Bessel Differential equation and Bessel Functions 10292020 Mathematical

Bessel Differential equation and Bessel Functions 10/29/2020 Mathematical Physics II-2019 Sem 1 1

Cylindrical polar coordinates Relations between the coordinates of a point in rectangular and cylindrical coordinate systems: Prove this Assignment 3 -Problem 1 The symbol ρ (rho) is often used instead of r. 10/29/2020 Mathematical Physics II-2019 Sem 1 2

Circular Cylindrical Coordinates With our unknown function ψ dependent on ρ, φ , and z , the Helmholtz equation becomes (3. 15) or (3. 16) As before, we assume a factored form for ψ , (3. 17) 10/29/2020 Mathematical Physics II-2019 Sem 1 3

Substituting into Eq. (3. 16), we have (3. 18) All the partial derivatives have become ordinary derivatives. Dividing by PΦZ and moving the z derivative to the right-hand side yields (3. 19) Then (3. 20) And (3. 21) 10/29/2020 Mathematical Physics II-2019 Sem 1 4

Setting , multiplying by , and rearranging terms, we obtain (3. 22) We may set the right-hand side to m 2 and (3. 23) Finally, for the ρ dependence we have (3. 24) This is Bessel’s differential equation. The original Helmholtz equation has been replaced by three ordinary differential equations. A solution of the Helmholtz equation is A general Sol. 10/29/2020 Mathematical Physics II-2019 Sem 1 (3. 26) 5

Bessel Functions of the First Kind 10/29/2020 Mathematical Physics II-2019 Sem 1 6

Bessel Functions of the second Kind 10/29/2020 Mathematical Physics II-2019 Sem 1 7

Bessel functions appear in a wide variety of physical problems. For example, separation of the Helmholtz or wave equation in circular cylindrical coordinates leads to Bessel’s equation. Bessel’s eq. The solutions of the Bessel’s eq. are called Bessel functions. In Chapter 3, we get the series solution of the above eq. 8

6. 1 Bessel Functions of the First Kind, * Generating function, integer order , J n (x) Although Bessel functions are of interest primarily as solutions of differential equations, it is instructive and convenient to develop them from a completely different approach, that of the generating function. Let us introduce a function of two variables, (6. 1) 9

Expanding it in a Laurent series, we obtain (6. 2) The coefficient of t n , J n , is defined to be Bessel function of the first kind of integer order n. Expanding the exponentials, we have (6. 3) Setting n =r- s , yields (6. 4) Since ( note: for a positive integer m) 10

we have The coefficient of t n is then (6. 5) This series form exhibits the behavior of the Bessel function J n for small x. The results for J 0 , J 1 , and J 2 are shown in Fig. 6. 1. The Bessel function s oscillate but are not periodic. . Figure 6. 1 Bessel function, , and 11

Eq. (6. 5) actually holds for n < 0 , also giving (6. 6) Since the terms for s<n (corresponding to the negative integer (s-n) ) vanish, the series can be considered to start with s=n. Replacing s by s + n , we obtain (6. 7 ) These series expressions may be used with n replaced by v to define J v and J -v for non-integer v. * Recurrence relations Differentiating Eq. (6. 1) partially with respect to t , we find that (6. 9) 12

and substituting Eq. (6. 2) for the exponential and equating the coefficients of t n- 1 , we obtain (6. 10) This is a three-term recurrence relation. On the other hand, differentiating Eq. (6. 1) partially with respect to x , we have (6. 11) Again, substituting in Eq. (6. 2) and equating the coefficients of t result n , we obtain the (6. 12) As a special case, (6. 13) 13

Adding Eqs. (6. 10) and (6. 12) and dividing by 2, we have (6. 14) Multiplying by x n and rearranging terms produces (6. 15) Subtracting Eq. (6. 12) from (6. 10) and dividing by 2 yields (6. 16) Multiplying by x -n and rearranging terms, we obtain (6. 17) 14

*Bessel’s differential equation Please verify the follow result in class. In particular, we have shown that the functions Jn defined by our generating functions, satisfy Bessel’s eq. , and thus are indeed Bessel functions 15

• Integral representation A particular useful and powerful way of treating Bessel functions employs integral representations. If we return to the generating function (Eq. (6. 2)), and substitute t = e iθ , (6. 23) in which we have used th e relations (6. 24) and so on. 16

In summation notation (6. 25) equating real and imaginary parts, respectively. It might be noted that angleθ (in radius) has no dimensions. Likewise sinθ has no dimensions and function cos(xsinθ) is perfectly proper from a dimensional point of view. By employing the orthogonality properties of cousine and sine, (6. 26 a) (6. 26 b) in which n and m are positive integers (zero is excluded), we obtain 17

