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28 Mathieu Functions and Hill’s EquationApplications

§28.32 Mathematical Applications

  1. §28.32(i) Elliptical Coordinates and an Integral Relationship
  2. §28.32(ii) Paraboloidal Coordinates

§28.32(i) Elliptical Coordinates and an Integral Relationship

If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. These are given by

28.32.1 x =ccoshξcosη,
y =csinhξsinη.

The two-dimensional wave equation

28.32.2 2Vx2+2Vy2+k2V=0

then becomes

28.32.3 2Vξ2+2Vη2+12c2k2(cosh(2ξ)cos(2η))V=0.

The separated solutions V(ξ,η)=v(ξ)w(η) can be obtained from the modified Mathieu’s equation (28.20.1) for v and from Mathieu’s equation (28.2.1) for w, where a is the separation constant and q=14c2k2.

This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting ζ=iξ, z=η in (28.32.3)).

Let u(ζ) be a solution of Mathieu’s equation (28.2.1) and K(z,ζ) be a solution of

28.32.4 2Kz22Kζ2=2q(cos(2z)cos(2ζ))K.

Also let be a curve (possibly improper) such that the quantity

28.32.5 K(z,ζ)du(ζ)dζu(ζ)K(z,ζ)ζ

approaches the same value when ζ tends to the endpoints of . Then

28.32.6 w(z)=K(z,ζ)u(ζ)dζ

defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to z uniformly on compact subsets of .

Kernels K can be found, for example, by separating solutions of the wave equation in other systems of orthogonal coordinates. See Schmidt and Wolf (1979).

§28.32(ii) Paraboloidal Coordinates

The general paraboloidal coordinate system is linked with Cartesian coordinates via

28.32.7 x1 =12c(cosh(2α)+cos(2β)cosh(2γ)),
x2 =2ccoshαcosβsinhγ,
x3 =2csinhαsinβcoshγ,

where c is a parameter, 0α<, π<βπ, and 0γ<. When the Helmholtz equation

28.32.8 2V+k2V=0

is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which A,B are separation constants. Two conditions are used to determine A,B. The first is the 2π-periodicity of the solutions; the second can be their asymptotic form. For further information see Arscott (1967) for k2<0, and Urwin and Arscott (1970) for k2>0.