What is a Bessel function used for?

What is a Bessel function used for?

Bessel functions are used to solve in 3D the wave equation at a given (harmonic) frequency. The solution is generally a sum of spherical bessels functions that gives the acoustic pressure at a given location of the 3D space. Bessel function is not only shown in acoustic field, but also in the heat transfer.

What is first kind Bessel function?

The Bessel functions of the first kind, denoted by J α ( x ) , are solutions of Bessel’s differential equation that are finite at the origin . The Bessel function J α ( x ) can be defined by the series. (1.13) Γ ( m + α + 1 ) ( x 2 ) 2 m + α .

What are Hankel functions?

Hankel Functions (14.91) H ν ( 1 ) ( x ) = J ν ( x ) + iY ν ( x ) , H ν ( 2 ) ( x ) = J ν ( x ) – iY ν ( x ) . These functions see use in problems involving incoming or outgoing waves, because the oscillation of J ν and Y ν is converted into a large- x behavior of e ix for H ( 1 ) ( x ) and e – ix for H ( 2 ) ( x ) .

What is spherical Bessel function?

When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form. The two linearly independent solutions to this equation are called the spherical Bessel functions jn and yn, and are related to the ordinary Bessel functions Jn and Yn by.

How is Bessel function calculated?

For cylindrical problems the order of the Bessel function is an integer value (ν = n) while for spherical problems the order is of half integer value (ν = n + 1/2).

What is a Bessel equation?

The Bessel differential equation is the linear second-order ordinary differential equation given by. (1) Equivalently, dividing through by , (2) The solutions to this equation define the Bessel functions and .

What is Laplace method?

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (often time) to a function of a complex variable. (complex frequency).

How do you form an Indicial equation?

y′=∞∑k=0(k+r)akxk+r−1,y″=∞∑k=0(k+r)(k+r−1)akxk+r−2. 4r(r−1)+1=0. This equation is called the indicial equation. This particular indicial equation has a double root at r=12.