BESSELY

Calculates the Bessel function of the second kind (the Neumann or Weber function).

Syntax:

BESSELY(x, n)

returns the Bessel function of the second kind, of order n, evaluated at x.

The Bessel functions of the second kind $Y_{n} (x)$ (also known as Neumann $N_{n} (x)$ or Weber functions) are solutions to the Bessel differential equation which are singular at the origin.

Example:

BESSELY(4, 1)

returns 0.397925709051.

Application:

Vibrations of a Circular Drumhead

Consider a circular drumhead with a radius of 1 meter. We want to model the vertical displacement of the drumhead, u(r,t), as a function of the radial distance from the center, r, and time, t. The drumhead is fixed at its edge (r=1), so u(1,t)=0 for all t.

The equation governing the vibrations is the wave equation in polar coordinates:

$\frac{\partial ^{2} u}{\partial t ^{2}} = c^{2} (\frac{\partial ^{2} u}{\partial r ^{2}} + \frac{1}{r} \frac{\partial u}{\partial r})$

where c is the wave speed on the drumhead.

When we separate variables, we find that the radial part of the solution is given by a linear combination of Bessel functions of the first and second kind:

$u (r, t) = (A J_{0} (k r) + B Y_{0} (k r)) cos (ω t + ϕ)$

where:

J0 is the Bessel function of the first kind of order 0.
Y0 is the Bessel function of the second kind of order 0 (this is the BESSELY function).
k is the wave number, related to the frequency ω by ω=kc.
A and B are constants determined by initial conditions.

Because the drumhead's displacement must be finite at the center (r=0), the coefficient for the Y0 term, B, must be zero. This is because Y0(0) approaches negative infinity, which is not physically possible for a drumhead.

Therefore, the physically meaningful solution for a drumhead vibrating in this mode is a linear combination of only J0 functions, and the BESSELY function is effectively "filtered out" by the physical boundary conditions.

However, if we were to consider a different physical system, such as a hollow cylinder or an annular membrane (like a washer), where the domain does not include the origin, the BESSELY function would be a crucial part of the solution.

For this example, let's look at the values of the BESSELY function of order 0, Y0(x), to see why it's not suitable for the center of a circular drumhead.

	x	Y0(x)
	A	B
1	0.1	-2.4851
2	0.2	-1.7812
3	0.3	-1.3854
4	0.4	-1.1165
5	0.5	-0.9245
6	0.6	-0.7812
7	0.7	-0.6706
8	0.8	-0.5822
9	0.9	-0.5103
10	1	-0.4502

The table above shows the values of Y0(x) for small values of x. As x approaches 0, the value of Y0(x) becomes increasingly negative and approaches negative infinity.

Example for Annular Domain:

Consider the vibration of an annular membrane with an inner radius of Rin and an outer radius of Rout. In this case, the domain does not include the origin, and the solution must be a linear combination of both J0 and Y0 functions:

$u (r, t) = (A J_{0} (k r) + B Y_{0} (k r)) cos (ω t + ϕ)$

The boundary conditions are now that the displacement is zero at both the inner and outer radii:

$u (R_{in}, t) = 0 ⟹ A J_{0} (k R_{in}) + B Y_{0} (k R_{in}) = 0$
$u (R_{out}, t) = 0 ⟹ A J_{0} (k R_{out}) + B Y_{0} (k R_{out}) = 0$

This system of equations has a non-trivial solution for A and B only if the determinant of the coefficients is zero:

$J_{0} (k R_{in}) Y_{0} (k R_{out}) - J_{0} (k R_{out}) Y_{0} (k R_{in}) = 0$

This equation must be solved numerically to find the allowed values of k, which in turn determine the natural frequencies of the annular membrane. The BESSELY function is essential for finding these frequencies because the domain excludes the origin, and thus, its singularity is not an issue.