Skip to main content

On the JWKB solution of the uniformly lengthening pendulum via change of independent variable in the Bessel’s equation

Pramana volume 88, Article number: 20 (2017) Cite this article

Abstract

Common recipe for the lengthening pendulum (LP) involves some change of variables to give a relationship with the Bessel’s equation. In this work, conventional semiclassical JWKB solution (named after Jeffreys, Wentzel, Kramers and Brillouin) of the LP is being obtained by first transforming the related Bessel’s equation into the normal form ‘via the suggested change of independent variable’. JWKB approximation of the first-order Bessel functions (ν=1) of both types along with their zeros are being obtained analytically with a very good accuracy as a result of the appropriately chosen associated initial values and they are extended to the neighbouring orders (ν=0 and 2) by the recursion relations. The required initial values are also being studied and a quantization rule regarding the experimental LP parameters is being determined. Although common numerical methods given in the literature require adiabatic LP systems where the lengthening rate is slow, JWKB solution presented here can safely be used for higher lengthening rates and a criterion for its validity is determined by the JWKB applicability criterion given in the literature. As a result, the semiclassical JWKB method which is normally used for the quantum mechanical and optical waveguide systems is applied to the classical LP system successfully.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6

References

  1. H Jeffreys, Proc. London Math. Soc. 23, 428 (1924)

    Google Scholar 

  2. G Z Wentzel, Z. Phys. 38, 518 (1926)

    ADS  Article  Google Scholar 

  3. H A Kramers, Z. Phys. 39, 828 (1926)

    ADS  Article  Google Scholar 

  4. L Brillouin, Comptes Rendus de l’Academie des Sciences Paris 183, 24 (1926)

    Google Scholar 

  5. A K Ghatak, R L Gallawa, and I C Goyal, Modified Airy functions and WKB solutions to the wave equation (NIST, Washington, 1991)

    Book  Google Scholar 

  6. C Deniz, Ann. Phys. 326, 1816 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  7. C M Bender and S A Orszag, Advanced mathematical methods for scientists and engineers asymptotic methods and perturbation theory (Springer-Verlag, NY, 1999) p. 484

    MATH  Google Scholar 

  8. L D Landau and E M Lifshitz, Quantum mechanics, non-relativistic theory, Course of theoretical physics, 2nd edn (Pergamon, NY, 1965) Vol. 3, p. 158

  9. R E Langer, Phys. Rev. 51, 669 (1937)

    ADS  Article  Google Scholar 

  10. M Hruska, W -Y Keung, and U Sukhatme, Phys. Rev. A 55, 3345 (1997)

    ADS  MathSciNet  Article  Google Scholar 

  11. N Fröman and P O Fröman, JWKB approximation contributions to the theory (North Holland Publ. Comp., Amsterdam, 1965)

    MATH  Google Scholar 

  12. A K Ghatak, E G Sauter, and I C Goyal, Eur. J. Phys. 18, 199 (1997)

    Article  Google Scholar 

  13. H Shen and H J Silverstone, Int. J. Quantum Chem. 99, 336 (2004)

    Article  Google Scholar 

  14. H Friedrich and J Trost, Phys. Rev. A 54, 1136 (1996)

    ADS  MathSciNet  Article  Google Scholar 

  15. C M Fabre and D Guéry-Odelin, Am. J. Phys. 79, 755 (2011)

    ADS  Article  Google Scholar 

  16. D W L Sprung, A Safari, and N Sator, Am. J. Phys. 79, 755 (2011) Erratum, Am. J. Phys. 80, 734 (2012)

    Article  Google Scholar 

  17. M L Boas, Mathematical methods in the physical sciences, 3rd edn (Wiley, NY, 2006)

    Google Scholar 

  18. A Werner and C J Eliezer, J. Aust. Math. Soc. 9, 331 (1969)

    Article  Google Scholar 

  19. M N Brearley, Proc. Edinburgh Math. Soc. (Series 2) 15, 61 (1966)

    Article  Google Scholar 

  20. C Deniz, Mathematical applications in modern science, Proceedings of the 19th International Conference on Applied Mathematics (AMATH’14) (Istanbul, 2014) Vol. 38, p. 22

  21. M Asadi-Zeydabadi, J. Appl. Math. Phys. 2, 26 (2014)

    Article  Google Scholar 

  22. M McMillan, D Blasing and H M Whitney, arXiv: [http: //arxiv.org/abs/1309.2907] (2013), also available in: M McMillan, D Blasing and H M Whitney, Am. J. Phys. 81, 682 (2013) [22a] Here we have the following alternative equivalent expressions: \(\begin {array}{ll}\theta (x)=\theta [x,y(x)]=x^{-1}y(x)=\theta [x(l)]=:\theta (l)\\ \hspace *{9.3pc} =\theta \{x[l(t)]\}=:\theta (t). \end {array}\)

