316:
2487:
number of mathematically special modes of vibration of a stretched string (see figure). The pure tone of lowest pitch or frequency is referred to as the fundamental and the multiples of that frequency are called its harmonic overtones. He assigned to one of the oscillators the frequency of the fundamental vibration of the whole block of solid. He assigned to the remaining oscillators the frequencies of the harmonics of that fundamental, with the highest of all these frequencies being limited by the motion of the smallest primary unit.
36:
149:
2455:
382:
304:) nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1. So the mode number of the system is 2–1 or 1–2, depending on which coordinate is considered the "first" and which is considered the "second" coordinate (so it is important to always indicate which mode number matches with each coordinate direction).
3511:, the cross-coupling approximation. Self-coupling will solely change the phase velocity and not the number of waves around a great circle, resulting in a stretching or shrinking of standing wave pattern. Modal cross-coupling occurs due to the rotation of the Earth, from aspherical elastic structure, or due to Earth's ellipticity and leads to a mixing of fundamental spheroidal and toroidal modes.
3454:. Thus a system like an atom consists of a linear combination of modes of definite energy. These energies are characteristic of the particular atom. The (complex) square of the probability amplitude at a point in space gives the probability of measuring an electron at that location. The spatial distribution of this probability is characteristic of the atom.
3061:
1142:
300:, we have a radial coordinate and an angular coordinate. If one measured from the center outward along the radial coordinate one would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because only half of the disk is considered due to the anti-symmetric (also called
739:
2106:
1910:
2870:
2434:
In any solid at any temperature, the primary particles (e.g. atoms or molecules) are not stationary, but rather vibrate about mean positions. In insulators the capacity of the solid to store thermal energy is due almost entirely to these vibrations. Many physical properties of the solid (e.g. modulus
268:
Most dynamical systems can be excited in several modes, possibly simultaneously. Each mode is characterized by one or several frequencies, according to the modal variable field. For example, a vibrating rope in 2D space is defined by a single-frequency (1D axial displacement), but a vibrating rope in
2486:
Debye subsequently recognized that each oscillator is intimately coupled to its neighboring oscillators at all times. Thus, by replacing
Einstein's identical uncoupled oscillators with the same number of coupled oscillators, Debye correlated the elastic vibrations of a one-dimensional solid with the
323:
In a one-dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero. These nodes correspond to points in the mode shape where the mode shape is zero. Since the vibration of a system is given by the mode shape multiplied by a time function, the
233:
In an electrical dynamical system, a resonant cavity made of thin metal walls, enclosing a hollow space, for a particle accelerator is a pure standing wave system, and thus an example of a mode, in which the hollow space of the cavity is the medium, the RF source (a
Klystron or another RF source) is
229:
In a structural dynamical system, a high tall building oscillating under its most flexural axis is a mode, in which all the material of the building -under the proper numerical simplifications- is the medium, the seismic/wind/environmental solicitations are the excitations and the displacements are
327:
When expanded to a two dimensional system, these nodes become lines where the displacement is always zero. If you watch the animation above you will see two circles (one about halfway between the edge and center, and the other on the edge itself) and a straight line bisecting the disk, where the
292:
A mode of vibration is characterized by a modal frequency and a mode shape. It is numbered according to the number of half waves in the vibration. For example, if a vibrating beam with both ends pinned displayed a mode shape of half of a sine wave (one peak on the vibrating beam) it would be
157:
889:
283:
mode of a system with multiple modes will be the mode storing the minimum amount of energy for a given amplitude of the modal variable, or, equivalently, for a given stored amount of energy, the dominant mode will be the mode imposing the maximum amplitude of the modal variable.
1448:
1667:
137:
1303:
880:
479:
1922:
1729:
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In a mechanical dynamical system, a vibrating rope is the most clear example of a mode, in which the rope is the medium, the stress on the rope is the excitation, and the displacement of the rope with respect to its static state is the modal
2435:
of elasticity) can be predicted given knowledge of the frequencies with which the particles vibrate. The simplest assumption (by
Einstein) is that all the particles oscillate about their mean positions with the same natural frequency
2494:
are exactly those acoustical vibrations which are considered in the theory of sound. Both longitudinal and transverse waves can be propagated through a solid, while, in general, only longitudinal waves are supported by fluids.
2861:
1316:
128:
of its normal modes. The modes are normal in the sense that they can move independently, that is to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are
3170:
2614:
162:
161:
158:
3056:{\displaystyle {\begin{aligned}S\left(\nu \right)&=\int _{0}^{T}{\frac {d}{dT}}E\left(\nu \right){\frac {dT}{T}}\\&={\frac {E\left(\nu \right)}{T}}-k\log \left(1-e^{-{\frac {h\nu }{kT}}}\right)\end{aligned}}}
163:
1580:
112:
with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. These fixed frequencies of the normal modes of a system are known as its
226:
In an acoustic dynamical system, a single sound pitch is a mode, in which the air is the medium, the sound pressure in the air is the excitation, and the displacement of the air molecules is the modal variable.
2329:
3256:
1151:
1137:{\displaystyle {\begin{aligned}-\omega ^{2}mA_{1}e^{i\omega t}&=-2kA_{1}e^{i\omega t}+kA_{2}e^{i\omega t}\\-\omega ^{2}mA_{2}e^{i\omega t}&=kA_{1}e^{i\omega t}-2kA_{2}e^{i\omega t}\end{aligned}}}
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752:
160:
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894:
757:
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In linear systems each mode is entirely independent of all other modes. In general all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes.
1567:
3419:
121:. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions.
