We review the theory of wave propagation in one dimension through a medium consisting of *N* identical “cells.” Surprisingly, exact closed-form results can be obtained for arbitrary *N*. Examples include the vibration of weighted strings, the acoustics of corrugated tubes, the optics of photonic crystals, and, of course, electron wave functions in the quantum theory of solids. As *N* increases, the band structure characteristic of waves in infinite periodic media emerges.

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**S**and

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6.

E. Merzbacher,

*Quantum Mechanics*(Wiley, New York, 1998), 3rd ed., Sec. 6.3.7.

See Ref. 6, Sec. 3.1.

8.

If the potential is

*symmetric*, $V(\u2212x)=V(x),$ one obtains the further condition that*z*is imaginary. See Ref. 6.9.

What follows is a variation on the method of

D. W. L.

Sprung

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One method is to diagonalize $P:\u2002D\u22121PD=(\u20090p+\u2009p\u2212\u20090),$ where $p\xb1$ are the eigenvalues and

**D**is the matrix of eigenvectors.Then $PN=D(\u20090p+N\u2009p\u2212N\u20090)D\u22121$.

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.

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Hua

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For a proof see S. Lang’s

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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

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D. J.

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T. M.

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and yet again by

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N. L.

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30

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D. J.

Griffiths

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F.

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32

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P.

Pereyra

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31

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M. G. E.

da Luz

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Heller

, and B. K.

Cheng

, “Exact form of Green functions for segmented potentials

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31

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It is related to the eigenfunctions of $P:\u2002p\xb1=exp(\xb1i\gamma ).$ In the limit $N\u2192\u221e$ there will be total reflection when ξ is outside of the range (−1, +1).

15.

Griffiths and Taussig, Ref. 13. Incidentally, the most general point interaction satisfies the boundary conditions $\psi (0+)+\gamma \u0304\psi (0\u2212)=\u2212\delta \u0304\psi \u2032(0\u2212);\u200a\psi \u2032(0+)+\u1fb1\psi \u2032(0\u2212)=\u2212\beta \u0304\psi (0\u2212),$ where $\u1fb1,$ $\beta \u0304,$ $\gamma \u0304,$ and $\delta \u0304$ are real parameters subject only to the constraint $\u1fb1\gamma \u0304\u2212\beta \u0304\delta \u0304=1.$

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 $\u1fb1=\gamma \u0304=\u22121,$ $\delta \u0304=0,$ $\beta \u0304=\u2212c.$ For the general case, $w=\u221212[(\u1fb1+\gamma \u0304)+i(\beta \u0304/k\u2212\delta \u0304k)],$ $z=\u221212[(\u1fb1\u2212\gamma \u0304)+i(\beta \u0304/k+\delta \u0304k)].$16.

Kiang, Ref. 4;

D.

Lessie

and J.

Spadaro

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54

, 909

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(1986

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Lee

, A.

Zysnarski

, and P.

Kerr

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57

, 729

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(1989

);Kalotas and Lee (Ref. 13);

Griffiths and Taussig (Ref. 13);

Sprung, Wu, and Martorell (Ref. 9).

17.

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

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(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

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36

, 1753

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(1995

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Sprung

, Hua

Wu

, and J.

Martorell

, “Addendum to ‘Periodic quantum wires and their quasi-one-dimensional nature,’

” J. Phys. D

32

, 2136

–2139

(1999

).18.

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).19.

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.

20.

For a subtle and illuminating perspective on some of these problems see H. Georgi,

*The Physics of Waves*(Prentice–Hall, Englewood Cliffs, NJ, 1993).21.

The fully periodic case $(N\u2192\u221e)$ 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).22.

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 $(k=2mE/\u210f),$ when $E<0,$ but not in the classical one $(k=\omega \mu /T),$ 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.)23.

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, seeS.

Parmley

et al., “Vibrational properties of a loaded string

,” Am. J. Phys.

63

, 547

–553

(1995

).24.

See, for example, W. C. Elmore and M. A. Heald,

*Physics of Waves*(Dover, New York, 1969), Sec. 4.1.25.

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.26.

See, for example, S. Temkin,

*Elements of Acoustics*(Wiley, New York, 1981), Chap. 2.27.

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.28.

Temkin, Ref. 24, Sec. 2.3.

29.

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), Secs. 6–8.30.

Transverse acoustic modes can be suppressed by maintaining the frequency below the “cutoff” $\u223cv/S.$ 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

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,” Am. J. Phys.

