§28.33 Physical Applications

Mathieu functions occur in practical applications in two main categories:

• •

  Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

• •

  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

with $W(x,y,t)={\mathrm{e}}^{\mathrm{i}\omega t}V(x,y)$, 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 $V_n(\xi ,\eta )$ must be $2\pi$-periodic in $\eta$, and have the form

where $q=\frac{1}{4}{c}^2{k}^2$ and $a_n(q)$ or $b_n(q)$ 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 $q$:

If we denote the positive solutions $q$ of (28.33.3) by $q_{n,m}$, then the vibration of the membrane is given by ${\omega }_{n,m}^2=4q_{n,m}\tau /(c^2\rho )$. 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.

• •

  Alhargan and Judah (1995), Bhattacharyya and Shafai (1988), and Shen (1981) for ring antennas.

• •

  Alhargan and Judah (1992), Germey (1964), Ragheb et al. (1991), and Sips (1967) for electromagnetic waves.

More complete bibliographies will be found in McLachlan (1947) and Meixner and Schäfke (1954).

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 $\cos \left(2\omega t\right)$. The equation of motion is given by

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 $(q,a)$ moves from $\mathrm{\infty }$ to $0$ along the ray $ℒ$ given by the part of the line $a=(2b/f)q$ that lies in the first quadrant of the $(q,a)$-plane. Hence from §28.17 the corresponding Mathieu equation is stable or unstable according as $(q,a)$ is in the intersection of $ℒ$ 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 $(q,a)$ that are at intersections of $ℒ$ 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.

• •

  Hunter and Kuriyan (1976) and Rushchitsky and Rushchitska (2000) for wave mechanics.

• •

  Fukui and Horiguchi (1992) for quantum theory.

• •

  Jager (1997, 1998) for relativistic oscillators.

• •

  Torres-Vega et al. (1998) for Mathieu functions in phase space.