§29.11 Lamé Wave Equation

Keywords:: Lamé wave equation, ellipsoidal wave equation
Permalink:: http://dlmf.nist.gov/29.11

The Lamé (or ellipsoidal) wave equation is given by

29.11.1 $\frac{{d}^{2}w}{{dz}^{2}}+(h-\nu(\nu+1)k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z,k\right)+k^{2}\omega^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{4}}}\left(z,k\right))w=0,$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter, $\omega$ : parameter and $w(z)$ : solution
Referenced by:: §29.11, §29.11, §29.18(ii), §29.7(ii)
Permalink:: http://dlmf.nist.gov/29.11.E1
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in which $\omega$ is another parameter. In the case $\omega=0$ , (29.11.1) reduces to Lamé’s equation (29.2.1).

For properties of the solutions of (29.11.1) see Arscott (1956, 1959), Arscott (1964b, Chapter X), Erdélyi et al. (1955, §16.14), Fedoryuk (1989), and Müller (1966a, b, c).