Waves in locally periodic media

American Journal of Physics

Volumn 69, Issue 2, 2001, Pages 137-154

  Griffiths, David J ^a   Steinke, Carl A ^a,b  

^a REED COLLEGE   (United States)

^b HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035587534     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1308266     Document Type: Article

References (119)

1. R. de L. Kronig and W. G. Penney, "Quantum Mechanics of Electrons in Crystal Lattices," Proc. R. Soc. London, Ser. A 130, 499-513 (1931).
2. See, for example, Neil W. Ashcroft and N. David Mermin, Solid State Physics (Saunders, Philadelphia, 1976), Chap. 8.
3. F. Abelès, "Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés," Ann. Phys. (Paris) 5, 706-782 (1950), especially pp. 777-781.
4. M. Cvetič and L. Pičman, "Scattering states for a finite chain in one dimension," J. Phys. A 14, 379-382 (1981). See also P. Erdös and R. C. Herndon, "Theories of electrons in one-dimensional disordered systems," Adv. Phys. 31, 65-163 (1982). The particular case of a delta function potential was solved earlier by D. Kiang, "Multiple scattering by a Dirac comb," Am. J. Phys. 42, 785-787 (1974).
5. M. Cvetič and L. Pičman, "Scattering states for a finite chain in one dimension," J. Phys. A 14, 379-382 (1981). See also P. Erdös and R. C. Herndon, "Theories of electrons in one-dimensional disordered systems," Adv. Phys. 31, 65-163 (1982). The particular case of a delta function potential was solved earlier by D. Kiang, "Multiple scattering by a Dirac comb," Am. J. Phys. 42, 785-787 (1974).
6. M. Cvetič and L. Pičman, "Scattering states for a finite chain in one dimension," J. Phys. A 14, 379-382 (1981). See also P. Erdös and R. C. Herndon, "Theories of electrons in one-dimensional disordered systems," Adv. Phys. 31, 65-163 (1982). The particular case of a delta function potential was solved earlier by D. Kiang, "Multiple scattering by a Dirac comb," Am. J. Phys. 42, 785-787 (1974).
7. -M12).
8. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1998), 3rd ed., Sec. 6.3.
9. See Ref. 6, Sec. 3.1.
10. If the potential is symmetric, V(-x) = V(x), one obtains the further condition that z is imaginary. See Ref. 6.
11. What follows is a variation on the method of D. W. L. Sprung, Hua Wu, and J. Martorell, "Scattering by a finite periodic potential," Am. J. Phys. 61, 1118-1124 (1993). See also D. W. L. Sprung, J. D. Sigetich, H. Hua, and J. Martorell, "Bound states of a finite periodic potential," Am. J. Phys. 68, 715-722 (2000).
12. What follows is a variation on the method of D. W. L. Sprung, Hua Wu, and J. Martorell, "Scattering by a finite periodic potential," Am. J. Phys. 61, 1118-1124 (1993). See also D. W. L. Sprung, J. D. Sigetich, H. Hua, and J. Martorell, "Bound states of a finite periodic potential," Am. J. Phys. 68, 715-722 (2000).
13. -1. Another method uses the Cauchy integral formula for matrices (Kiang, Ref. 4). For details see C. A. Steinke, "Scattering from a finite periodic potential and the classical analogs," senior thesis, Reed College, 1998. For a particularly elegant approach see Hua Wu, D. W. L. Sprung, and J. Martorell, "Periodic quantum wires and their quasi-one- dimensional nature, " J. Phys. D 26, 798-803 (1993).
14. -1. Another method uses the Cauchy integral formula for matrices (Kiang, Ref. 4). For details see C. A. Steinke, "Scattering from a finite periodic potential and the classical analogs," senior thesis, Reed College, 1998. For a particularly elegant approach see Hua Wu, D. W. L. Sprung, and J. Martorell, "Periodic quantum wires and their quasi-one-dimensional nature," J. Phys. D 26, 798-803 (1993).