  23. H J Arfken and G B Weber, Mathematical methods for physicists, 6th edn (Elsevier Academic Press, 2005) p. 675

  24. A Abramowitz and I A Stegun, Handbook of mathematical functions, with formulas, graphs, and mathematical tables, 3rd printing with corrections, Vol. 55 of NBS Applied Mathematics Series, Superintendent of Documents (US Government Printing Office, Washington DC, 1965)

  25. W E Boyce and R C Diprima, Elementary differential equations and boundary value problems, 7th edn (John Wiley, NY, 2001)

    Google Scholar 

  26. C L Byrne, Selected topics in applied mathematics (http:// faculty.uml.edu/cbyrne/cbyrne.html, http://faculty.uml.edu/cbyrne/bookDraft.htm, 2012)

  27. N Sekeljic, Dynamics at the Horsetooth, Focussed Issue: Asymptotics and Perturbations 2A, 2010

  28. J Niedziela, Bessel function and their applications (University of Tennessee, Knoxville, 2008)

    Google Scholar 

  29. C L Beattie, Bell System Tech. J. 37, 689 (1958)

    Article  Google Scholar 

  30. S Wolfram, The mathematica book, 5th edn (Wolfram Media Inc., 2003)

  31. http://reference.wolfram.com/mathematica/ref/BesselJ.html [31a] See eq. (3.9) for the representation of the corresponding terms without tildes (c 1 and c 2) in the exact solution in our notation.

  32. R E Langer, Trans. Am. Math. Soc. 33, 23 (1931)

    Article  Google Scholar 

  33. R E Langer, Trans. Am. Math. Soc. 36, 90 (1934)

    Article  Google Scholar 

  34. W W Bell, Special functions for scientists and engineers (D. Van Nostrand Compant Ltd., London, 1968) [34a] Note that the locations of the CAR and CIR are the same, i.e., CAR lies on the LHS and CIR lies on the RHS, for both SLP in [6] and LP-BDE system here. So, the conventional JWKB connection formulas give the same form of solutions for both systems in both regions. [34b] However, we shall pay for it by missing the first zero of Y ν=0,n=0(x) (see table 3). [34c] Note that, from (3.12), x is dimensionless. [34d] Remember that we preferred the following two experiments for the same LP system in (6.2): LP ={g,l 0,v} and LP ′= \(\{g^{\prime },l_{0}^{\prime },v^{\prime }\}\) as \(\{g=g^{\prime }, l_{0}=l_{0}^{\prime },v\neq v^{\prime }\}\) and we determined the necessary experimental v and v ′ values in eqs. (6.4) and (6.6), respectively.

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering, Department of Electrical and Electronics Engineering, Adnan Menderes University, Aytepe Central Campus, 09100, Aydın, Turkey

    COŞKUN DENIZ

Authors
  1. COŞKUN DENIZ
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to COŞKUN DENIZ.

Appendix A

Appendix A

To illustrate, as an alternative two-experiment set (one for finding J 1 and the other for finding Y 1) where only l 0 is allowed to change among the set {g,l 0,v}, let us consider the same system in (6.2). But now we have: {g = g ′ ∧\(l_{0}^{\prime }\) ≠ l 0 ∧ v ′ = v} [34d], and let us consider different initial values as different alternatives, say: \(x_{0}^{\prime }\neq x_{0}\Rightarrow \{ \alpha _{1}^{\prime }\neq \alpha _{1}\neq 0, \beta _{1}^{\prime }\) = β 1 = 0}. According to Lemma 4.2, we do not have to choose the experimental initial values necessarily as the nth zeros of Y 1 and J 1 as we did in (4.17f)–(4.18f) to obtain the desired Bessel functions: J 1 ≈\(\tilde {J}_{1}\) and Y 1 ≈\(\tilde {Y}_{1}\) entirely. Here, although we now choose different initial values in our alternative two-experiment set, in effect, we should have the same JWKB-approximated Bessel functions if Lemma 4.2 without requiring the nth root of Y 1 and J 1 is true. Choosing such an x 0 point on J 1 in part (i) and such an \(x_{0}^{\prime }\) point on Y 1 in part (ii) according to Lemma 4.2 with (4.16a) and (4.16b), we have the following experimental details giving the same results as we obtained above:

(i) For finding \(\tilde {J}_{1}(x)~({\approx } J_{1}(x))\) via Lemma 4.2:

figure i

(ii) For finding \(\tilde {Y}_{1}(x)~({\approx } Y_{1}(x))\) via Lemma 4.2:

figure j

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

DENIZ, C. On the JWKB solution of the uniformly lengthening pendulum via change of independent variable in the Bessel’s equation. Pramana - J Phys 88, 20 (2017). https://doi.org/10.1007/s12043-016-1319-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-016-1319-2

Keywords

  • Jeffreys–Wentzel–Kramers–Brillouin
  • Wentzel–Kramers–Brillouin
  • semiclassical approximation
  • linear differential equations
  • initial value problems
  • the lengthening pendulum.

PACS Nos

  • 02.30.Hq
  • 02.30.Mv
  • 03.65.Sq; 45.10. −b