(6. 27) (6. 28) If these two equations are added together (6. 29) As a special case, (6. 30) 18

Nothing that repeats itself in all four quadrants ( ), we may write Eq. (6. 30) as , (6. 30 a) On the other hand, so that reverses its sign in the third and fourth quadrants (6. 30 b) Adding Eq. (6. 30 a) and i times Eq. (6. 30 b), we obtain the complex exponential representation (6. 30 c) This integral representation (Eq. (6. 30 c)) may be obtained somewhat more directly by employing contour integration. 19

• Example 6. 11 Fraunhofer Diffraction, Circular Aperture In theory of diffraction through a circular aperture we encounter the integral (6. 31) for , the amplitude of the diffracted wave. Here is an azimuth angle in the , is the angle defined by a plane of the circular aperture of radius a, and point on a screen below the circular aperture relative to the normal through the center point. The parameter b is given by (6. 32) with the wavelength of the incident wave. The other symbols are defined by Fig. 6. 2 From Eq. (6. 30 c) , we get (6. 33) 20

Figure 6. 2 Fraunhofer diffraction –circular aperture 21

Equation (6. 15) enables us to integrate Eq. (6. 33) immediately to obtain (6. 34) The intensity of the light in the diffraction pattern is proportional to and (6. 35) 6. 2 Orthogonality 22

For v > 0, J v (0)=0. Thus, for a finite interval [0, a ], when ), we are able to have zero of J v (i. e. if m ≠ n , is the m th (6. 49) This gives us orthogonality over the interval [0, a ]. * Normalization The normalization result may be written as (6. 50) * Bessel series If we assume that the set of Bessel functions is complete, then any well-behaved function Bessel series ( v fixed, m =1, 2, … ) may be expanded in a 23

, (6. 51) The coefficients c vm are determined by using Eq. (6. 50), (6. 52) * Continuum form If a → ∞, then the series forms may be expected to go over into integrals. The discrete roots become a continuous variable. A key relation is the Bessel function closure equation (6. 59) 24

Figure 6. 3 Neumann functions , , , and 25

6. 3 Neumann function, Bessel function of the second kind, From theory of the differential equations it is known that Bessel’s equation has two independent solutions, Indeed, for non-integral order v we have already found two solutions and labeled them and , using the infinite series (Eq. (6. 5)). The trouble is that when v is integral Eq. (6. 8) holds and we have but one independent solution. A second solution may be developed by the method of Section 3. 6. This yields a perfectly good solution of Bessel’s equation but is not the usual standard form. Definition and series form As an alternate approach, we have the particular linear combination of ' and (6. 60) 26

This is Neumann function (Fig. 6. 3). For nonintegral v clearly satisfies , Bessel’s equation, for i t is a linear combination of known solutions, and , our Neumann function or Bessel function of the second To verify that kind, actually does satisfy Bessel’s equation for integral n , we may process as follows. L’Hospital’s rule applied to Eq. (6. 60) yields (6. 65) 27

Differentiating Bessel’f equation for with repect to v , we have (6. 66) Multiplying the equation for by (-1) v , substracting from the equation (as suggested by Eq. (6. 65)), and taking the limit , we obtain (6. 67) , an integer, the right-hand side vanishes by Eq. (6. 8) and For is seen to be a solution of Bessel’s equation. The most general solution for any v can be written as (6. 68) Example Coaxial Wave Guides We are interested in an electromagnetic wave confined between concentric , . Most of the mathematics is the conducting cylindrical surfaces and worked out in Section 3. 3. From EM knowledge, 28

( : electrical field along z axis) Let , we have This is the Bessel equation. If , the solution is with. But, for the coaxial wave guide one generalization is needed. The origin is now excluded ( ). Hence the Neumann function may not be excluded. becomes (6. 79) With the condition (6. 80) we have the basic equatios for a TM (transverse magnetic ) wave. 29

The (tangential) electric field must vanish at the conducting surfaces (Direchlet boundary condition) or (6. 81) (6. 82) these transcendental equations may be solved for From the relation and the ratio . (6. 83) and since must be positive for a real wave, the minimum frequency that will be propagated (in this TM mode) is (6. 84) with fixed by the boundary conditions, Eqs. (6. 81) and (6. 82). This is the cutoff frequency of the wave guide. 30