15. For a proof see S. Lang's Linear Algebra (Springer, New York, 1987), 3rd ed., p. 241.
16. M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 777, 782.
17. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
18. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
19. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
20. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
21. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
22. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
23. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez-Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
24. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one-dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
25. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
26. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883-888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
27. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
28. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
29. This extraordinary result was apparently first obtained (in the quantum context) by Cvetič and Pičman (Ref. 4), though a quite different analytical solution was given by C. Rorres, "Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 27, 303-321 (1974). It was rediscovered by D. J. Vezzetti and M. M. Cahay, "Transmission resonances in finite, repeated structures," J. Phys. D 19, L53-L55 (1986): again by T. M. Kalotas and A. R. Lee, "One-dimensional quantum interference," Eur. J. Phys. 12, 275-282 (1991), by Sprung, Wu, and Martorell (Ref. 9), and yet again by M. G. Rozman, P. Reineker, and R. Tehver, "Scattering by locally periodic one-dimensional potentials," Phys. Lett. A 187, 127-131 (1994). See also N. L. Chuprikov, "Tunneling in a one-dimensional system of N identical potential barriers," Semiconductors 30, 246-251 (1996): R. Pérez- Alvarez and H. Rodriguez-Coppola, "Transfer Matrix in ID Schrödinger Problems with Constant and Position-Dependent Mass," Phys. Status Solidi B 145, 493-500 (1988); E. Liviotti. "Transmission through one- dimensional periodic media," Helv. Phys. Acta 67, 767-768 (1994); and P. Erdös, E. Liviotti, and R. C. Herndon, "Wave transmission through lattices, superlattices and layered media," J. Phys. D 30, 338-345 (1997). These authors used a variety of different notations, and not all of them realized that the polynomials in question were Chebychev's. Others found the transmission coefficient for particular potentials, but did not realize that the result generalizes: Kiang (Ref. 4). D. J. Griffiths and N. F. Taussig, "Scattering from a locally periodic potential," Am. J. Phys. 60, 883- 888 (1992). Asymptotic forms, and transmission times, were considered by F. Barra and P. Gaspard, "Scattering in periodic systems: From resonances to band structure," J. Phys. A 32, 3357-3375 (1999). For generalizations to three-dimensional and multichannel systems, see, for example, P. Pereyra, "Non-commutative polynomials and the transport properties in multichannel-multilayer systems," ibid. 31, 4521-4531 (1998). For a radically different approach to the whole problem see M. G. E. da Luz, E. J. Heller, and B. K. Cheng, "Exact form of Green functions for segmented potentials," ibid. 31, 2975-2990 (1998).
30. ± = exp(±iγ). In the limit N →∞ there will be total reflection when ξ is outside of the range (-1, +1).
31. -), where ᾱ, β̄, γ̄, and δ̄ are real parameters subject only to the constraint ᾱγ̄-β̄δ̄= 1. See F. A. B. Coutinho, Y. Nogami, and J. F. Perez, "Generalized point interactions in one-dimensional quantum mechanics," J. Phys. A 30, 3937-3945 (1997). The delta-function potential [Eq. (36)] is the special case ᾱ=γ̄= -1, δ̄ = 0, β̄=-c. For the general case, w= -1/2[(ᾱ+γ̄) + i(β̄/k- δ̄k)], z= -1/2[(ᾱ-γ̄) + i(β̄/k + δ̄k)].
32. Kiang, Ref. 4; D. Lessie and J. Spadaro, "One-dimensional multiple scattering in quantum mechanics." Am. J. Phys. 54, 909-913 (1986); H.-W. Lee, A. Zysnarski, and P. Kerr, "One-dimensional scattering by a locally periodic potential," ibid. 57, 729-734 (1989); Kalotas and Lee (Ref. 13); Griffiths and Taussig (Ref. 13); Sprung. Wu, and Martorell (Ref. 9).
33. Kiang, Ref. 4; D. Lessie and J. Spadaro, "One-dimensional multiple scattering in quantum mechanics." Am. J. Phys. 54, 909-913 (1986); H.-W. Lee, A. Zysnarski, and P. Kerr, "One-dimensional scattering by a locally periodic potential," ibid. 57, 729-734 (1989); Kalotas and Lee (Ref. 13); Griffiths and Taussig (Ref. 13); Sprung. Wu, and Martorell (Ref. 9).
34. Kiang, Ref. 4; D. Lessie and J. Spadaro, "One-dimensional multiple scattering in quantum mechanics." Am. J. Phys. 54, 909-913 (1986); H.-W. Lee, A. Zysnarski, and P. Kerr, "One-dimensional scattering by a locally periodic potential," ibid. 57, 729-734 (1989); Kalotas and Lee (Ref. 13); Griffiths and Taussig (Ref. 13); Sprung. Wu, and Martorell (Ref. 9).
35. Kiang, Ref. 4; D. Lessie and J. Spadaro, "One-dimensional multiple scattering in quantum mechanics." Am. J. Phys. 54, 909-913 (1986); H.-W. Lee, A. Zysnarski, and P. Kerr, "One-dimensional scattering by a locally periodic potential," ibid. 57, 729-734 (1989); Kalotas and Lee (Ref. 13); Griffiths and Taussig (Ref. 13); Sprung. Wu, and Martorell (Ref. 9).
36. Kiang, Ref. 4; D. Lessie and J. Spadaro, "One-dimensional multiple scattering in quantum mechanics." Am. J. Phys. 54, 909-913 (1986); H.-W. Lee, A. Zysnarski, and P. Kerr, "One-dimensional scattering by a locally periodic potential," ibid. 57, 729-734 (1989); Kalotas and Lee (Ref. 13); Griffiths and Taussig (Ref. 13); Sprung. Wu, and Martorell (Ref. 9).
