38:
1646:(an infinitely long cylinder is an example.) A large number of interesting results can be proven from these general conditions. It turns out that any tube with a bulge (where the width of the tube increases) admits at least one bound state that exist inside the mode gaps. The frequencies of all the bound states can be identified by using a pulse short in time. This can be shown using the variational principles. An interesting result by
156:
1526:
22:
244:
1079:. In other words, the impedance indicates the ratio of voltage to current of the circuit component (in this case a waveguide) during propagation of the wave. This description of the waveguide was originally intended for alternating current, but is also suitable for electromagnetic and sound waves, once the wave and material properties (such as
1101:
is important when components of an electric circuit are connected (waveguide to antenna for example): The impedance ratio determines how much of the wave is transmitted forward and how much is reflected. In connecting a waveguide to an antenna a complete transmission is usually required, so an effort
331:
The development of radio communication initially occurred at the lower frequencies because these could be more easily propagated over large distances. The long wavelengths made these frequencies unsuitable for use in hollow metal waveguides because of the impractically large diameter tubes required.
1645:
Waveguides are interesting objects of study from a strictly mathematical perspective. A waveguide (or tube) is defined as type of boundary condition on the wave equation such that the wave function must be equal to zero on the boundary and that the allowed region is finite in all dimensions but one
453:
Immediately after World War II waveguide was the technology of choice in the microwave field. However, it has some problems; it is bulky, expensive to produce, and the cutoff frequency effect makes it difficult to produce wideband devices. Ridged waveguide can increase bandwidth beyond an octave,
122:
The geometry of a waveguide reflects its function; in addition to more common types that channel the wave in one dimension, there are two-dimensional slab waveguides which confine waves to two dimensions. The frequency of the transmitted wave also dictates the size of a waveguide: each waveguide
1529:
In this military radar, microwave radiation is transmitted between the source and the reflector by a waveguide. The figure suggests that microwaves leave the box in a circularly symmetric mode (allowing the antenna to rotate), then they are converted to a linear mode, and pass through a flexible
507:
alongside a set of boundary conditions depending on the geometrical shape and materials bounding the region. The usual assumption for infinitely long uniform waveguides allows us to assume a propagating form for the wave, i.e. stating that every field component has a known dependency on the
471:). However, planar technologies really started to take off when printed circuits were introduced. These methods are significantly cheaper than waveguide and have largely taken its place in most bands. However, waveguide is still favoured in the higher microwave bands from around
2527:
Oliner, Arthur A. (January 30, 2006). "The evolution of electromagnetic waveguides: from hollow metallic guides to microwave integrated circuits". In Sarkar, T. K.; Mailloux, Robert; Oliner, Arthur A.; Salazar-Palma, Magdalena; Sengupta, Dipak L. (eds.).
1412:
222:), in which case the waveguide ensures that the power of the testing wave is conserved, or the specimen may be put inside the waveguide (as in a dielectric constant measurement, so that smaller objects can be tested and the accuracy is better.
783:
term represents the propagation constant (still unknown) along the direction along which the waveguide extends to infinity. The
Helmholtz equation can be rewritten to accommodate such form and the resulting equality needs to be solved for
942:
of the guided wave is complex, in general. For a lossless case, the propagation constant might be found to take on either real or imaginary values, depending on the chosen solution of the eigenvalue equation and on the angular frequency
348:. Southworth at first took the theory from papers on waves in dielectric rods because the work of Lord Rayleigh was unknown to him. This misled him somewhat; some of his experiments failed because he was not aware of the phenomenon of
170:
The uses of waveguides for transmitting signals were known even before the term was coined. The phenomenon of sound waves guided through a taut wire have been known for a long time, as well as sound through a hollow pipe such as a
462:
conductors since TEM does not have a cutoff frequency. A shielded rectangular conductor can also be used and this has certain manufacturing advantages over coax and can be seen as the forerunner of the planar technologies
431:, first developed at Rad Lab. The German side, on the other hand, largely ignored the potential of waveguides in radar until very late in the war. So much so that when radar parts from a downed British plane were sent to
439:
At that time, microwave techniques were badly neglected in
Germany. It was generally believed that it was of no use for electronic warfare, and those who wanted to do research work in this field were not allowed to do
2363:
1475:, and the VSWR is the voltage standing wave ratio, which value of 1 denotes full transmission, without reflection and thus no standing wave, while very large values mean high reflection and standing wave pattern.
332:
Consequently, research into hollow metal waveguides stalled and the work of Lord
Rayleigh was forgotten for a time and had to be rediscovered by others. Practical investigations resumed in the 1930s by
1184:
761:
147:
across enormous distances. Any shape of cross section of waveguide can support EM waves. Irregular shapes are difficult to analyse. Commonly used waveguides are rectangular and circular in shape.
1469:
2382:
Institute of
Electrical and Electronics, Engineers; Radatz, Jane; Standards Coordinating Committee, Terms and Definitions; IEEE Computer Society, Standards Coordinating Committee (1997).