6. 4 Hankel function Many authors perfer to introduce the Hankel functions by means of integral representations and then use them to define the Neumann function, . We here introduce them a simple way as follows. As we have already obtained the Neumann function by more elementary (and less powerful) techniques, we may use it to define the Hankel functions, and : (6. 85) (6. 86) This is exactly analogous to taking (6. 87) 31

For real arguments and are complex conjugates. The extent of the analogy will be seen better when the asymptotic forms are considered. Indeed, it is their asymptotic behavior that makes the Hankel functuions useful! 6. 5 Modified Bessel function , and The Helmholtz equation, separated in circular cylindrical coordinates, leads to Eq. (6. 22 a), the Bessel equation. Equation (6. 22 a) is satisfied by the Bessel and Neumann functions and any linear combination such as the Hankel functions and. Now the Helmholtz equation describes the space part of wave phenomena. If instesd we have a diffusion problem, then the Helmholtz equation is replaced by. (6. 88) 32

The analog to Eq. (6. 22 a) is (6. 89) The Helmholtz equation may be transformed into the diffusion equation by the transformation. Similarly, changes Eq. (6. 22 a) into Eq. (6. 89) and shows that The solution of Eq. (6. 89) are Bessel function of imaginary argument. To obtain a solution that is regular at the origin, we take as the regular Bessel function. It is customary (and convenient) to choose the normalization so that (6. 90) (Here the variable is written as is being replaced by x for simplicity. ) Often this (6. 91) 33

Series form In the terms of infinite series this is equivalent to removing the sign in Eq. (6. 5) and writing (6. 92) The extra normalization cancels the from each term and leaves real. For integral v this yields (6. 93) Recurrence relations may be developed from the series The recurrence relations satisfied by expansions, but it is easier to work from the existing recurrence relations for. Let us replace x by –ix and rewrite Eq. (6. 90) as (6. 94) 34

Then Eq. (6. 10) becomes Repalcing x by ix , we have a recurrence relation for , (6. 95) Equation (6. 12) transforms to (6. 96) From Eq. (6. 93) it is seen that we have but one independent solution when v is an integer, exactly as in the Bessel function solution of Eq. (6. 108) is essentially a matter od convenience. by We choose to define a second solution in terms of the Hankel function (6. 97) 35

The factor makes real when x is real. Using Eqs. (6. 60) and (6. 90), we may transform Eq. (6. 97) to (6. 98) analogous to Eq. (6. 60) for The. choice of Eq. (6. 97) as a definition is somewhat unfortunate in that the function does not satisfy the same recurrence relations as. To avoid this annoyance other authors have included an additional factor of cos. This permits satisfy the same recurrence relations as , but it has the disadvantage of making for To put the modified Bessel functions we introduce them here because: and in proper perspective, 1. These functions are solutions of the frequently encountered modified Bessel equation. 2. They are needed for specific physical problems such as diffusion problems. 36

Figure 6. 4 Modified Bessel functions 6. 6 Asmptotic behaviors Frequently in physical problems there is a need to know how a given Bessel or modified Bessel functions for large values of argument, that is, the asymptotic behavior. Using the method of stepest descent studied in Chapter 2, we are able to derive the asymptotic behaviors of Hankel functions ( see page 450 in the text book for details) and related functions: 37

(6. 99) 1. 2. The second kind Hankel function is just the complex conjugate of the first (for real argument), (6. 100) 3. Since is the real part of (6. 101) 4. The Neumann function is the imaginary part of , or (6. 102) 5. Finally, the regular hyperbolic or modified Bessel function is given by or (6. 1 0 3) (6. 104) 38

Application 1: of Fig. below, is fixed. If the membrane is struck so that its initial displacement is F(ρ, φ) and is then released, find the displacement at anytime. The boundary value problem for the displacement z(ρ, ϕ, t) from the equilibrium or rest position (the xy palne) is, z(1, ϕ, t)=0, z(0, ϕ, 0)=0, zt(ρ, ϕ, 0)=0, z(ρ, ϕ, 0)= F(ρ, φ) 10/29/2020 Mathematical Physics II-2019 Sem 1 39

Application 2: A circular plate of unit radius, whose faces are insulated, has half of its boundary kept at constant temperature u 1, and the other half at constant temperature u 2. Find the steady-state temperature of the plate. In polar coordinates (ρ, φ) the partial differential equation for steady state flow is, 10/29/2020 Mathematical Physics II-2019 Sem 1 40

Bonus Activity 4 Try to solve the following Problems and submit your answers for bonus credits. 10/29/2020 Mathematical Physics II-2019 Sem 1 41

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