37. Kiang, Ref. 4; D. Lessie and J. Spadaro, "One-dimensional multiple scattering in quantum mechanics." Am. J. Phys. 54, 909-913 (1986); H.-W. Lee, A. Zysnarski, and P. Kerr, "One-dimensional scattering by a locally periodic potential," ibid. 57, 729-734 (1989); Kalotas and Lee (Ref. 13); Griffiths and Taussig (Ref. 13); Sprung. Wu, and Martorell (Ref. 9).
38. The delta-function example is explored in Griffiths and Taussig (Ref. 13); delta functions and rectangular barriers are treated in P. Carpena, V. Gasparian, and M. Ortuño, "Number of bound states of a Kronig-Penney finite-periodic superlattice," Euro. Phys. J. B 8, 635-641 (1999); for the general case see M. Sassoli de Bianchi and M. Di Ventra, "On the number of states bound by one-dimensional finite periodic potentials," J. Math. Phys. 36, 1753-1764 (1995); D. W. L. Sprung, Hua Wu, and J. Martorell, "Addendum to 'Periodic quantum wires and their quasi-one-dimensional nature,' " J. Phys. D 32, 2136-2139 (1999).
39. The delta-function example is explored in Griffiths and Taussig (Ref. 13); delta functions and rectangular barriers are treated in P. Carpena, V. Gasparian, and M. Ortuño, "Number of bound states of a Kronig-Penney finite-periodic superlattice," Euro. Phys. J. B 8, 635-641 (1999); for the general case see M. Sassoli de Bianchi and M. Di Ventra, "On the number of states bound by one-dimensional finite periodic potentials," J. Math. Phys. 36, 1753-1764 (1995); D. W. L. Sprung, Hua Wu, and J. Martorell, "Addendum to 'Periodic quantum wires and their quasi-one-dimensional nature,' " J. Phys. D 32, 2136-2139 (1999).
40. The delta-function example is explored in Griffiths and Taussig (Ref. 13); delta functions and rectangular barriers are treated in P. Carpena, V. Gasparian, and M. Ortuño, "Number of bound states of a Kronig-Penney finite-periodic superlattice," Euro. Phys. J. B 8, 635-641 (1999); for the general case see M. Sassoli de Bianchi and M. Di Ventra, "On the number of states bound by one-dimensional finite periodic potentials," J. Math. Phys. 36, 1753-1764 (1995); D. W. L. Sprung, Hua Wu, and J. Martorell, "Addendum to 'Periodic quantum wires and their quasi-one-dimensional nature,' " J. Phys. D 32, 2136-2139 (1999).
41. This case, and also the finite well, were explored numerically by E. Cota, J. Flores, and G. Monsivais, "A simple way to understand the origin of the electron band structure," Am. J. Phys. 56, 366-372 (1988). It was studied analytically by Gregory Elliott (personal communication).
42. A program called "ENERGY BAND CREATOR" (with a graphical interface) has been written by a group at Kansas State University to calculate these energies for up to 50 cells. It is available at http://www.phys.ksu.edu/perg/vqm/programs.
43. For a subtle and illuminating perspective on some of these problems see H. Georgi, The Physics of Waves (Prentice-Hall, Englewood Cliffs, NJ, 1993).
44. The fully periodic case (N→∞) was studied by U. Oseguera, "Classical Kronig-Penney model," Am. J. Phys. 60, 127-130 (1992). For interesting historical commentary see I. B. Crandall, Theory of Vibrating Systems and Sound (van Nostrand, New York, 1926), Sec. 26; L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, New York, 1946).
45. The fully periodic case (N→∞) was studied by U. Oseguera, "Classical Kronig-Penney model," Am. J. Phys. 60, 127-130 (1992). For interesting historical commentary see I. B. Crandall, Theory of Vibrating Systems and Sound (van Nostrand, New York, 1926), Sec. 26; L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, New York, 1946).
46. The fully periodic case (N→∞) was studied by U. Oseguera, "Classical Kronig-Penney model," Am. J. Phys. 60, 127-130 (1992). For interesting historical commentary see I. B. Crandall, Theory of Vibrating Systems and Sound (van Nostrand, New York, 1926), Sec. 26; L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, New York, 1946).
47. The minus sign indicates that this system is analogous to the delta-function well (negative c), and one might wonder whether there exist classical analogs to quantum bound states-standing waves in the weighted zone, with exponential attenuation outside. But this would require an imaginary k, which is possible in the quantum case (k= √2mE/ℏ), when E<0, but not in the classical one (k = ω √μ/T). unless we are prepared to countenance strings with negative mass or negative tension. (Actually, the latter is realizable, if we use a long stiff watch spring under compression.)
48. This makes a nice demonstration - we used fishing weights of about half a gram, and measured the normal mode frequencies for various N (Steinke, Ref. 10). For a numerical and experimental study, see S. Parmley et al., "Vibrational properties of a loaded string," Am. J. Phys. 63, 547-553 (1995).
