In mathematics, prolate spheroidal wave functions (PSWFs) are eigenfunctions of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions on certain ellipsoids of revolution (an ellipse rotated around its long axis, cigar shape). Related are the oblate spheroidal wave functions (pancake shaped ellipsoid).
Solutions to the wave equation Solve the Helmholtz equation,
∇ 2 Φ + k 2 Φ = 0 {\displaystyle \nabla ^{2}\Phi +k^{2}\Phi =0} , by the method of separation of variables in prolate spheroidal coordinates, ( ξ , η , φ ) {\displaystyle (\xi ,\eta ,\varphi )} , with:
x = a ( ξ 2 − 1 ) ( 1 − η 2 ) cos φ , {\displaystyle \ x=a{\sqrt {(\xi ^{2}-1)(1-\eta ^{2})}}\cos \varphi ,}
y = a ( ξ 2 − 1 ) ( 1 − η 2 ) sin φ , {\displaystyle \ y=a{\sqrt {(\xi ^{2}-1)(1-\eta ^{2})}}\sin \varphi ,}
z = a ξ η , {\displaystyle \ z=a\,\xi \,\eta ,}
and ξ ≥ 1 {\displaystyle \xi \geq 1} , | η | ≤ 1 {\displaystyle |\eta |\leq 1} , and 0 ≤ φ ≤ 2 π {\displaystyle 0\leq \varphi \leq 2\pi } . Here, 2 a > 0 {\displaystyle 2a>0} is the interfocal distance of the elliptical cross section of the prolate spheroid. Setting c = k a {\displaystyle c=ka} , the solution Φ ( ξ , η , φ ) {\displaystyle \Phi (\xi ,\eta ,\varphi )} can be written as the product of e i m φ {\displaystyle e^{{\rm {i}}m\varphi }} , a radial spheroidal wave function R m n ( c , ξ ) {\displaystyle R_{mn}(c,\xi )} and an angular spheroidal wave function S m n ( c , η ) {\displaystyle S_{mn}(c,\eta )} . The radial wave function R m n ( c , ξ ) {\displaystyle R_{mn}(c,\xi )} satisfies the linear ordinary differential equation:
( ξ 2 − 1 ) d 2 R m n ( c , ξ ) d ξ 2 + 2 ξ d R m n ( c , ξ ) d ξ − ( λ m n ( c ) − c 2 ξ 2 + m 2 ξ 2 − 1 ) R m n ( c , ξ ) = 0 {\displaystyle \ (\xi ^{2}-1){\frac {d^{2}R_{mn}(c,\xi )}{d\xi ^{2}}}+2\xi {\frac {dR_{mn}(c,\xi )}{d\xi }}-\left(\lambda _{mn}(c)-c^{2}\xi ^{2}+{\frac {m^{2}}{\xi ^{2}-1}}\right){R_{mn}(c,\xi )}=0}
The angular wave function satisfies the differential equation:
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