2405:
there is no single or finite number of normal modes, but there are infinitely many normal modes. If the problem is bounded (i.e. it is defined on a finite section of space) there are
2502:, the displacement of particles from their positions of equilibrium coincides with the propagation direction of the wave. Mechanical longitudinal waves have been also referred to as
3354:
2786:
2490:
The normal modes of vibration of a crystal are in general superpositions of many overtones, each with an appropriate amplitude and phase. Longer wavelength (low frequency)
2795:
734:{\displaystyle {\begin{aligned}m{\ddot {x}}_{1}&=-kx_{1}+k(x_{2}-x_{1})=-2kx_{1}+kx_{2}\\m{\ddot {x}}_{2}&=-kx_{2}+k(x_{1}-x_{2})=-2kx_{2}+kx_{1}\end{aligned}}}
65:
466:
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Each normal coordinate corresponds to a single vibrational frequency of the system and the corresponding motion of the system is called the normal mode of vibration.
2101:{\displaystyle {\vec {\eta }}_{2}={\begin{pmatrix}x_{1}^{2}(t)\\x_{2}^{2}(t)\end{pmatrix}}=c_{2}{\begin{pmatrix}1\\-1\end{pmatrix}}\cos {(\omega _{2}t+\varphi _{2})}}
1905:{\displaystyle {\vec {\eta }}_{1}={\begin{pmatrix}x_{1}^{1}(t)\\x_{2}^{1}(t)\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\1\end{pmatrix}}\cos {(\omega _{1}t+\varphi _{1})}}
3070:
2524:
3432:
are analogous to modes. The waves in quantum systems are oscillations in probability amplitude rather than material displacement. The frequency of oscillation,
159:
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For an elastic, isotropic, homogeneous sphere, spheroidal, toroidal and radial (or breathing) modes arise. Spheroidal modes only involve P and SV waves (like
2227:
3189:
3738:
3260:
In order to calculate the internal energy and the specific heat, we must know the number of normal vibrational modes a frequency between the values
2110:
This corresponds to the masses moving in the opposite directions, while the center of mass remains stationary. This mode is called symmetric.
3758:
3636:
3609:
200:
state of excitation, in which all the components of the system will be affected sinusoidally at a fixed frequency associated with that mode.
2217:
3504:. The degeneracy does not exist on Earth as it is broken by rotation, ellipticity and 3D heterogeneous velocity and density structure.
141:
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3665:
1443:{\displaystyle {\begin{bmatrix}\omega ^{2}m-2k&k\\k&\omega ^{2}m-2k\end{bmatrix}}{\begin{pmatrix}A_{1}\\A_{2}\end{pmatrix}}=0}
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272:
For a given amplitude on the modal variable, each mode will store a specific amount of energy because of the sinusoidal excitation.
87:
1501:
3367:
2447:. The spectrum of waveforms can be described mathematically using a Fourier series of sinusoidal density fluctuations (or thermal
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According to quantum theory, the mean energy of a normal vibrational mode of a crystalline solid with characteristic frequency
2356:
1662:{\displaystyle {\begin{aligned}\omega _{1}&={\sqrt {\frac {k}{m}}}\\\omega _{2}&={\sqrt {\frac {3k}{m}}}\end{aligned}}}
2363:
of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the
3311:
148:
3507:
It may be assumed that each mode can be isolated, the self-coupling approximation, or that many modes close in frequency
3743:
1298:{\displaystyle {\begin{aligned}(\omega ^{2}m-2k)A_{1}+kA_{2}&=0\\kA_{1}+(\omega ^{2}m-2k)A_{2}&=0\end{aligned}}}
207:
concept is taken as a general characterization of specific states of oscillation, thus treating the dynamic system in a
48:
3570:
2645:
2443:. Einstein also assumed that the allowed energy states of these oscillations are harmonics, or integral multiples of
58:
52:
44:
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1914:
Which corresponds to both masses moving in the same direction at the same time. This mode is called antisymmetric.
315:
140:
Vibration of a single normal mode of a circular disc with a pinned boundary condition along the entire outer edge.
296:
In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using
2359:(superposition) of waves and their reflections (although one may also say the opposite; that a moving wave is a
875:{\displaystyle {\begin{aligned}x_{1}(t)&=A_{1}e^{i\omega t}\\x_{2}(t)&=A_{2}e^{i\omega t}\end{aligned}}}
69:
328:
displacement is close to zero. In an idealized system these lines equal zero exactly, as shown to the right.
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2114:
403:
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vibrating in mode 1. If it had a full sine wave (one peak and one trough) it would be vibrating in mode 2.
2209:
3753:
3540:
2459:
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represents the dependence of amplitude on location and the cosine/sine are the oscillations in time.
2165:
1308:
258:
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3530:
471:
337:
2625:
represents the "zero-point energy", or the energy which an oscillator will have at absolute zero.
349:
262:
168:
3497:) and do not exist in fluid outer core. Radial modes are just a subset of spheroidal modes with
2454:
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3661:
3632:
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3429:
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381:
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114:
2439:. This is equivalent to the assumption that all atoms vibrate independently with a frequency
442:
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1146:
Omitting the exponential factor (because it is common to all terms) and simplifying yields:
378:. They are attached in the following manner, forming a system that is physically symmetric:
193:
105:
3358:
The integration is performed over all frequencies of the crystal. Then the internal energy
3451:
2509:
1489:
372:
3631:(The new millennium edition, paperback first published ed.). New York: Basic Books.
3555:
3525:
3470:
2475:
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2406:
2164:
The process demonstrated here can be generalized and formulated using the formalism of
249:
The concept of normal modes also finds application in other dynamical systems, such as
2184:
is a continuous form of normal mode. In a standing wave, all the space elements (i.e.
241:, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "
136:
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3520:
3463:
2205:
2181:
1452:
If the matrix on the left is invertible, the unique solution is the trivial solution
197:
130:
2856:{\displaystyle \left({\frac {\partial S}{\partial E}}\right)_{N,V}={\frac {1}{T}}}
2416:). If the problem is not bounded, there is a continuous spectrum of normal modes.