218:
Waveguides are used in scientific instruments to measure optical, acoustic and elastic properties of materials and objects. The waveguide can be put in contact with the specimen (as in a
1987:
917:
1270:
846:
179:. Other uses of waveguides are in transmitting power between the components of a system such as radio, radar or optical devices. Waveguides are the fundamental principle of
674:
496:, there are only limited frequencies and forms for the wave function which can propagate in the waveguide. The lowest frequency in which a certain mode can propagate is the
391:
at the
University of Birmingham in the United Kingdom, provided a good power source and made microwave radar feasible. The most important centre of US research was at the
215:
Rectangular and circular waveguides are commonly used to connect feeds of parabolic dishes to their electronics, either low-noise receivers or power amplifier/transmitters.
87:
There are different types of waveguides for different types of waves. The original and most common meaning is a hollow conductive metal pipe used to carry high frequency
573:
622:
1204:
1069:
1001:
981:
961:
940:
866:
802:
781:
1258:
1231:
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1263:
An impedance mismatch creates a reflected wave, which added to the incoming waves creates a standing wave. An impedance mismatch can be also quantified with the
1021:
642:
526:
983:
is purely real, the mode is said to be "below cutoff", since the amplitude of the field phasors tends to exponentially decrease with propagation; an imaginary
2199:
1574:, which guides waves by any of several distinct mechanisms. Guides in the form of a hollow tube with a highly reflective inner surface have also been used as
500:
of that mode. The mode with the lowest cutoff frequency is the fundamental mode of the waveguide, and its cutoff frequency is the waveguide cutoff frequency.
364:
mode in circular waveguide losses go down with frequency and at one time this was a serious contender for the format for long-distance telecommunications.
320:
The study of dielectric waveguides (such as optical fibers, see below) began as early as the 1920s, by several people, most famous of which are
Rayleigh,
400:
80:
Without the physical constraint of a waveguide, waves would expand into three-dimensional space and their intensities would decrease according to the
2404:
2132:
1206:(Gamma) is the reflection coefficient (0 denotes full transmission, 1 full reflection, and 0.5 is a reflection of half the incoming voltage),
2842:
2799:
2691:
2670:
2539:
2517:
2496:
2464:
2393:
2352:
2188:
2167:
2142:
1831:
1040:
450:
German academics were even allowed to continue publicly publishing their research in this field because it was not felt to be important.
1003:, instead, represents modes said to be "in propagation" or "above cutoff", as the complex amplitude of the phasors does not change with
1578:
for illumination applications. The inner surfaces may be polished metal, or may be covered with a multilayer film that guides light by
2852:
2020:
2290:
2239:
2036:
676:), and rewrite the Helmholtz equation and boundary conditions accordingly. Then, every unknown field is forced to have a form like
1741:
1587:
1112:
49:
is a structure that guides waves by restricting the transmission of energy to one direction. Common types of waveguides include
1628:
492:
in a waveguide is one solution of the wave equations, or, in other words, the form of the wave. Due to the constraints of the
302:
1702:
1627:
is also used to describe elastic waves guided in micro-scale devices, like those employed in piezoelectric delay lines and in
2837:
317:
wavelengths using waveguides, and in 1897 described to the Royal
Institution in London his research carried out in Kolkata.
679:
1589:—such confinement is necessarily imperfect, however, since total internal reflection can never truly guide light within a
194:
Optical fibers transmit light and signals for long distances with low attenuation and a wide usable range of wavelengths.
1611:
is a physical structure for guiding sound waves. Sound in an acoustic waveguide behaves like electromagnetic waves on a
259:
328:. Optical fiber began to receive special attention in the 1960s due to its importance to the communications industry.
2847:
1663:
1489:
576:
160:
1417:
392:
349:
1881:
269:
1558:
66:
1651:
1499:
1260:
are the impedance of the first component (from which the wave enters) and the second component, respectively.
1076:
871:
435:
for analysis, even though they were recognised as microwave components, their purpose could not be identified.
384:
314:
1530:
stage. Their polarisation is then rotated in a twisted stage and finally they irradiate the parabolic antenna.
427:
was also briefly at Rad Lab, but while there he produced his small aperture theory which proved important for
2787:
1407:{\displaystyle \mathrm {VSWR} ={\frac {|V|_{\rm {max}}}{|V|_{\rm {min}}}}={\frac {1+|\Gamma |}{1-|\Gamma |}}}
489:
219:
1571:
1106:
376:
184:
163:
2713:
423:
of waveguide structures so that components in waveguide could be analysed with standard circuit theory.
