Spheroidal wave equation explained

In mathematics, the spheroidal wave equation is given by

( 1- t ^{2 ) (d ^{2y})/(dt ^{2})} -2 (b +1 ) t (d y)/(dt) + (c - 4 qt ^{2)} y =0

It is a generalization of the Mathieu differential equation.^[1] If

y(t) is a solution to this equation and we define S(t): = ( 1- t ^{2)} ^{b/2} y(t) , then S(t) is a prolate spheroidal wave function in the sense that it satisfies the equation^[2] ( 1- t ^{2 ) (d ^{2S})/(dt ^{2})} -2 t (d S)/(dt) + (c - 4 q + b + b ^{2} + 4 q( 1- t ^{2)} - (b ^{2})/(1- t ^{2}) ) S =0

See also

• Wave equation

References

Bibliography

• M. Abramowitz and I. Stegun, Handbook of Mathematical function (US Gov. Printing Office, Washington DC, 1964)
• H. Bateman, Partial Differential Equations of Mathematical Physics (Dover Publications, New York, 1944)

Notes and References

1. see Abramowitz and Stegun, page 722
2. see Bateman, page 442