49. See. for example, W. C. Elmore and M. A. Heald, Physics of Waves (Dover, New York, 1969), Sec. 4.1.
50. In the case of varying S, we shall assume that the change is gradual enough that the force remains uniformly distributed over the cross section.
51. See, for example, S. Temkin, Elements of Acoustics (Wiley, New York, 1981), Chap. 2.
52. There are related applications to underwater acoustics, but these involve a more complicated fluid, and nonperiodic variations. L. M. Brekhovskikh, Waves in Layered Media (Academic. Orlando, FL, 1980), 2nd ed.
53. Temkin, Ref. 24, Sec. 2.3.
54. Brekhovskikh, Ref. 25, Chap. II; A. Alippi, A. Bettucci, and F. Craciun, "Ultrasonic waves in monodimensional periodic composites," in Physical Acoustics: Fundamentals and Applications, edited by O. Leroy and M. A. Breazeale (Plenum, New York, 1990). For the classic papers on acoustic filters see R. B. Lindsay, Physical Acoustics (Dowden, Hutchinson, and Ross, Stroudsburg, PA, 1974), Sees. 6-8.
55. Brekhovskikh, Ref. 25, Chap. II; A. Alippi, A. Bettucci, and F. Craciun, "Ultrasonic waves in monodimensional periodic composites," in Physical Acoustics: Fundamentals and Applications, edited by O. Leroy and M. A. Breazeale (Plenum, New York, 1990). For the classic papers on acoustic filters see R. B. Lindsay, Physical Acoustics (Dowden, Hutchinson, and Ross, Stroudsburg, PA, 1974), Sees. 6-8.
56. Brekhovskikh, Ref. 25, Chap. II; A. Alippi, A. Bettucci, and F. Craciun, "Ultrasonic waves in monodimensional periodic composites," in Physical Acoustics: Fundamentals and Applications, edited by O. Leroy and M. A. Breazeale (Plenum, New York, 1990). For the classic papers on acoustic filters see R. B. Lindsay, Physical Acoustics (Dowden, Hutchinson, and Ross, Stroudsburg, PA, 1974), Sees. 6-8.
57. Transverse acoustic modes can be suppressed by maintaining the frequency below the "cutoff" ∼v1√S. See Temkin, Ref. 26, Chap. 3. For an interesting experimental approach, which could easily be adapted to the systems discussed here, see C. L. Adler, K. Mita, and D. Phipps, "Quantitative measurement of acoustic whistlers," Am. J. Phys. 66, 607-612 (1998).
58. Transverse acoustic modes can be suppressed by maintaining the frequency below the "cutoff" ∼v1√S. See Temkin, Ref. 26, Chap. 3. For an interesting experimental approach, which could easily be adapted to the systems discussed here, see C. L. Adler, K. Mita, and D. Phipps, "Quantitative measurement of acoustic whistlers," Am. J. Phys. 66, 607-612 (1998).
59. The horn equation is traditionally attributed to Webster, who published it in 1919, but it was in fact studied by many others, from Daniel Bernoulli to Lord Rayleigh. For a fascinating account, with extensive bibliography, see E. Eisner, "Complete Solutions of the 'Webster' Horn Equation," J. Acoust. Soc. Am. 41, 1126-1146 (1967).
60. Temkin, Ref. 26, treats the exponential and power-law profiles in Sec. 3.8; for a complete listing see Eisner, Ref. 31, p. 1128.
61. Temkin, Ref. 26, Sec. 3.9.
62. See, for instance, L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics (Wiley, New York, 1982), pp. 231-243. For interesting related work see J. V. Sánchez-Pérez et al., "Sound Attenuation by a Two-Dimensional Array of Rigid Cylinders," Phys. Rev. Lett. 80, 5325-5328 (1998).
63. See, for instance, L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics (Wiley, New York, 1982), pp. 231- 243. For interesting related work see J. V. Sánchez-Pérez et al., "Sound Attenuation by a Two-Dimensional Array of Rigid Cylinders," Phys. Rev. Lett. 80, 5325-5328 (1998).
64. F. S. Crawford, "Singing corrugated pipes," Am. J. Phys. 42, 278-288 (1974). This article is a real gem.
65. n = 2L/n.
66. Crawford (Ref. 35) takes the corrugations to be sinusoidal, but a comparison of Figs. 11 and 13 tends to confirm one's intuition that the details of the profile are not terribly critical.
67. Crawford's measured value was 175 Hz; presumably the end corrections are about the same for smooth and corrugated pipes. Incidentally, the corrugahorn is ill-tempered (the overtones are not perfect harmonics), which perhaps accounts for the fact that it has always been more popular with physicists than musicians-though according to our figures its bad temper is rather mild.
68. See, for example, M. Rahman, Water Waves: Relating Modern Theory to Advanced Engineering Practice (Oxford U.P., Oxford, 1995), Chap. 4.