17:
3628:
The
Feynman lectures on physics. Volume 1: Mainly mechanics, radiation, and heat
2429:
1716:
1493:
203:
Because no real system can perfectly fit under the standing wave framework, the
3545:
3494:
2216:
1720:
424:
109:
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2471:
2201:
341:
242:
118:
3165:{\displaystyle F(\nu )=E-TS=kT\log \left(1-e^{-{\frac {h\nu }{kT}}}\right)}
2609:{\displaystyle E(\nu )={\frac {1}{2}}h\nu +{\frac {h\nu }{e^{h\nu /kT}-1}}}
2512:, individual particles move perpendicular to the propagation of the wave.
3702:(2nd ed.). Cambridge: Cambridge University Press. pp. 231–237.
3508:
3493:
this tends to
Rayleigh waves. Toroidal modes only involve SH waves (like
2463:
2491:
2448:
319:
A mode shape of a drum membrane, with nodal lines shown in pale green
250:
234:
the excitation and the electromagnetic field is the modal variable.
2324:{\displaystyle \Psi (t)=f(x,y,z)(A\cos(\omega t)+B\sin(\omega t))}
314:
238:
155:
147:
135:
1484:. The non trivial solutions are to be found for those values of
365:
269:
3D space is defined by two frequencies (2D axial displacement).
185:
152:
A flash photo of a cup of black coffee vibrating in normal modes
3489:
concentrates fundamental branch closer to surface and at large
3251:{\displaystyle F(\nu )=kT\log \left({\frac {h\nu }{kT}}\right)}
3600:
Goldstein, Herbert; Poole, Charles P.; Safko, John L. (2008).
29:
3604:(3rd ed., ed.). San Francisco, Munich: Addison Wesley.
3462:
Normal modes are generated in the Earth from long wavelength
2382:
form of the standing wave. This space-dependence is called a
364:
Consider two equal bodies (not affected by gravity), each of
108:
is a pattern of motion in which all parts of the system move
380:
3466:
from large earthquakes interfering to form standing waves.
743:
Since we expect oscillatory motion of a normal mode (where
884:
Substituting these into the equations of motion gives us:
352:, the system can be transformed to new coordinates called
324:
displacement of the node points remain zero at all times.
167:
Excitation of normal modes in a drop of water during the
340:
with small displacements from equilibrium, important in
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point together), but each has a different amplitude.
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1154:
892:
755:
482:
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where the edge points are fixed and cannot move. Let
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density fluctuations (or atomic displacement waves).
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Usually, for problems with continuous dependence on
3660:(Reprint ed.). Malabar, Florida: Krieger Pub.
3413:
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124:The most general motion of a linear system is a
57:but its sources remain unclear because it lacks
2470:in crystalline solids consists of treating the
1492:; i.e. is not invertible. It follows that the
3658:Formulas for natural frequency and mode shape
2355:Physically, standing waves are formed by the
1562:{\displaystyle (\omega ^{2}m-2k)^{2}-k^{2}=0}
8:
3414:{\displaystyle U=\int f(\nu )E(\nu )\,d\nu }
3288:. Since the total number of normal modes is
3677:Dynamics and Control of Distributed Systems
420:denote the displacement of the right mass.
2466:of a vibrating string. The mathematics of
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2200:coordinates) are oscillating in the same
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88:Learn how and when to remove this message
27:Pattern of oscillating motion in a system
3675:Tzou, H.S.; Bergman, L.A., eds. (2008).
2453:
2222:The general form of a standing wave is:
3592:
3481:but have degeneracy of azimuthal order
371:, attached to three springs, each with
3349:{\displaystyle \int f(\nu )\,d\nu =3N}
2790:By knowing the thermodynamic formula,
1496:of the matrix must be equal to 0, so:
747:is the same for both masses), we try:
3724:Harvard lecture notes on normal modes
7:
2815:
2807:
2231:
1575:, the two positive solutions are:
1488:whereby the matrix on the left is
423:Denoting acceleration (the second
25:
3581:Vibrations of a circular membrane
3473:) and depend on overtone number
3436:, relates to the mode energy by
2865:the entropy per normal mode is:
2215:
1680:into the matrix and solving for
34:
3739:Ordinary differential equations
2781:{\displaystyle E(\nu )=kT\left}
2409:normal modes (usually numbered
3401:
3395:
3389:
3383:
3327:
3321:
3202:
3196:
3083:
3077:
2658:
2652:
2537:
2531:
2318:
2315:
2306:
2291:
2282:
2270:
2267:
2249:
2240:
2234:
2094:
2065:
2005:
1999:
1977:
1971:
1933:
1898:
1869:
1812:
1806:
1784:
1778:
1740:
1531:
1505:
1268:
1243:
1184:
1159:
832:
826:
776:
770:
686:
660:
564:
538:
188:of physics and engineering, a
1:
3759:Singular value decomposition
3625:Feynman, Richard P. (2011).
215:of states can be performed.
3656:Blevins, Robert D. (2001).
2636:tends to the classic value
1917:The second normal mode is:
3775:
3700:Introduction to seismology
3698:Shearer, Peter M. (2009).
3681:Cambridge University Press
3274:. Allow this number to be
2423:
2113:The general solution is a
1726:The first normal mode is:
1719:, and the frequencies are
218:Typical examples include:
438:with respect to time) as
211:fashion, in which linear
461:{\textstyle {\ddot {x}}}
43:This article includes a
3561:Mode (electromagnetism)
3536:Harmonic series (music)
72:more precise citations.