66

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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

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41

, 1126

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(1967

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Temkin, Ref. 26, treats the exponential and power-law profiles in Sec. 3.8;

for a complete listing see Eisner, Ref. 31, p. 1128.

33.

Temkin, Ref. 26, Sec. 3.9.

34.

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

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80

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F. S.

Crawford

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42

, 278

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(1974

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If there is no corrugation at all $(S1=S2),$ then $\epsilon +=1,$ $\epsilon \u2212=0,$ and $\xi =cos(4ka),$ so $\gamma =4ka$ [Eq. (34)], and Eq. (112) reduces to $sin[(N+1)ks]=0.$ But $(N+1)s=L,$ the length of the tube, and (happily) we recover the familiar standing wave formula $\lambda n=2L/n.$

37.

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.

38.

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.

39.

See, for example, M. Rahman,

*Water Waves: Relating Modern Theory to Advanced Engineering Practice*(Oxford U.P., Oxford, 1995), Chap. 4.40.

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.

41.

C. C. Mei,

*The Applied Dynamics of Ocean Surface Waves*(Wiley, New York, 1983), Chap. 4.42.

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.

43.

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. 2.6.44.

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. 142.45.

For an amusing historical commentary see W. Bascom,

*Waves and Beaches: The Dynamics of the Ocean Surface*(Anchor, Garden City, NY, 1980), p. 142.46.

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

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,” 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

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(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.47.

R. B.

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Manley

, and B. S. H.

Connell

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64

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).48.

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. 7.1.49.

If the space between the conductors is filled with linear insulating material of permittivity ε and permeability μ, then $LC=\epsilon \mu ,$ and $v=1/\epsilon \mu .$ For an air-filled transmission line $v=1/\epsilon 0\mu 0=c,$ the speed of light.

50.

See, for example, D. J. Griffiths,

*Introduction to Electrodynamics*(Prentice Hall, Upper Saddle River, NJ, 1999), 3rd ed., Sec. 9.3.51.

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

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Hong

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67

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101

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11

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Bendickson

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53

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Kazanskiy

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Podlozny

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21

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Layered optical media have found important applications as lens coatings, x-ray mirrors, and photonic crystals. See A. G. Michette,

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Boher

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Houdy

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Yablonovitch

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58

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See, for example, C. Itzykson and J.-B. Zuber,

*Quantum Field Theory*(McGraw–Hill, New York, 1980), Sec. 2-1-2.54.

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.55.

In the relativistic context,

*E*includes the rest energy: $E=Enr+mc2.$ Thus $k=Enr2+2Enrmc2/\u210fc,$ and in the nonrelativistic limit $(Enr\u226amc2),$ $k\u22482mEnr/\u210f,$ consistent with Eq. (4).56.

See

D. J.

Griffiths

and S.

Walborn

, “Dirac deltas and discontinuous functions

,” Am. J. Phys.

67

, 446

–447

(1999

), and references therein.57.

M. G.

Calkin

and D.

Kiang

, “Proper treatment of the delta function potential in the one-dimensional Dirac equation

,” Am. J. Phys.

55

, 737

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(1987

);B. H. J.

McKellar

and G. J.

Stephenson

, Jr., “Relativistic quarks in one-dimensional periodic structures

,” Phys. Rev. C

35

, 2262

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(1987

).58.

See Ref. 15.

59.

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.60.

McKellar and Stephenson, Ref. 57.

See also

Barry R.

Holstein

, “Klein’s paradox

,” Am. J. Phys.

66

, 507

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(1968

).61.

J. O.

Vasseur

et al., “Absolute band gaps and electromagnetic transmission in quasi-one-dimensional comb structures

,” Phys. Rev. B

55

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(1997

).62.

N. Kashima,

*Passive Optical Components for Optical Fiber Transmission*(Artech House, Boston, MA, 1995), Chap. 11.63.

J.

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, “Reflection of neutrons by periodic stratifications

,” Physica B

202

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(1994

).64.

F.

Meseguer

et al., “Raleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal

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59

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).65.

J.

Adam

and J. C.

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, “A Simple Mathematical Model and Alternative Paradigm for Certain Chemotherapeutic Regimens

,” Math. Comput. Modelling

22

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R.

Tsu

and L.

Esaki

, “Tunneling in a finite superlattice

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(1973

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and N.

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Kouwenhoven

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85

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M. O.

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, J.

Lee

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P. K. H.

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Abelès may not have realized that his method applies to

*arbitrary*variations within each cell. Earlier work ofW.

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,” Phys. Rev.

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