69. Deep water waves are not so interesting, from our present perspective, since v is independent of the depth, and there is no realistic way to provide for local periodicity.
70. C. C. Mei, The Applied Dynamics of Ocean Surface Waves (Wiley, New York, 1983), Chap. 4.
71. A set of artificial shoals could conceivably be constructed outside a harbor to exclude waves in a particularly destructive frequency range, but as far as we know this has not been tried. Nor do we know of any naturally occurring examples.
72. Physical oceanographers typically invoke the Wentzel-Kramers-Brillouin approximation to handle Eq. (120). See Mei, Ref. 41, Sec. 4.5, or M. W. Dingemans, Water Wave Propagation over Uneven Bottoms (World Scientific, Singapore, 1997), Sec. 2.6.
73. Physical oceanographers typically invoke the Wentzel-Kramers-Brillouin approximation to handle Eq. (120). See Mei, Ref. 41, Sec. 4.5, or M. W. Dingemans, Water Wave Propagation over Uneven Bottoms (World Scientific, Singapore, 1997), Sec. 2.6.
74. For shallow waves the horizontal velocity is effectively independent of the vertical coordinate. See A. Defant, Physical Oceanography (Pergamon. New York, 1961), Vol. 2, p. 142.
75. For an amusing historical commentary see W. Bascom, Waves and Beaches: The Dynamics of the Ocean Surface (Anchor, Garden City, NY, 1980). p. 142.
76. Lord Kelvin analyzed the case of water flowing down a channel with sinusoidal depth [W. Thomson, "On Stationary Waves in Flowing Water," Philos. Mag. 22, 353-357 (1886)]; see also D. E. Hewgill, J. Reeder, and M. Shinbrot, "Some Exact Solutions of the Nonlinear Problem of Water Waves," Pac. J. Math. 92, 87-109 (1981). This system, which corresponds to the played corrugahorn, was studied experimentally by T. Shinbrot, "On Salient Phenomena of Stationary Waves in Water," senior thesis, Reed College, 1978. But only modes with wavelengths comparable to the corrugation distance were explored, and it would be interesting to see whether other modes could be stimulated by Crawford's mechanism. B. J. Korgen ["Seiches," Am. Sci. 83, 331-341 (1995)] discusses a related phenomenon, in which standing waves are generated when the upper layer of ocean water flows over a sequence of internal solitons at the thermocline level.
77. Lord Kelvin analyzed the case of water flowing down a channel with sinusoidal depth [W. Thomson, "On Stationary Waves in Flowing Water," Philos. Mag. 22, 353-357 (1886)]; see also D. E. Hewgill, J. Reeder, and M. Shinbrot, "Some Exact Solutions of the Nonlinear Problem of Water Waves," Pac. J. Math. 92, 87-109 (1981). This system, which corresponds to the played corrugahorn, was studied experimentally by T. Shinbrot, "On Salient Phenomena of Stationary Waves in Water," senior thesis, Reed College, 1978. But only modes with wavelengths comparable to the corrugation distance were explored, and it would be interesting to see whether other modes could be stimulated by Crawford's mechanism. B. J. Korgen ["Seiches," Am. Sci. 83, 331-341 (1995)] discusses a related phenomenon, in which standing waves are generated when the upper layer of ocean water flows over a sequence of internal solitons at the thermocline level.
78. Lord Kelvin analyzed the case of water flowing down a channel with sinusoidal depth [W. Thomson, "On Stationary Waves in Flowing Water," Philos. Mag. 22, 353-357 (1886)]; see also D. E. Hewgill, J. Reeder, and M. Shinbrot, "Some Exact Solutions of the Nonlinear Problem of Water Waves," Pac. J. Math. 92, 87-109 (1981). This system, which corresponds to the played corrugahorn, was studied experimentally by T. Shinbrot, "On Salient Phenomena of Stationary Waves in Water," senior thesis, Reed College, 1978. But only modes with wavelengths comparable to the corrugation distance were explored, and it would be interesting to see whether other modes could be stimulated by Crawford's mechanism. B. J. Korgen ["Seiches," Am. Sci. 83, 331-341 (1995)] discusses a related phenomenon, in which standing waves are generated when the upper layer of ocean water flows over a sequence of internal solitons at the thermocline level.
79. Lord Kelvin analyzed the case of water flowing down a channel with sinusoidal depth [W. Thomson, "On Stationary Waves in Flowing Water," Philos. Mag. 22, 353-357 (1886)]; see also D. E. Hewgill, J. Reeder, and M. Shinbrot, "Some Exact Solutions of the Nonlinear Problem of Water Waves," Pac. J. Math. 92, 87-109 (1981). This system, which corresponds to the played corrugahorn, was studied experimentally by T. Shinbrot, "On Salient Phenomena of Stationary Waves in Water," senior thesis, Reed College, 1978. But only modes with wavelengths comparable to the corrugation distance were explored, and it would be interesting to see whether other modes could be stimulated by Crawford's mechanism. B. J. Korgen ["Seiches," Am. Sci. 83, 331-341 (1995)] discusses a related phenomenon, in which standing waves are generated when the upper layer of ocean water flows over a sequence of internal solitons at the thermocline level.