3571:Sturm–Liouville theory
3415:
3350:
3252:
3166:
3057:
2857:
2782:
2610:
2483:
2325:
2157:are determined by the
2102:
1906:
1663:
1563:
1444:
1299:
1138:
876:
735:
462:
406:of the left mass, and
402:denote the horizontal
385:
320:
171:
153:
145:
3541:Infrared spectroscopy
3416:
3351:
3253:
3167:
3058:
2858:
2783:
2640:at high temperatures
2611:
2457:
2326:
2170:Hamiltonian mechanics
2103:
1907:
1715:. (These vectors are
1664:
1564:
1445:
1300:
1139:
877:
736:
463:
384:
332:In mechanical systems
318:
166:
151:
139:
3551:Mechanical resonance
3424:In quantum mechanics
3368:
3312:
3190:
3071:
3065:The free energy is:
2871:
2796:
2646:
2525:
2228:
2166:Lagrangian mechanics
1923:
1730:
1581:
1502:
1317:
1152:
890:
753:
480:
443:
338:conservative systems
259:atmospheric dynamics
119:resonant frequencies
3744:Classical mechanics
3602:Classical mechanics
3576:Torsional vibration
3531:Harmonic oscillator
2913:
1998:
1970:
1805:
1777:
472:equations of motion
360:Coupled oscillators
354:normal coordinates.
350:electrical circuits
336:In the analysis of
230:the modal variable.
175:General definitions
115:natural frequencies
3477:and angular order
3411:
3362:will be given by:
3346:
3248:
3162:
3053:
3051:
2899:
2853:
2778:
2606:
2484:
2462:and the first six
2321:
2159:initial conditions
2098:
2052:
2010:
1984:
1956:
1902:
1856:
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1293:
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1132:
872:
870:
731:
729:
458:
386:
321:
263:molecular dynamics
172:
169:Leidenfrost effect
154:
146:
45:list of references
3749:Quantum mechanics
3638:978-0-465-04085-8
3611:978-0-201-65702-9
3430:quantum mechanics
3428:Bound states in
3242:
3153:
3040:
2989:
2956:
2927:
2851:
2822:
2755:
2714:
2689:
2604:
2551:
2505:compression waves
2500:longitudinal mode
1936:
1743:
1653:
1652:
1615:
1614:
622:
500:
455:
346:molecular spectra
298:polar coordinates
255:quantum mechanics
237:When relating to
164:
98:
97:
90:
18:Mode of vibration
16:(Redirected from
3766:
3713:
3694:
3671:
3643:
3642:
3622:
3616:
3615:
3597:
3566:Quasinormal mode
3503:
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2563:
2552:
2544:
2518:
2510:transverse modes
2468:wave propagation
2446:
2442:
2438:
2415:
2404:
2381:
2351:
2330:
2328:
2327:
2322:
2219:
2199:
2161:of the problem.
2156:
2147:
2138:
2129:
2107:
2105:
2104:
2099:
2097:
2093:
2092:
2077:
2076:
2057:
2056:
2028:
2027:
2015:
2014:
1997:
1992:
1969:
1964:
1944:
1943:
1938:
1937:
1929:
1911:
1909:
1908:
1903:
1901:
1897:
1896:
1881:
1880:
1861:
1860:
1835:
1834:
1822:
1821:
1804:
1799:
1776:
1771:
1751:
1750:
1745:
1744:
1736:
1714:
1710:
1701:
1697:
1679:
1668:
1666:
1665:
1660:
1658:
1654:
1648:
1640:
1639:
1630:
1629:
1616:
1607:
1606:
1597:
1596:
1574:
1568:
1566:
1565:
1560:
1552:
1551:
1539:
1538:
1517:
1516:
1487:
1483:
1449:
1447:
1446:
1441:
1433:
1432:
1425:
1424:
1411:
1410:
1393:
1392:
1373:
1372:
1337:
1336:
1311:representation:
1304:
1302:
1301:
1296:
1294:
1280:
1279:
1255:
1254:
1239:
1238:
1212:
1211:
1196:
1195:
1171:
1170:
1143:
1141:
1140:
1135:
1133:
1129:
1128:
1113:
1112:
1094:
1093:
1078:
1077:
1058:
1057:
1042:
1041:
1029:
1028:
1012:
1011:
996:
995:
980:
979:
964:
963:
938:
937:
922:
921:
909:
908:
881:
879:
878:
873:
871:
867:
866:
851:
850:
825:
824:
811:
810:
795:
794:
769:
768:
746:
740:
738:
737:
732:
730:
726:
725:
710:
709:
685:
684:
672:
671:
653:
652:
630:
629:
624:
623:
615:
604:
603:
588:
587:
563:
562:
550:
549:
531:
530:
508:
507:
502:
501:
493:
469:
467:
465:
464:
459:
457:
456:
448:
437:
419:
401:
377:
370:
194:dynamical system
165:
106:dynamical system
93:
86:
82:
79:
73:
68:this article by
59:inline citations
38:
37:
30:
21:
3774:
3773:
3769:
3768:
3767:
3765:
3764:
3763:
3729:
3728:
3720:
3710:
3697:
3691:
3674:
3668:
3655:
3652:
3650:Further reading
3647:
3646:
3639:
3624:
3623:
3619:
3612:
3599:
3598:
3594:
3589:
3517:
3498:
3490:
3486:
3482:
3478:
3474:
3460:
3452:Planck constant
3447:
3437:
3433:
3426:
3366:
3365:
3359:
3310:
3309:
3296:
3295:, the function
3289:
3275:
3265:
3261:
3234:
3226:
3220:
3188:
3187:
3175:
3145:
3137:
3127:
3120:
3116:
3069:
3068:
3050:
3049:
3032:
3024:
3014:
3007:
3003:
2974:
2970:
2959:
2958:
2945:
2932:
2919:
2892:
2881:
2869:
2868:
2814:
2806:
2800:
2799:
2794:
2793:
2747:
2739:
2733:
2732:
2706:
2698:
2692:
2691:
2674:
2670:
2644:
2643:
2637:
2626:
2619:
2573:
2572:
2564:
2523:
2522:
2516:
2506:
2444:
2440:
2436:
2432:
2424:Main articles:
2422:
2410:
2390:
2364:
2334:
2226:
2225:
2185:
2178:
2155:
2149:
2146:
2140:
2137:
2131:
2128:
2122:
2084:
2068:
2051:
2050:
2041:
2040:
2030:
2019:
2009:
2008:
1981:
1980:
1949:
1926:
1921:
1920:
1888:
1872:
1855:
1854:
1848:
1847:
1837:
1826:
1816:
1815:
1788:
1787:
1756:
1733:
1728:
1727:
1712:
1709:
1703:
1702:. Substituting
1699:
1695:
1688:
1681:
1678:
1672:
1656:
1655:
1641:
1631:
1621:
1618:
1617:
1598:
1588:
1579:
1578:
1572:
1543:
1530:
1508:
1500:
1499:
1485:
1481:
1474:
1467:
1460:
1453:
1427:
1426:
1416:
1413:
1412:
1402:
1395:
1387:
1386:
1364:
1362:
1356:
1355:
1350:
1328:
1321:
1315:
1314:
1292:
1291:
1281:
1271:
1246:
1230:
1224:
1223:
1213:
1203:
1187:
1162:
1150:
1149:
1131:
1130:
1114:
1104:
1079:
1069:
1059:
1043:
1033:
1020:
1014:
1013:
997:
987:
965:
955:
939:
923:
913:
900:
888:
887:
869:
868:
852:
842:
835:
816:
813:
812:
796:
786:
779:
760:
751:
750:
744:
728:
727:
717:
701:
676:
663:
644:
631:
612:
606:
605:
595:
579:
554:
541:
522:
509:
490:
478:
477:
441:
440:
439:
428:
413:
407:
395:
389:
375:
373:spring constant
368:
362:
334:
313:
290:
182:
177:
156:
142:See other modes
133:to each other.
94:
83:
77:
74:
63:
49:related reading
39:
35:
28:
23:
22:
15:
12:
11:
5:
3772:
3770:
3762:
3761:
3756:
3751:
3746:
3741:
3731:
3730:
3727:
3726:
3719:
3718:External links
3716:
3715:
3714:
3708:
3695:
3690:978-0521033749
3689:
3679:. Cambridge :
3672:
3667:978-1575241845
3666:
3651:
3648:
3645:
3644:
3637:
3617:
3610:
3591:
3590:
3588:
3585:
3584:
3583:
3578:
3573:
3568:
3563:
3558:
3556:Modal analysis
3553:
3548:
3543:
3538:
3533:
3528:
3526:Critical speed
3523:
3516:
3513:
3471:Rayleigh waves
3459:
3456:
3425:
3422:
3410:
3407:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3345:
3342:
3339:
3336:
3333:
3329:
3326:
3323:
3320:
3317:
3246:
3240:
3237:
3232:
3229:
3223:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3160:
3151:
3148:
3143:
3140:
3134:
3130:
3126:
3123:
3119:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3047:
3038:
3035:
3030:
3027:
3021:
3017:
3013:
3010:
3006:
3002:
2999:
2996:
2993:
2988:
2983:
2980:
2977:
2973:
2967:
2964:
2962:
2960:
2955:
2951:
2948:
2941:
2938:
2935:
2931:
2925:
2922:
2918:
2911:
2906:
2902:
2898:
2895:
2893:
2890:
2887:
2884:
2880:
2877:
2876:
2850:
2847:
2842:
2837:
2834:
2831:
2826:
2820:
2817:
2812:
2809:
2803:
2776:
2772:
2769:
2764:
2759:
2753:
2750:
2745:
2742:
2736:
2731:
2728:
2723:
2718:
2712:
2709:
2704:
2701:
2695:
2688:
2685:
2680:
2677:
2673:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2602:
2599:
2594:
2591:
2587:
2583:
2580:
2576:
2570:
2567:
2561:
2558:
2555:
2550:
2547:
2542:
2539:
2536:
2533:
2530:
2504:
2476:Fourier series
2426:Einstein solid
2421:
2420:Elastic solids
2418:
2414:= 1, 2, 3, ...
