09-23-2021-0616 - d'Alembert's formula & Bessel functions

 

In mathematics, and specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the one-dimensional wave equation  (where subscript indices indicate partial differentiation, using the d'Alembert operator, the PDE becomes: ).

The solution depends on the initial conditions at  :  and . It consists of separate terms for the initial conditions  and  : 

It is named after the mathematician Jean le Rond d'Alembert, who derived it in 1747 as a solution to the problem of a vibrating string.^[1]

https://en.wikipedia.org/wiki/D%27Alembert%27s_formula

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation

for an arbitrary complex number α, the order of the Bessel function. Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α.

The most important cases are when α is an integer or half-integer. Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer α are obtained when the Helmholtz equation is solved in spherical coordinates.

Bessel functions are the radial part of the modes of vibration of a circular drum.

https://en.wikipedia.org/wiki/Bessel_function

https://en.wikipedia.org/wiki/Spherical_harmonics

https://en.wikipedia.org/wiki/Inhomogeneous_electromagnetic_wave_equation

https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

https://en.wikipedia.org/wiki/Zonal_spherical_harmonics

https://en.wikipedia.org/wiki/Spin-weighted_spherical_harmonics

https://en.wikipedia.org/wiki/Ladder_operator

https://en.wikipedia.org/wiki/Creation_and_annihilation_operators