139:
which have a much larger wavelength. Some naturally occurring structures can also act as waveguides. The
1697:
1602:
1511:
310:
2219:
2118:
1619:, are a simple example of an acoustic waveguide. Another example are pressure waves in the pipes of an
301:
in 1894. The first mathematical analysis of electromagnetic waves in a metal cylinder was performed by
2615:
2562:
2433:
2313:
2270:
1717:
1712:
1036:
807:
420:
404:
333:
209:
116:
96:
70:
2148:
1267:(SWR or VSWR for voltage), which is connected to the impedance ratio and reflection coefficient by:
647:
1707:
1554:
1264:
1088:
1044:
493:
432:
180:
2301:
1545:
Waveguides used at optical frequencies are typically dielectric waveguides, structures in which a
2774:
2647:
2245:
2114:
1685:
1669:
1540:
1098:
504:
81:
50:
531:
273:
253:
2795:
2766:
2733:
2687:
2666:
2639:
2631:
2590:
2535:
2513:
2492:
2460:
2389:
2348:
2329:
2286:
2235:
2184:
2163:
2138:
1681:
1647:
1616:
1612:
1493:
1072:
586:
528:). More specifically, the common approach is to first replace all unknown time-varying fields
321:
226:
124:
58:
2819:
Sophocles J. Orfanidis, Department of
Electrical and Computer Engineering, Rutgers University
2258:
1557:, is surrounded by a material with lower permittivity. The structure guides optical waves by
1189:
1054:
986:
966:
946:
925:
851:
787:
766:
2758:
2725:
2623:
2580:
2570:
2441:
2321:
2278:
2227:
1677:
1673:
1579:
1092:
497:
428:
412:
408:
357:
341:
306:
208:
In a radar, a waveguide transfers radio frequency energy to and from the antenna, where the
30:
2551:"Review of Current Guided Wave Ultrasonic Testing (GWUT) Limitations and Future Directions"
1236:
1209:
1722:
1519:
455:
416:
108:
37:
2619:
2566:
2437:
2317:
2274:
2627:
2585:
2550:
2388:(6th ed.). New York, New York: Institute of Electrical and Electronics Engineers.
1583:
1472:
1032:
1006:
627:
511:
353:
265:
198:
155:
2826:
2729:
1640:
1563:
294:
140:
128:
100:
41:
Electric field Ex component of the TE31 mode inside an x-band hollow metal waveguide.
2816:
2651:
2341:
Han, C.C.; Hwang, Y. (2012). "Satellite attenas". In Lo, Y. T.; Lee, S. W. (eds.).
2249:
1550:
624:, sufficient to fully describe any infinitely long single-tone signal at frequency
372:
305:
in 1897. For sound waves, Lord
Rayleigh published a full mathematical analysis of
298:
1848:
1510:
frequency ranges. Depending on the frequency, they can be constructed from either
2702:
2681:
2660:
2529:
2507:
2486:
2475:
2454:
2383:
2342:
2178:
2157:
352:
already found in Lord
Rayleigh's work. Serious theoretical work was taken up by
325:
176:
144:
112:
99:
are used at higher radio frequencies, and transparent dielectric waveguides and
2421:
2445:
2325:
2231:
1620:
1582:(this is a special case of a photonic-crystal fiber). One can also use small
1575:
1546:
1525:
1515:
468:
424:
388:
136:
88:
74:
21:
2770:
2737:
2635:
1593:-index core (in the prism case, some light leaks out at the prism corners).
1503:
464:
380:
337:
272:
to the section by replacing the section with a link and a summary or by
202:
127:
determined by its size and will not conduct waves of greater wavelength; an
104:
92:
2603:
2594:
2333:
2643:
2259:"The work of Jagadis Chandra Bose: 100 years of millimeter-wave research"
1080:
2604:"Orientation by means of long range acoustic signaling in baleen whales"
2120:
Transmission/Reflection and short-Circuit Line Permittivity Measurements
1498:
Waveguides can be constructed to carry waves over a wide portion of the
2778:
2746:
1654:
is that any tube of constant width with a twist, admits a bound state.
1084:
472:
459:
2575:
2282:
2409:
International Journal of Emerging Technology and Advanced Engineering
1507:
580:
2762:
2549:
Olisa, Samuel Chukwuemeka; Khan, Muhammad A.; Starr, Andrew (2021).
2126:. Boulder, Colorado: National Institute of Standards and Technology.
25:
An example of a waveguide: A section of flexible waveguide used for
2714:"Prismatic film light guides: Performance and recent developments"
1586:
around the pipe which reflect light via total internal reflection
1524:
368:
154:
132:
62:
54:
36:
26:
20:
2422:"A History of Microwave Filter Research, Design, and Development"
2405:"Review of Impedance Matching Networks for Bandwidth Enhancement"
2220:"The work of Jagadis Chandra Bose: 100 years of mm-wave research"
212:
needs to be matched for efficient power transmission (see below).
2385:
The IEEE Standard Dictionary of Electrical and Electronics Terms
403:. The head of the Fundamental Development Group at Rad Lab was
172:
399:
but many others took part in the US, and in the UK such as the
2477:
Theory and Application of Mathieu Functions, By N.W. Mclachlan
2364:"Optical Fiber: A waveguide for light and internal reflection"
1048:
396:
345:
237:
115:, and specially-shaped metal rods conduct ultrasonic waves in
2680:
Ramo, Simon; Whinnery, John R.; Van Duzer, Theodore (1994).