80. R. B. Prigo, T. O. Manley, and B. S. H. Connell, "Linear, one-dimensional models of the surface and internal standing waves for a long and narrow lake," Am. J. Phys. 64, 288-300 (1996).
81. For an introduction to the theory of transmission lines, see (for instance) N. N. Rao, Elements of Engineering Electromagnetics (Prentice Hall, Upper Saddle River, NJ, 1991), 3rd ed., Sec. 7.1.
82. If the space between the conductors is filled with linear insulating material of permittivity ∈ and permeability μ, then LC=∈μ, and v=1/√∈μ. For an air-filled transmission line v=1/√∈0μ0=c, the speed of light.
83. See, for example, D. J. Griffiths, Introduction to Electrodynamics (Prentice Hall, Upper Saddle River, NJ, 1999), 3rd ed., Sec. 9.3.
84. The transfer matrix for this problem (including oblique incidence) was obtained by Abelès (Ref. 3) in 1950. The most accessible reference in English is M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), 6th ed., Sec. 1.6.5. See also P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977); U. Bandelow and U. Leonhardt, "Light propagation in one-dimensional lossless dielectrica: Transfer matrix method and coupled mode theory," Opt. Commun. 101, 92-99 (1993); J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892-2899 (1994); J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996); Vadim B. Kazanskiy and Vladimir V. Podlozny, "Resonance Phenomena in Finite-Periodic Multilayer Sequence of Identical Gratings of Rectangular Bars," Microwave Opt. Technol. Lett. 21, 299-304 (1999).
85. The transfer matrix for this problem (including oblique incidence) was obtained by Abelès (Ref. 3) in 1950. The most accessible reference in English is M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), 6th ed., Sec. 1.6.5. See also P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977); U. Bandelow and U. Leonhardt, "Light propagation in one-dimensional lossless dielectrica: Transfer matrix method and coupled mode theory," Opt. Commun. 101, 92-99 (1993); J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892-2899 (1994); J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996); Vadim B. Kazanskiy and Vladimir V. Podlozny, "Resonance Phenomena in Finite-Periodic Multilayer Sequence of Identical Gratings of Rectangular Bars," Microwave Opt. Technol. Lett. 21, 299-304 (1999).
86. The transfer matrix for this problem (including oblique incidence) was obtained by Abelès (Ref. 3) in 1950. The most accessible reference in English is M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), 6th ed., Sec. 1.6.5. See also P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977); U. Bandelow and U. Leonhardt, "Light propagation in one-dimensional lossless dielectrica: Transfer matrix method and coupled mode theory," Opt. Commun. 101, 92-99 (1993); J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892-2899 (1994); J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996); Vadim B. Kazanskiy and Vladimir V. Podlozny, "Resonance Phenomena in Finite-Periodic Multilayer Sequence of Identical Gratings of Rectangular Bars," Microwave Opt. Technol. Lett. 21, 299-304 (1999).
87. The transfer matrix for this problem (including oblique incidence) was obtained by Abelès (Ref. 3) in 1950. The most accessible reference in English is M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), 6th ed., Sec. 1.6.5. See also P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977); U. Bandelow and U. Leonhardt, "Light propagation in one-dimensional lossless dielectrica: Transfer matrix method and coupled mode theory," Opt. Commun. 101, 92-99 (1993); J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892-2899 (1994); J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996); Vadim B. Kazanskiy and Vladimir V. Podlozny, "Resonance Phenomena in Finite-Periodic Multilayer Sequence of Identical Gratings of Rectangular Bars," Microwave Opt. Technol. Lett. 21, 299-304 (1999).
88. The transfer matrix for this problem (including oblique incidence) was obtained by Abelès (Ref. 3) in 1950. The most accessible reference in English is M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), 6th ed., Sec. 1.6.5. See also P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977); U. Bandelow and U. Leonhardt, "Light propagation in one-dimensional lossless dielectrica: Transfer matrix method and coupled mode theory," Opt. Commun. 101, 92-99 (1993); J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892-2899 (1994); J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996); Vadim B. Kazanskiy and Vladimir V. Podlozny, "Resonance Phenomena in Finite-Periodic Multilayer Sequence of Identical Gratings of Rectangular Bars," Microwave Opt. Technol. Lett. 21, 299-304 (1999).
