch (1973) for elliptic membranes.
If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as $\mathrm{cos}\left(2\omega t\right)$. The equation of motion is given by
28.33.4 | $${w}^{\mathrm{\prime \prime}}\left(t\right)+\left(b-f\mathrm{cos}\left(2\omega t\right)\right)w\left(t\right)=0,$$ | ||
with $b$, $f$, and $\omega $ positive constants. Substituting $z=\omega t$, $a=b/{\omega}^{2}$, and $2q=f/{\omega}^{2}$, we obtain Mathieu’s standard form (28.2.1).
As $\omega $ runs from $0$ to $+\mathrm{\infty}$, with $b$ and $f$ fixed, the point $\left(q,a\right)$ moves from $\mathrm{\infty}$ to $0$ along the ray $\mathcal{L}$ given by the part of the line $a=\left(2b/f\right)q$ that lies in the first quadrant of the $\left(q,a\right)$-plane. Hence from §28.17 the corresponding Mathieu equation is stable or unstable according as $\left(q,a\right)$ is in the intersection of $\mathcal{L}$ with the colored or the uncolored open regions depicted in Figure 28.17.1. In particular, the equation is stable for all sufficiently large values of $\omega $.
For points $\left(q,a\right)$ that are at intersections of $\mathcal{L}$ with the characteristic curves $a={a}_{n}\left(q\right)$ or $a={b}_{n}\left(q\right)$, a periodic solution is possible. However, in response to a small perturbation at least one solution may become unbounded.
References for other initial-value problems include:
McLachlan (1947, Chapter XV) for amplitude distortion in moving-coil loud-speakers, frequency modulation, dynamical systems, and vibration of stretched strings.
Vedeler (1950) for ships rolling among waves.
Meixner and Schäfke (1954, §§4.1, 4.2, and 4.7) for quantum mechanical problems and rotation of molecules.
Aly et al. (1975) for scattering theory.
Fukui and Horiguchi (1992) for quantum theory.
Torres-Vega et al. (1998) for Mathieu functions in phase space.