375:
gave a great impetus to waveguide research, at least on the
111:
are used as waveguides for sound in musical instruments and
1570:
Other types of optical waveguide are also used, including
1179:{\displaystyle \Gamma ={\frac {Z_{2}-Z_{1}}{Z_{2}+Z_{1}}}}
2224:
1997 IEEE MTT-S International Microwave Symposium Digest
2000:
1998:
1996:
454:
but a better solution is to use a technology working in
445:
H. Mayer, wartime vice-president of Siemens & Halske
1861:
1859:
1857:
293:
The first structure for guiding waves was proposed by
1959:
1931:
1929:
1892:
1890:
1518:
materials. Waveguides are used for transferring both
1420:
1273:
1239:
1212:
1192:
1115:
1057:
1009:
989:
969:
949:
928:
874:
854:
810:
790:
769:
756:{\displaystyle U(x,y,z)={\hat {U}}(x,y)e^{-\gamma z}}
682:
650:
630:
589:
534:
514:
2426:
IEEE Transactions on Microwave Theory and Techniques
2263:
IEEE Transactions on Microwave Theory and Techniques
1071:). A waveguide in circuit theory is described by a
419:. Much of the Rad Lab work concentrated on finding
2786:Zhang, Hanqiao; Krooswyk, Steven; Ou, Jeff (2015).
2686:. New York: John Wiley and Sons. pp. 321–324.
1742:
Institute of Electrical and Electronics et al. 1997
575:(assuming for simplicity to describe the fields in
1471:are the minimum and maximum values of the voltage
1463:
1406:
1252:
1225:
1198:
1178:
1063:
1015:
995:
975:
955:
934:
911:
860:
840:
796:
775:
755:
668:
636:
616:
567:
520:
848:, yielding in the end an eigenvalue equation for
360:. This work led to the discovery that for the TE
205:, where waves are formed, to the cooking chamber.
1091:) are properly converted into electrical terms (
297:in 1893, and was first experimentally tested by
2032:
437:
2817:Electromagnetic Waves and Antennas: Waveguides
2084:
1765:
1464:{\displaystyle \left|V\right|_{\rm {min/max}}}
503:Propagation modes are computed by solving the
229:is a commonly used specific type of waveguide.
2683:Fields and Waves in Communication Electronics
2016:
1983:
309:in his seminal work, "The Theory of Sound".
8:
2788:"Chapter 1 - Transmission line fundamentals"
2701:Rayleigh, John William Strutt Baron (1894).
1777:
2534:. John Wiley & Sons. pp. 543–566.
1830:sfn error: no target: CITEREFEmerson1997a (
2608:Annals of the New York Academy of Sciences
2453:Lo, Y. T.; Lee, S. W. (December 6, 2012).
143:layer in the ocean can guide the sound of
2584:
2574:
2459:. Springer Science & Business Media.
2456:Antenna Handbook: Volume III Applications
2347:. Springer Science & Business Media.
2344:Antenna Handbook: Volume III Applications
2004:
1971:
1801:
1753:
1444:
1434:
1433:
1419:
1396:
1388:
1375:
1367:
1358:
1339:
1338:
1333:
1324:
1309:
1308:
1303:
1294:
1291:
1274:
1272:
1244:
1238:
1217:
1211:
1191:
1167:
1154:
1142:
1129:
1122:
1114:
1056:
1008:
988:
968:
948:
927:
912:{\displaystyle {\hat {U}}(x,y)_{\gamma }}
903:
876:
875:
873:
853:
812:
811:
809:
789:
768:
741:
711:
710:
681:
649:
629:
588:
533:
513:
401:Telecommunications Research Establishment
2156:Beranek, Leo Leroy; Mellow, Tim (2012).
1920:
1908:
1844:
1825:
1813:
1789:
1615:. Waves on a string, like the ones in a
1561:. An example of an optical waveguide is
484:Propagation modes and cutoff frequencies
2159:Acoustics: Sound Fields and Transducers
2048:
1865:
1734:
16:Structure that guides waves efficiently
1935:
1896:
1877:
1672:as computational elements to simulate
2833:Applied and interdisciplinary physics
2096:
1947:
201:a waveguide transfers power from the
7:
2747:"Physical Modeling Synthesis Update"
2403:Khare, Rashmi; Nema, Rajesh (2012).
2300:Goldstone, J.; Jaffe, R. L. (1992).
2072:
260:Waveguide (electromagnetism)#History
103:serve as waveguides for light. In
1960:Ramo, Whinnery & Van Duzer 1994
1502:, but are especially useful in the
1102:is made to match their impedances.
2794:. Morgan Kaufmann. pp. 1–26.
2628:10.1111/j.1749-6632.1971.tb13093.x
2509:Fundamentals of Optical Waveguides
2474:McLachlan, Norman William (1964).
2060:
1455:
1452:
1449:
1441:
1438:
1435:
1393:
1372:
1346:
1343:
1340:
1316:
1313:
1310:
1284:
1281:
1278:
1275:
1193:
1116:
1058:
868:and a corresponding eigenfunction
183:(GWT), one of the many methods of
159:Waveguide supplying power for the
14:
2226:. Vol. 2. pp. 553–556.
919:for each solution of the former.
458:(that is, non-waveguide) such as
2302:"Bound states in twisting tubes"
2180:Microwave and Optical Waveguides
2131:Balanis, Constantine A. (1989).