89. The transfer matrix for this problem (including oblique incidence) was obtained by Abelès (Ref. 3) in 1950. The most accessible reference in English is M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), 6th ed., Sec. 1.6.5. See also P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977); U. Bandelow and U. Leonhardt, "Light propagation in one-dimensional lossless dielectrica: Transfer matrix method and coupled mode theory," Opt. Commun. 101, 92-99 (1993); J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892-2899 (1994); J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996); Vadim B. Kazanskiy and Vladimir V. Podlozny, "Resonance Phenomena in Finite-Periodic Multilayer Sequence of Identical Gratings of Rectangular Bars," Microwave Opt. Technol. Lett. 21, 299-304 (1999).
90. Layered optical media have found important applications as lens coatings, x-ray mirrors, and photonic crystals. See A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986); P. Boher and P. Houdy, "Multicouches nanométriques pour l'optique X: Quelques exemples applicables aux lasers X," Ann. Phys. (Paris) 17, 141-150 (1992); E. Spiller, Soft X-Ray Optics (SPIE, Bellingham, WA, 1994), Chap. 7; E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987); J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U.P., Princeton, NJ, 1995).
91. Layered optical media have found important applications as lens coatings, x-ray mirrors, and photonic crystals. See A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986); P. Boher and P. Houdy, "Multicouches nanométriques pour l'optique X: Quelques exemples applicables aux lasers X," Ann. Phys. (Paris) 17, 141-150 (1992); E. Spiller, Soft X-Ray Optics (SPIE, Bellingham, WA, 1994), Chap. 7; E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987); J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U.P., Princeton, NJ, 1995).
92. Layered optical media have found important applications as lens coatings, x-ray mirrors, and photonic crystals. See A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986); P. Boher and P. Houdy, "Multicouches nanométriques pour l'optique X: Quelques exemples applicables aux lasers X," Ann. Phys. (Paris) 17, 141-150 (1992); E. Spiller, Soft X-Ray Optics (SPIE, Bellingham, WA, 1994), Chap. 7; E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987); J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U.P., Princeton, NJ, 1995).
93. Layered optical media have found important applications as lens coatings, x-ray mirrors, and photonic crystals. See A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986); P. Boher and P. Houdy, "Multicouches nanométriques pour l'optique X: Quelques exemples applicables aux lasers X," Ann. Phys. (Paris) 17, 141-150 (1992); E. Spiller, Soft X-Ray Optics (SPIE, Bellingham, WA, 1994), Chap. 7; E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987); J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U.P., Princeton, NJ, 1995).
94. Layered optical media have found important applications as lens coatings, x-ray mirrors, and photonic crystals. See A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986); P. Boher and P. Houdy, "Multicouches nanométriques pour l'optique X: Quelques exemples applicables aux lasers X," Ann. Phys. (Paris) 17, 141-150 (1992); E. Spiller, Soft X-Ray Optics (SPIE, Bellingham, WA, 1994), Chap. 7; E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987); J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U.P., Princeton, NJ, 1995).
95. See, for example, C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980), Sec. 2-1-2.
96. From now on ψ will always stand for the two-component spinor [Eq. (141)]. There are other representations for the gamma matrices, and they lead to different expressions for α and β; but they are all equivalent, corresponding to different choices for the basis spinors.
97. nr/ℏ, consistent with Eq. (4).
98. See D. J. Griffiths and S. Walborn, "Dirac deltas and discontinuous functions," Am. J. Phys. 67, 446-447 (1999), and references therein.
99. M. G. Calkin and D. Kiang, "Proper treatment of the delta function potential in the one-dimensional Dirac equation," Am. J. Phys. 55, 737-739 (1987); B. H. J. McKellar and G. J. Stephenson, Jr., "Relativistic quarks in one-dimensional periodic structures," Phys. Rev. C 35, 2262-2271 (1987).
100. M. G. Calkin and D. Kiang, "Proper treatment of the delta function potential in the one-dimensional Dirac equation," Am. J. Phys. 55, 737-739 (1987); B. H. J. McKellar and G. J. Stephenson, Jr., "Relativistic quarks in one-dimensional periodic structures," Phys. Rev. C 35, 2262-2271 (1987).
101. See Ref. 15.
102. For related work see M. L. Glasser, "A class of one-dimensional relativistic band models," Am. J. Phys. 51, 936-939 (1983), and references cited therein.
103. McKellar and Stephenson, Ref. 57. See also Barry R. Holstein, "Klein's paradox," Am. J. Phys. 66, 507-512 (1968).
104. McKellar and Stephenson, Ref. 57. See also Barry R. Holstein, "Klein's paradox," Am. J. Phys. 66, 507-512 (1968).
105. J. O. Vasseur et al., "Absolute band gaps and electromagnetic transmission in quasi-one-dimensional comb structures," Phys. Rev. B 55, 10434-10442 (1997).
106. N. Kashima, Passive Optical Components for Optical Fiber Transmission (Artech House, Boston, MA, 1995), Chap. 11.
107. J. Lekner, "Reflection of neutrons by periodic stratifications," Physica B 202, 16-22 (1994).
108. F. Meseguer et al, "Raleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal," Phys. Rev. B 59, 12169-12172 (1999).