2407:countably many
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2208:(reaching the
2177:
2176:Standing waves
2174:
2153:
2144:
2135:
2126:
2096:
2091:
2087:
2083:
2080:
2075:
2071:
2067:
2063:
2060:
2055:
2049:
2046:
2043:
2042:
2039:
2036:
2035:
2033:
2026:
2022:
2018:
2013:
2007:
2004:
2001:
1996:
1991:
1987:
1983:
1982:
1979:
1976:
1973:
1968:
1963:
1959:
1955:
1954:
1952:
1947:
1942:
1935:
1932:
1900:
1895:
1891:
1887:
1884:
1879:
1875:
1871:
1867:
1864:
1859:
1853:
1850:
1849:
1846:
1843:
1842:
1840:
1833:
1829:
1825:
1820:
1814:
1811:
1808:
1803:
1798:
1794:
1790:
1789:
1786:
1783:
1780:
1775:
1770:
1766:
1762:
1761:
1759:
1754:
1749:
1742:
1739:
1707:
1693:
1686:
1676:
1651:
1647:
1644:
1637:
1634:
1632:
1628:
1624:
1620:
1619:
1613:
1610:
1604:
1601:
1599:
1595:
1591:
1587:
1586:
1558:
1555:
1550:
1546:
1542:
1537:
1533:
1529:
1526:
1523:
1520:
1515:
1511:
1507:
1479:
1472:
1465:
1458:
1439:
1436:
1431:
1423:
1419:
1415:
1414:
1409:
1405:
1401:
1400:
1398:
1391:
1385:
1382:
1379:
1376:
1371:
1367:
1363:
1361:
1358:
1357:
1354:
1351:
1349:
1346:
1343:
1340:
1335:
1331:
1327:
1326:
1324:
1290:
1287:
1284:
1282:
1278:
1274:
1270:
1267:
1264:
1261:
1258:
1253:
1249:
1245:
1242:
1237:
1233:
1229:
1226:
1225:
1222:
1219:
1216:
1214:
1210:
1206:
1202:
1199:
1194:
1190:
1186:
1183:
1180:
1177:
1174:
1169:
1165:
1161:
1158:
1157:
1127:
1124:
1121:
1117:
1111:
1107:
1103:
1100:
1097:
1092:
1089:
1086:
1082:
1076:
1072:
1068:
1065:
1062:
1060:
1056:
1053:
1050:
1046:
1040:
1036:
1032:
1027:
1023:
1019:
1016:
1015:
1010:
1007:
1004:
1000:
994:
990:
986:
983:
978:
975:
972:
968:
962:
958:
954:
951:
948:
945:
942:
940:
936:
933:
930:
926:
920:
916:
912:
907:
903:
899:
896:
895:
865:
862:
859:
855:
849:
845:
841:
838:
836:
834:
831:
828:
823:
819:
815:
814:
809:
806:
803:
799:
793:
789:
785:
782:
780:
778:
775:
772:
767:
763:
759:
758:
724:
720:
716:
713:
708:
704:
700:
697:
694:
691:
688:
683:
679:
675:
670:
666:
662:
659:
656:
651:
647:
643:
640:
637:
634:
632:
628:
621:
618:
611:
608:
607:
602:
598:
594:
591:
586:
582:
578:
575:
572:
569:
566:
561:
557:
553:
548:
544:
540:
537:
534:
529:
525:
521:
518:
515:
512:
510:
506:
499:
496:
489:
486:
485:
454:
451:
411:
393:
361:
358:
333:
330:
312:
309:
289:
286:
247:
246:
235:
231:
227:
224:
181:
178:
176:
173:
96:
95:
53:external links
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3771:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3736:
3734:
3725:
3722:
3721:
3717:
3711:
3709:9780521882101
3705:
3701:
3696:
3692:
3686:
3682:
3678:
3673:
3669:
3663:
3659:
3654:
3653:
3649:
3640:
3634:
3630:
3629:
3621:
3618:
3613:
3607:
3603:
3596:
3593:
3586:
3582:
3579:
3577:
3574:
3572:
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3544:
3542:
3539:
3537:
3534:
3532:
3529:
3527:
3524:
3522:
3521:Antiresonance
3519:
3518:
3514:
3512:
3510:
3505:
3501:
3496:
3485:. Increasing
3472:
3467:
3465:
3464:seismic waves
3458:In seismology
3457:
3455:
3453:
3444:
3440:
3431:
3423:
3421:
3408:
3405:
3398:
3392:
3386:
3380:
3377:
3374:
3371:
3363:
3356:
3343:
3340:
3337:
3334:
3331:
3324:
3318:
3315:
3307:
3306:is given by:
3303:
3299:
3293:
3286:
3282:
3278:
3272:
3268:
3258:
3244:
3238:
3235:
3230:
3227:
3221:
3217:
3214:
3211:
3208:
3205:
3199:
3193:
3185:
3182:
3178:
3172:
3158:
3149:
3146:
3141:
3138:
3132:
3128:
3124:
3121:
3117:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3080:
3074:
3066:
3063:
3045:
3036:
3033:
3028:
3025:
3019:
3015:
3011:
3008:
3004:
3000:
2997:
2994:
2991:
2986:
2981:
2978:
2975:
2971:
2965:
2963:
2953:
2949:
2946:
2939:
2936:
2933:
2929:
2923:
2920:
2916:
2909:
2904:
2900:
2896:
2894:
2888:
2885:
2882:
2878:
2866:
2863:
2848:
2845:
2840:
2835:
2832:
2829:
2824:
2818:
2810:
2801:
2791:
2788:
2774:
2770:
2767:
2762:
2757:
2751:
2748:
2743:
2740:
2734:
2729:
2726:
2721:
2716:
2710:
2707:
2702:
2699:
2693:
2686:
2683:
2678:
2675:
2671:
2667:
2664:
2661:
2655:
2649:
2641:
2633:
2629:
2623:
2616:
2600:
2597:
2592:
2589:
2585:
2581:
2578:
2574:
2568:
2565:
2559:
2556:
2553:
2548:
2545:
2540:
2534:
2528:
2520:
2513:
2511:
2507:
2501:
2496:
2493:
2488:
2481:
2477:
2473:
2469:
2465:
2461:
2456:
2452:
2450:
2431:
2427:
2419:
2417:
2413:
2408:
2402:
2398:
2394:
2387:
2385:
2379:
2375:
2371:
2367:
2362:
2361:superposition
2358:
2353:
2349:
2345:
2341:
2337:
2331:
2312:
2309:
2303:
2300:
2297:
2294:
2288:
2285:
2279:
2276:
2273:
2264:
2261:
2258:
2255:
2252:
2246:
2243:
2237:
2223:
2220:
2218:
2213:
2211:
2207:
2203:
2197:
2193:
2189:
2183:
2182:standing wave
2175:
2173:
2171:
2167:
2162:
2160:
2152:
2143:
2134:
2125:
2120:
2116:
2115:superposition
2111:
2108:
2089:
2085:
2081:
2078:
2073:
2069:
2061:
2058:
2053:
2047:
2044:
2037:
2031:
2024:
2020:
2016:
2011:
2002:
1994:
1989:
1985:
1974:
1966:
1961:
1957:
1950:
1945:
1940:
1930:
1918:
1915:
1912:
1893:
1889:
1885:
1882:
1877:
1873:
1865:
1862:
1857:
1851:
1844:
1838:
1831:
1827:
1823:
1818:
1809:
1801:
1796:
1792:
1781:
1773:
1768:
1764:
1757:
1752:
1747:
1737:
1724:
1722:
1718:
1706:
1692:
1685:
1675:
1671:Substituting
1669:
1649:
1645:
1642:
1635:
1633:
1626:
1622:
1611:
1608:
1602:
1600:
1593:
1589:
1576:
1569:
1556:
1553:
1548:
1544:
1540:
1535:
1527:
1524:
1521:
1518:
1513:
1509:
1497:
1495:
1491:
1478:
1471:
1464:
1457:
1450:
1437:
1434:
1429:
1421:
1417:
1407:
1403:
1396:
1389:
1383:
1380:
1377:
1374:
1369:
1365:
1359:
1352:
1347:
1344:
1341:
1338:
1333:
1329:
1322:
1312:
1310:
1305:
1288:
1285:
1283:
1276:
1272:
1265:
1262:
1259:
1256:
1251:
1247:
1240:
1235:
1231:
1227:
1220:
1217:
1215:
1208:
1204:
1200:
1197:
1192:
1188:
1181:
1178:
1175:
1172:
1167:
1163:
1147:
1144:
1125:
1122:
1119:
1115:
1109:
1105:
1101:
1098:
1095:
1090:
1087:
1084:
1080:
1074:
1070:
1066:
1063:
1061:
1054:
1051:
1048:
1044:
1038:
1034:
1030:
1025:
1021:
1017:
1008:
1005:
1002:
998:
992:
988:
984:
981:
976:
973:
970:
966:
960:
956:
952:
949:
946:
943:
941:
934:
931:
928:
924:
918:
914:
910:
905:
901:
897:
885:
882:
863:
860:
857:
853:
847:
843:
839:
837:
829:
821:
817:
807:
804:
801:
797:
791:
787:
783:
781:
773:
765:
761:
748:
741:
722:
718:
714:
711:
706:
702:
698:
695:
692:
689:
681:
677:
673:
668:
664:
657:
654:
649:
645:
641:
638:
635:
633:
626:
619:
616:
609:
600:
596:
592:
589:
584:
580:
576:
573:
570:
567:
559:
555:
551:
546:
542:
535:
532:
527:
523:
519:
516:
513:
511:
504:
497:
494:
487:
475:
473:
452:
449:
435:
431:
426:
421:
417:
410:
405:
399:
392:
383:
379:
374:
367:
359:
357:
355:
351:
347:
343:
339:
331:
329:
325:
317:
310:
308:
305:
303:
302:skew-symmetry
299:
294:
287:
285:
282:
278:
273:
270:
266:
264:
260:
256:
252:
244:
240:
236:
232:
228:
225:
221:
220:
219:
216:
214:
213:superposition
210:
206:
201:
199:
198:standing wave
195:
191:
187:
179:
174:
170:
150:
143:
138:
134:
132:
127:
126:superposition
122:
120:
116:
111:
107:
103:
92:
89:
81:
78:December 2010
71:
67:
61:
60:
54:
50:
46:
41:
32:
31:
19:
3754:Spectroscopy
3699:
3676:
3657:
3627:
3620:
3601:
3595:
3506:
3499:
3468:
3461:
3442:
3438:
3427:
3364:
3357:
3308:
3301:
3297:
3291:
3284:
3280:
3276:
3270:
3266:
3259:
3186:
3184:, tends to:
3180:
3176:
3173:
3067:
3064:
2867:
2864:
2792:
2789:
2642:
2631:
2627:
2621:
2617:
2521:
2514:
2503:
2497:
2489:
2485:
2474:as an ideal
2433:
2411:
2400:
2396:
2392:
2388:
2383:
2377:
2373:
2369:
2365:
2357:interference
2354:
2347:
2343:
2339:
2335:
2332:
2224:
2221:
2214:
2195:
2191:
2187:
2179:
2163:
2150:
2141:
2132:
2123:
2119:normal modes
2118:
2112:
2109:
1919:
1916:
1913:
1725:
1717:eigenvectors
1704:
1690:
1683:
1673:
1670:
1577:
1571:Solving for
1570:
1498:
1476:
1469:
1462:
1455:
1451:
1313:
1306:
1148:
1145:
886:
883:
749:
742:
476:
433:
429:
422:
415:
408:
404:displacement
397:
390:
387:
363:
353:
335:
326:
322:
306:
295:
291:
288:Mode numbers
280:
276:
274:
271:
267:
248:
217:
208:
204:
202:
189:
183:
123:
110:sinusoidally
101:
99:
84:
75:
64:Please help
56:
3174:which, for
2460:fundamental
2430:Debye model
2384:normal mode
2210:equilibrium
1721:eigenvalues
1711:results in
1494:determinant
186:wave theory
102:normal mode
70:introducing
3733:Categories
3587:References
3546:Leaky mode
3495:Love waves
2480:sinusoidal
1482:) = (0, 0)
425:derivative
131:orthogonal
3409:ν
3399:ν
3387:ν
3378:∫
3335:ν
3325:ν
3316:∫
3231:ν
3218:
3200:ν
3142:ν
3133:−
3125:−
3114:
3093:−
3081:ν
3029:ν
3020:−
3012:−
3001:
2992:−
2979:ν
2937:ν
2901:∫
2886:ν
2816:∂
2808:∂
2771:⋯
2744:ν
2703:ν
2656:ν
2618:The term
2598:−
2582:ν
2569:ν
2557:ν
2535:ν
2472:harmonics
2464:overtones
2310:ω
2304:
2286:ω
2280:
2232:Ψ
2202:frequency
2086:φ
2070:ω
2062:
2045:−
1934:→
1931:η
1890:φ
1874:ω
1866:
1741:→
1738:η
1698:, yields
1623:ω
1590:ω
1541:−
1522:−
1510:ω
1378:−
1366:ω
1342:−
1330:ω
1260:−
1248:ω
1176:−
1164:ω
1123:ω
1096:−
1088:ω
1052:ω
1022:ω
1018:−
1006:ω
974:ω
947:−
932:ω
902:ω
898:−
861:ω
805:ω
693:−
674:−
639:−
620:¨
571:−
552:−
517:−
498:¨
453:¨
342:acoustics
243:overtones
223:variable.