242:
1629:stimulated Brillouin scattering
841:{\displaystyle {\hat {U}}(x,y)}
579:components) with their complex
1397:
1389:
1376:
1368:
1334:
1325:
1304:
1295:
900:
887:
881:
835:
823:
817:
734:
722:
716:
704:
686:
669:{\displaystyle \omega =2\pi f}
611:
593:
562:
538:
1:
2420:Levy, R.; Cohn, S.B. (1984).
2033:Zhang, Krooswyk & Ou 2015
415:, Carol Gray Montgomery, and
2843:Telecommunications equipment
2730:10.1016/0165-1633(89)90026-9
2602:Payne, R.; Webb, D. (1971).
2480:. New York, New York: Dover.
2134:Engineering Electromagnetics
2021:Pressure and density effects
1766:Olisa, Khan & Starr 2021
1095:and impedance for example).
508:propagation direction (i.e.
407:. His researchers included
2506:Okamoto, Katsunari (2010).
2368:Test & Measurement Tips
1664:Digital waveguide synthesis
1522:and communication signals.
1490:Waveguide (radio frequency)
256:the scope of other articles
161:Argonne National Laboratory
2869:
2485:Marcuvitz, Nathan (1951).
2085:Goldstone & Jaffe 1992
1703:Earth–ionosphere waveguide
1661:
1638:
1600:
1538:
1487:
1484:Radio-frequency waveguides
1479:Electromagnetic waveguides
568:{\displaystyle u(x,y,z,t)}
350:waveguide cutoff frequency
185:non-destructive evaluation
67:radio-frequency waveguides
2853:Electromagnetic radiation
2792:High Speed Digital Design
2745:Smith, Julius O. (1996).
2665:. John Wiley & Sons.
2446:10.1109/TMTT.1984.1132817
2326:10.1103/PhysRevB.45.14100
2232:10.1109/MWSYM.1997.602853
2183:. CRC Press. p. 38.
2017:Beranek & Mellow 2012
1984:Beranek & Mellow 2012
1559:total internal reflection
1109:can be calculated using:
922:The propagation constant
2712:Saxe, Steven G. (1989).
2659:Pozar, David M. (2012).
1988:Characteristic Impedance
1500:electromagnetic spectrum
1077:characteristic impedance
617:{\displaystyle U(x,y,z)}
429:waveguide cavity filters
2257:Emerson, D.T. (1997b).
1635:Mathematical waveguides
1199:{\displaystyle \Gamma }
1064:{\displaystyle \Omega }
1039:is a generalization of
996:{\displaystyle \gamma }
976:{\displaystyle \gamma }
956:{\displaystyle \omega }
935:{\displaystyle \gamma }
861:{\displaystyle \gamma }
797:{\displaystyle \gamma }
776:{\displaystyle \gamma }
383:, developed in 1940 by
220:medical ultrasonography
2751:Computer Music Journal
2718:Solar Energy Materials
2218:Emerson, D.T. (1997).
2177:Cronin, N. J. (1995).
2037:Reflection coefficient
1572:photonic-crystal fiber
1531:
1465:
1408:
1254:
1227:
1200:
1180:
1107:reflection coefficient
1065:
1017:
997:
977:
957:
936:
913:
862:
842:
798:
777:
757:
670:
638:
618:
569:
522:
448:
167:
164:Advanced Photon Source
73:other than light like
42:
34:
2838:Electrical components
2662:Microwave Engineering
2005:Khare & Nema 2012
1754:Payne & Webb 1971
1698:Circular polarization
1668:Sound synthesis uses
1603:Waveguide (acoustics)
1528:
1466:
1409:
1255:
1253:{\displaystyle Z_{2}}
1228:
1226:{\displaystyle Z_{1}}
1201:
1181:
1066:
1047:, and is measured in
1041:electrical resistance
1018:
998:
978:
958:
937:
914:
863:
843:
799:
778:
758:
671:
644:, (angular frequency
639:
619:
570:
523:
421:lumped element models
311:Jagadish Chandra Bose
274:splitting the content
268:and help introduce a
158:
97:Dielectric waveguides
71:electromagnetic waves
40:
24:
1921:Han & Hwang 2012
1909:Levy & Cohn 1984
1713:Orthomode transducer
1418:
1271:
1237:
1210:
1190:
1113:
1075:having a length and
1055:
1007:
987:
967:
947:
926:
872:
852:
808:
788:
767:
680:
648:
628:
587:
532:
512:
433:Siemens & Halske
405:Edward Mills Purcell
393:Radiation Laboratory
334:George C. Southworth
117:ultrasonic machining
2704:The Theory of Sound
2620:1971NYASA.188..110P
2567:2021Senso..21..811O
2531:History of Wireless
2438:1984ITMTT..32.1055L
2318:1992PhRvB..4514100G
2312:(24): 14100–14107.
2275:1997ITMTT..45.2267E
2198:EETech Media, LLC.
2115:Baker-Jarvis, James
1708:Linear polarization
1670:digital delay lines
1597:Acoustic waveguides
1555:index of refraction
1549:material with high
1265:standing wave ratio
1089:dielectric constant
1045:alternating current
494:boundary conditions
276:into a new article.