109. J. Adam and J. C. Panetta, "A Simple Mathematical Model and Alternative Paradigm for Certain Chemotherapeutic Regimens," Math. Comput. Modelling 22, 49-60 (1995).
110. xAs/GaAs heterostructures," J. Appl. Phys. 54, 5206-5213 (1983); P. Yeh, "Resonant tunneling of electromagnetic radiation in superlattice structures," J. Opt. Soc. Am. A 2, 568-571 (1985); H. Wu et al, "Quantum wire with periodic serial structure," Phys. Rev. B 44, 6351-6360 (1991); P. S. Deo and A. M. Jayannavar, "Quantum waveguide transport in serial stub and loop structures," Phys. Rev. B 50, 11629-11639 (1994).
111. xAs/GaAs heterostructures," J. Appl. Phys. 54, 5206-5213 (1983); P. Yeh, "Resonant tunneling of electromagnetic radiation in superlattice structures," J. Opt. Soc. Am. A 2, 568-571 (1985); H. Wu et al, "Quantum wire with periodic serial structure," Phys. Rev. B 44, 6351-6360 (1991); P. S. Deo and A. M. Jayannavar, "Quantum waveguide transport in serial stub and loop structures," Phys. Rev. B 50, 11629-11639 (1994).
112. xAs/GaAs heterostructures," J. Appl. Phys. 54, 5206-5213 (1983); P. Yeh, "Resonant tunneling of electromagnetic radiation in superlattice structures," J. Opt. Soc. Am. A 2, 568-571 (1985); H. Wu et al, "Quantum wire with periodic serial structure," Phys. Rev. B 44, 6351-6360 (1991); P. S. Deo and A. M. Jayannavar, "Quantum waveguide transport in serial stub and loop structures," Phys. Rev. B 50, 11629-11639 (1994).
113. xAs/GaAs heterostructures," J. Appl. Phys. 54, 5206-5213 (1983); P. Yeh, "Resonant tunneling of electromagnetic radiation in superlattice structures," J. Opt. Soc. Am. A 2, 568-571 (1985); H. Wu et al, "Quantum wire with periodic serial structure," Phys. Rev. B 44, 6351-6360 (1991); P. S. Deo and A. M. Jayannavar, "Quantum waveguide transport in serial stub and loop structures," Phys. Rev. B 50, 11629-11639 (1994).
114. xAs/GaAs heterostructures," J. Appl. Phys. 54, 5206-5213 (1983); P. Yeh, "Resonant tunneling of electromagnetic radiation in superlattice structures," J. Opt. Soc. Am. A 2, 568-571 (1985); H. Wu et al, "Quantum wire with periodic serial structure," Phys. Rev. B 44, 6351-6360 (1991); P. S. Deo and A. M. Jayannavar, "Quantum waveguide transport in serial stub and loop structures," Phys. Rev. B 50, 11629-11639 (1994).
115. xAs/GaAs heterostructures," J. Appl. Phys. 54, 5206-5213 (1983); P. Yeh, "Resonant tunneling of electromagnetic radiation in superlattice structures," J. Opt. Soc. Am. A 2, 568-571 (1985); H. Wu et al, "Quantum wire with periodic serial structure," Phys. Rev. B 44, 6351-6360 (1991); P. S. Deo and A. M. Jayannavar, "Quantum waveguide transport in serial stub and loop structures," Phys. Rev. B 50, 11629-11639 (1994).
116. xAs/GaAs heterostructures," J. Appl. Phys. 54, 5206-5213 (1983); P. Yeh, "Resonant tunneling of electromagnetic radiation in superlattice structures," J. Opt. Soc. Am. A 2, 568-571 (1985); H. Wu et al, "Quantum wire with periodic serial structure," Phys. Rev. B 44, 6351-6360 (1991); P. S. Deo and A. M. Jayannavar, "Quantum waveguide transport in serial stub and loop structures," Phys. Rev. B 50, 11629-11639 (1994).
117. xAs/GaAs heterostructures," J. Appl. Phys. 54, 5206-5213 (1983); P. Yeh, "Resonant tunneling of electromagnetic radiation in superlattice structures," J. Opt. Soc. Am. A 2, 568-571 (1985); H. Wu et al, "Quantum wire with periodic serial structure," Phys. Rev. B 44, 6351-6360 (1991); P. S. Deo and A. M. Jayannavar, "Quantum waveguide transport in serial stub and loop structures," Phys. Rev. B 50, 11629-11639 (1994).
118. P. K. H. Ho et al., "All-Polymer Optoelectronic Devices," Science 285, 233-236 (1999).
119. Abelès may not have realized that his method applies to arbitrary variations within each cell. Earlier work of W. Shockley ["On the Surface States Associated with a Periodic Potential," Phys. Rev. 56, 317-323 (1939)] treats the general case (but only for specific small N).