_{1}

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A glance at Bessel functions shows they behave similar to the damped sinusoidal function. In this paper two physical examples (pendulum and spring-mass system with linearly increasing length and mass respectively) have been used as evidence for this observation. It is shown in this paper how Bessel functions can be approximated by the damped sinusoidal function. The numerical method that is introduced works very well in adiabatic condition (slow change) or in small time (independent variable) intervals. The results are also compared with the Lagrange polynomial.

A brief view of the graphs of Bessel and sinusoidal functions shows they are very similar. Bessel functions look like damped sinusoidal functions. Sinusoidal functions are well known for all of us and we have seen the foot prints of them almost everywhere. We knew them from trigonometry but Bessel functions are new for college students and seem more complicated and the students get familiar with them usually in differential equation. Bessel and sinusoidal functions are solution of Bessel and harmonic differential equations. We know these differential equations belong to the family of Sturm-Liouville equation. Bessel and sinusoidal functions are orthogonal function and they appear in the solution of some partial differential equations. The type of orthogonal function that appears in the solution depends on the geometry, physics (the form of the differential equation) and the boundary conditions. In spite of the similarity between them still for the students dealing with Bessel function is more difficult than sinusoidal function.

The purpose of this paper is using a pedagogical method to show the similarity between them through two physical examples. Basically this is an attempt to understand the mathematics through physics. This method is based on the similarity between the form of Bessel and sinusoidal functions and their similarity has been interpreted by these examples. This paper is not aiming to discuss or prove fundamental similarities or differences between these two groups of functions. Basically our observation shows Bessel functions behave like sinusoidal functions with decreasing amplitude and varying period. In this paper two examples are given to understand the root of these behaviors. These examples are the lengthening pendulum and spring-mass system with variable mass. Both length and mass in the pendulum and the spring-mass system respectively increase linearly with time. Finally for these examples the results of the exact solution (Bessel function) are compared with the approximation method (damped sinusoidal function).

In addition to this pedagogical method (physical perspective of Bessel equation) the damped sinusoidal function is a good numerical approximation for Bessel function. It is compared with the Lagrange polynomial fitting; this method provides results better than the Lagrange polynomial fitting.

The lengthening pendulum which is known also as Lorentz’s pendulum is similar to a simple pendulum with increasing length

where

The solution of (2) is given by Bessel function as following

where

where

To understand the solution in terms of the sinusoidal function, Equation (1) for the small angle approximation can be written as

where

The equation of motion can also be written in terms of

In this case the damping coefficient is

In this case the mass is increased in steady rate:

where

The solution of (8) is given by Bessel function as following:

where

is a zero of

where

where

The equation of motion in terms of

In this case the damping coefficient is

are given by the conditions of the system at

In general the damped sinusoidal function provides a good approximation for Bessel functions. It can be compared with the quadratic Lagrange polynomial fitting which is given by [

For the quadratic case

In this section some results are shown for both cases. The numerical values are used are not based on any physical reason and they are used just for comparison of these two methods.

In both of these problems there are two independent variables (

The amplitude decays as

Suppose the pendulum problem has been solved exactly for some initial conditions. This solution is given in (4) by Bessel functions and at a given

(the first zero of

match with each other through the condition at

In the Lagrange polynomial fitting three points are needed but for the damped sinusoidal function only the value of Bessel function and its derivative at a point are needed. This is a big advantage of this method compare to the Lagrange polynomial fitting. The Bessel function is oscillatory therefore the order of polynomial in a large interval should be higher and more points are needed.

The results from

The graphs of Bessel functions are similar to the damped sinusoidal solution. In this paper this observation has been investigated by two physical examples. The Bessel’s equation has been compared with the equation of the damped simple harmonic motion. The solutions of these methods are compared at the neighborhood of an arbitrary point. The results are shown the approximation method works very well particularly when u is large (i.e. for large value of independent variable for example time in this paper for two examples). The damped sinusoidal

function not only is a good way to interpret the property of the Bessel equation but also is a good numerical approximation. The comparison with the Lagrange polynomial shows the numerical advantage of the damped sinusoidal function.

In this paper we use two physical examples but this method can be generalized for any Bessel’s equation. The general form of Bessel’s equation is