190:Specific examples:
181:guided wave testing
51:acoustic waveguides
2848:British inventions
2488:Waveguide Handbook
2204:All About Circuits
2162:. Academic Press.
1686:string instruments
1625:acoustic waveguide
1609:acoustic waveguide
1541:Waveguide (optics)
1535:Optical waveguides
1532:
1461:
1404:
1250:
1223:
1196:
1176:
1099:Impedance matching
1061:
1027:Impedance matching
1013:
993:
973:
953:
932:
909:
858:
838:
794:
773:
753:
666:
634:
614:
565:
518:
505:Helmholtz equation
367:The importance of
266:discuss this issue
168:
135:will not transmit
82:inverse square law
59:optical waveguides
43:
35:
2801:978-0-12-418663-7
2693:978-0-471-58551-0
2672:978-0-470-63155-3
2576:10.3390/s21030811
2541:978-0-471-78301-5
2519:978-0-08-045506-8
2498:978-0-86341-058-1
2466:978-1-4615-2638-4
2395:978-1-55937-833-8
2354:978-1-4615-2638-4
2306:Physical Review B
2283:10.1109/22.643830
2269:(12): 2267–2273.
2190:978-0-7503-0216-6
2169:978-0-12-391421-7
2144:978-0-471-62194-2
1778:Baker-Jarvis 1990
1682:vibrating strings
1648:Jeffrey Goldstone
1617:tin can telephone
1613:transmission line
1494:Transmission line
1402:
1353:
1174:
1073:transmission line
1016:{\displaystyle z}
884:
820:
719:
637:{\displaystyle f}
521:{\displaystyle z}
307:propagation modes
291:
290:
227:transmission line
125:cutoff wavelength
2860:
2805:
2782:
2741:
2708:
2697:
2676:
2655:
2598:
2588:
2578:
2545:
2523:
2502:
2481:
2470:
2449:
2432:(9): 1055–1067.
2416:
2399:
2378:
2376:
2374:
2358:
2337:
2296:
2253:
2214:
2212:
2210:
2194:
2173:
2152:
2151:on May 14, 2009.
2147:. Archived from
2127:
2125:
2100:
2094:
2088:
2082:
2076:
2070:
2064:
2058:
2052:
2046:
2040:
2030:
2024:
2014:
2008:
2002:
1991:
1981:
1975:
1969:
1963:
1957:
1951:
1945:
1939:
1933:
1924:
1918:
1912:
1906:
1900:
1894:
1885:
1875:
1869:
1863:
1852:
1842:
1836:
1835:
1823:
1817:
1811:
1805:
1799:
1793:
1787:
1781:
1775:
1769:
1763:
1757:
1751:
1745:
1739:
1678:wind instruments
1674:wave propagation
1580:Bragg reflection
1553:, and thus high
1470:
1468:
1467:
1462:
1460:
1459:
1458:
1448:
1432:
1413:
1411:
1410:
1405:
1403:
1401:
1400:
1392:
1380:
1379:
1371:
1359:
1354:
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1351:
1350:
1349:
1337:
1328:
1322:
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1307:
1298:
1292:
1287:
1259:
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1256:
1251:
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1232:
1230:
1229:
1224:
1222:
1221:
1205:
1203:
1202:
1197:
1185:
1183:
1182:
1177:
1175:
1173:
1172:
1171:
1159:
1158:
1148:
1147:
1146:
1134:
1133:
1123:
1070:
1068:
1067:
1062:
1022:
1020:
1019:
1014:
1002:
1000:
999:
994:
982:
980:
979:
974:
962:
960:
959:
954:
941:
939:
938:
933:
918:
916:
915:
910:
908:
907:
886:
885:
877:
867:
865:
864:
859:
847:
845:
844:
839:
822:
821:
813:
803:
801:
800:
795:
782:
780:
779:
774:
762:
760:
759:
754:
752:
751:
721:
720:
712:
675:
673:
672:
667:
643:
641:
640:
635:
623:
621:
620:
615:
574:
572:
571:
566:
527:
525:
524:
519:
498:cutoff frequency
446:
413:Nathan Marcuvitz
409:Julian Schwinger
342:Wilmer L. Barrow
286:
283:
277:
246:
245:
238:
107:, air ducts and
2868:
2867:
2863:
2862:
2861:
2859:
2858:
2857:
2823:
2822:
2813:
2808:
2802:
2785:
2763:10.2307/3681331
2744:
2711:
2700:
2694:
2679:
2673:
2658:
2601:
2548:
2542:
2526:
2520:
2505:
2499:
2484:
2473:
2467:
2452:
2419:
2402:
2396:
2381:
2372:
2370:
2362:Herres, David.
2361:
2355:
2340:
2299:
2293:
2256:
2242:
2217:
2208:
2206:
2197:
2191:
2176:
2170:
2155:
2145:
2130:
2123:
2113:
2109:
2104:
2103:
2095:
2091:
2083:
2079:
2071:
2067:
2059:
2055:
2047:
2043:
2031:
2027:
2015:
2011:
2003:
1994:
1982:
1978:
1970:
1966:
1958:
1954:
1946:
1942:
1934:
1927:
1919:
1915:
1907:
1903:
1895:
1888:
1876:
1872:
1864:
1855:
1843:
1839:
1829:
1824:
1820:
1812:
1808:
1800:
1796:
1788:
1784:
1776:
1772:
1764:
1760:
1752:
1748:
1740:
1736:
1731:
1723:Flap attenuator
1694:
1666:
1660:
1658:Sound synthesis
1643:
1637:
1605:
1599:
1543:
1537:
1496:
1488:Main articles:
1486:
1481:
1422:
1421:
1416:
1415:
1381:
1360:
1332:
1323:
1302:
1293:
1269:
1268:
1240:
1235:
1234:
1213:
1208:
1207:
1188:
1187:
1163:
1150:
1149:
1138:
1125:
1124:
1111:
1110:
1053:
1052:
1043:in the case of
1029:
1005:
1004:
985:
984:
965:
964:
945:
944:
924:
923:
899:
870:
869:
850:
849:
806:
805:
786:
785:
765:
764:
737:
678:
677:
646:
645:
626:
625:
585:
584:
583:representation
530:
529:
510:
509:
486:
481:
447:
444:
417:Robert H. Dicke
363:
287:
281:
278:
263:
258:, specifically
247:
243:
236:
153:
91:, particularly
17:
12:
11:
5:
2866:
2864:
2856:
2855:
2850:
2845:
2840:
2835:
2825:
2824:
2821:
2820:
2812:
2811:External links
2809:
2807:
2806:
2800:
2783:
2742:
2709:
2698:
2692:
2677:
2671:
2656:
2614:(1): 110–141.
2599:
2546:
2540:
2524:
2518:
2503:
2497:
2482:
2471:
2465:
2450:
2417:
2400:
2394:
2379:
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2254:
2240:
2215:
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2128:
2110:
2108:
2105:
2102:
2101:
2089:
2077:
2065:
2053:
2041:
2025:
2009:
1992:
1976:
1972:Marcuvitz 1951
1964:
1952:
1940:
1925:
1913:
1901:
1886:
1870:
1853:
1837:
1818:
1806:
1802:McLachlan 1964
1794:
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1770:
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1746:
1733:
1732:
1730:
1727:
1726:
1725:
1720:
1715:
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1690:
1662:Main article:
1659:
1656:
1639:Main article:
1636:
1633:
1601:Main article:
1598:
1595:
1539:Main article:
1536:
1533:
1485:
1482:
1480:
1477:
1473:absolute value
1457:
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1216:
1195:
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1162:
1157:
1153:
1145:
1141:
1137:
1132:
1128:
1121:
1118:
1060:
1033:circuit theory
1028:
1025:
1012:
992:
972:
952:
931:
906:
902:
898:
895:
892:
889:
883:
880:
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834:
831:
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825:
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793:
772:
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747:
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724:
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709:
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592:
564:
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558:
555:
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549:
546:
543:
540:
537:
517:
488:A propagation
485:
482:
480:
477:
442:
361:
358:Sallie P. Mead
354:John R. Carson
289:
288:
250:
248:
241:
235:
232:
231:
230:
223:
216:
213:
206:
199:microwave oven
195:
152:
149:
101:optical fibers
15:
13:
10:
9:
6:
4:
3:
2:
2865:
2854:
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2776:
2772:
2768:
2764:
2760:
2756:
2752:
2748:
2743:
2739:
2735:
2731:
2727:
2724:(1): 95–109.
2723:
2719:
2715:
2710:
2706:
2705:
2699:
2695:
2689:
2685:
2684:
2678:
2674:
2668:
2664:
2663:
2657:
2653:
2649:
2645:
2641:
2637:
2633:
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2621:
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2582:
2577:
2572:
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2537:
2533:
2532:
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2521:
2515:
2511:
2510:
2504:
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2479:
2478:
2472:
2468:
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2439:
2435:
2431:
2427:
2423:
2418:
2414:
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2401:
2397:
2391:
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2386:
2380:
2369:
2365:
2360:
2356:
2350:
2346:
2345:
2339:
2335:
2331:
2327:
2323:
2319:
2315:
2311:
2307:
2303:
2298:
2294:
2292:9780986488511
2288:
2284:
2280:
2276:
2272:
2268:
2264:
2260:
2255:
2251:
2247:
2243:
2241:0-7803-3814-6
2237:
2233:
2229:
2225:
2221:
2216:
2205:
2201:
2196:
2192:
2186:
2182:
2181:
2175:
2171:
2165:
2161:
2160:
2154:
2150:
2146:
2140:
2136:
2135:
2129:
2122:
2121:
2116:
2112:
2111:
2106:
2098:
2093:
2090:
2086:
2081:
2078:
2074:
2069:
2066:
2062:
2057:
2054:
2050:
2045:
2042:
2038:
2034:
2029:
2026:
2022:
2018:
2013:
2010:
2006:
2001:
1999:
1997:
1993:
1989:
1985:
1980:
1977:
1973:
1968:
1965:
1961:
1956:
1953:
1949:
1944:
1941:
1937:
1932:
1930:
1926:
1922:
1917:
1914:
1910:
1905:
1902:
1898:
1893:
1891:
1887:
1883:
1879:
1874:
1871:
1867:
1862:
1860:
1858:
1854:
1850:
1846:
1845:Emerson 1997b
1841:
1838:
1833:
1827:
1826:Emerson 1997a
1822:
1819:
1815:
1814:Rayleigh 1894
1810:
1807:
1803:
1798:
1795:
1791:
1786:
1783:
1779:
1774:
1771:
1767:
1762:
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1687:
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1679:
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1665:
1657:
1655:
1653:
1649:
1642:
1641:Wave equation
1634:
1632:
1630:
1626:
1622:
1618:
1614:
1610:
1604:
1596:
1594:
1592:
1588:
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1581:
1577:
1573:
1568:
1566:
1565:
1564:optical fiber
1560:
1556:
1552:
1548:
1542:
1534:
1527:
1523:
1521:
1517:
1513:
1509:
1505:
1501:
1495:
1491:
1483:
1478:
1476:
1474:
1445:
1429:
1426:
1423:
1385:
1382:
1364:
1361:
1355:
1329:
1299:
1288:
1266:
1261:
1245:
1241:
1218:
1214:
1168:
1164:
1160:
1155:
1151:
1143:
1139:
1135:
1130:
1126:
1119:
1108:
1103:
1100:
1096:
1094:
1090:
1086:
1082:
1078:
1074:
1050:
1046:
1042:
1038:
1034:
1026:
1024:
1010:
990:
970:
950:
929:
920:
904:
896:
893:
890:
878:
855:
832:
829:
826:
814:
791:
770:
748:
745:
742:
738:
731:
728:
725:
713:
707:
701:
698:
695:
692:
689:
683:
663:
660:
657:
654:
651:
631:
608:
605:
602:
599:
596:
590:
582:
578:
559:
556:
553:
550:
547:
544:
541:
535:
515:
506:
501:
499:
495:
491:
483:
478:
476:
474:
470:
466:
461:
457:
451:
441:
436:
434:
430:
426:
422:
418:
414:
410:
406:
402:
398:
395:(Rad Lab) at
394:
390:
386:
382:
378:
374:
370:
365:
359:
355:
351:
347:
343:
339:
335:
329:
327:
323:
318:
316:
312:
308:
304:
303:Lord Rayleigh
300:
296:
295:J. J. Thomson
285:
282:November 2020
275:
271:
270:summary style
267:
261:
257:
255:
251:This section
249:
240:
239:
233:
228:
224:
221:
217:
214:
211:
207:
204:
200:
196:
193:
192:
191:
188:
186:
182:
178:
174:
165:
162:
157:
150:
148:
146:
142:
141:SOFAR channel
138:
134:
130:
129:optical fiber
126:
120:
118:
114:
110:
106:
102:
98:
94:
90:
85:
83:
78:
76:
72:
69:which direct
68:
64:
61:which direct
60:
56:
53:which direct
52:
48:
39:
32:
28:
23:
19:
2791:
2757:(2): 44–56.
2754:
2750:
2721:
2717:
2707:. Macmillan.
2703:
2682:
2661:
2611:
2607:
2558:
2554:
2530:
2512:. Elsevier.
2508:
2487:
2476:
2455:
2429:
2425:
2412:
2408:
2384:
2371:. Retrieved
2367:
2343:
2309:
2305:
2266:
2262:
2223:
2209:December 31,
2207:. Retrieved
2203:
2200:"Waveguides"
2179:
2158:
2149:the original
2133:
2119:
2092:
2080:
2068:
2056:
2049:Okamoto 2010
2044:
2028:
2012:
1979:
1967:
1955:
1943:
1916:
1904:
1873:
1866:Balanis 1989
1840:
1821:
1809:
1797:
1790:EETech Media
1785:
1773:
1761:
1749:
1737:
1718:Polarization
1676:in tubes of
1667:
1652:Robert Jaffe
1644:
1624:
1608:
1606:
1590:
1569:
1562:
1551:permittivity
1544:
1497:
1262:
1104:
1097:
1030:
921:
763:, where the
502:
487:
452:
449:
438:
385:John Randall
373:World War II
366:
330:
319:
299:Oliver Lodge
292:
279:
252:
189:
169:
131:that guides
121:
113:loudspeakers
86:
79:
46:
44:
18:
2415:(1): 92–96.
1936:Cronin 1995
1897:Oliner 2006
1878:Oliner 2006
1623:. The term
1576:light pipes
379:side. The
313:researched
177:stethoscope
175:or medical
89:radio waves
75:radio waves
29:that has a
2827:Categories
2561:(3): 811.
2373:January 1,
2107:References
2097:Smith 1996
1948:Pozar 2012
1547:dielectric
1516:dielectric
1512:conductive
479:Properties
469:microstrip
425:Hans Bethe
389:Harry Boot
322:Sommerfeld
315:millimeter
254:duplicates
145:whale song
137:microwaves
93:microwaves
2771:0148-9267
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