3515:See also
3509:resonate
1490:singular
281:dominant
3450:is the
2498:In the
2492:phonons
2449:phonons
2204:and in
2117:of the
1713:(1, −1)
1307:And in
184:In the
66:improve
3706:
3687:
3664:
3635:
3608:
3446:where
2508:. For
2333:where
2148:, and
2121:where
1700:(1, 1)
1309:matrix
348:, and
277:normal
251:optics
209:linear
2620:(1/2)
2206:phase
1468:) = (
474:are:
311:Nodes
239:music
196:is a
192:in a
104:of a
51:, or
3704:ISBN
3685:ISBN
3662:ISBN
3633:ISBN
3606:ISBN
3264:and
2519:is:
2458:The
2428:and
470:the
366:mass
275:The
261:and
205:mode
190:mode
180:Mode
3502:= 0
3215:log
3111:log
2998:log
2478:of
2451:).
2301:sin
2277:cos
2168:or
2059:cos
1863:cos
1723:.)
427:of
279:or
117:or
3735::
3683:.
3443:hf
3441:=
3285:dν
3271:dν
3269:+
3181:hν
3179:≫
3177:kT
2687:12
2638:kT
2622:hν
2445:hν
2399:,
2395:,
2386:.
2376:,
2372:,
2346:,
2342:,
2194:,
2190:,
2180:A
2172:.
2139:,
2130:,
1689:,
1475:,
1461:,
344:,
265:.
257:,
253:,
245:".
100:A
55:,
47:,
3712:.
3693:.
3670:.
3641:.
3614:.
3500:l
3491:l
3487:l
3483:m
3479:l
3475:n
3448:h
3439:E
3434:f
3406:d
3402:)
3396:(
3393:E
3390:)
3384:(
3381:f
3375:=
3372:U
3360:U
3344:N
3341:3
3338:=
3332:d
3328:)
3322:(
3319:f
3304:)
3302:ν
3300:(
3298:f
3292:N
3290:3
3283:)
3281:ν
3279:(
3277:f
3267:ν
3262:ν
3245:)
3239:T
3236:k
3228:h
3222:(
3212:T
3209:k
3206:=
3203:)
3197:(
3194:F
3159:)
3150:T
3147:k
3139:h
3129:e
3122:1
3118:(
3108:T
3105:k
3102:=
3099:S
3096:T
3090:E
3087:=
3084:)
3078:(
3075:F
3046:)
3037:T
3034:k
3026:h
3016:e
3009:1
3005:(
2995:k
2987:T
2982:)
2976:(
2972:E
2966:=
2954:T
2950:T
2947:d
2940:)
2934:(
2930:E
2924:T
2921:d
2917:d
2910:T
2905:0
2897:=
2889:)
2883:(
2879:S
2849:T
2846:1
2841:=
2836:V
2833:,
2830:N
2825:)
2819:E
2811:S
2802:(
2775:]
2768:+
2763:4
2758:)
2752:T
2749:k
2741:h
2735:(
2730:O
2727:+
2722:2
2717:)
2711:T
2708:k
2700:h
2694:(
2684:1
2679:+
2676:1
2672:[
2668:T
2665:k
2662:=
2659:)
2653:(
2650:E
2634:)
2632:ν
2630:(
2628:E
2601:1
2593:T
2590:k
2586:/
2579:h
2575:e
2566:h
2560:+
2554:h
2549:2
2546:1
2541:=
2538:)
2532:(
2529:E
2517:ν
2441:ν
2437:ν
2412:n
2403:)
2401:z
2397:y
2393:x
2391:(
2380:)
2378:z
2374:y
2370:x
2368:(
2366:f
2350:)
2348:z
2344:y
2340:x
2338:(
2336:f
2319:)
2316:)
2313:t
2307:(
2298:B
2295:+
2292:)
2289:t
2283:(
2274:A
2271:(
2268:)
2265:z
2262:,
2259:y
2256:,
2253:x
2250:(
2247:f
2244:=
2241:)
2238:t
2235:(
2198:)
2196:z
2192:y
2188:x
2186:(
2154:2
2151:φ
2145:1
2142:φ
2136:2
2133:c
2127:1
2124:c
2095:)
2090:2
2082:+
2079:t
2074:2
2066:(
2054:)
2048:1
2038:1
2032:(
2025:2
2021:c
2017:=
2012:)
2006:)
2003:t
2000:(
1995:2
1990:2
1986:x
1978:)
1975:t
1972:(
1967:2
1962:1
1958:x
1951:(
1946:=
1941:2
1899:)
1894:1
1886:+
1883:t
1878:1
1870:(
1858:)
1852:1
1845:1
1839:(
1832:1
1828:c
1824:=
1819:)
1813:)
1810:t
1807:(
1802:1
1797:2
1793:x
1785:)
1782:t
1779:(
1774:1
1769:1
1765:x
1758:(
1753:=
1748:1
1708:2
1705:ω
1696:)
1694:2
1691:A
1687:1
1684:A
1682:(
1677:1
1674:ω
1650:m
1646:k
1643:3
1636:=
1627:2
1612:m
1609:k
1603:=
1594:1
1573:ω
1557:0
1554:=
1549:2
1545:k
1536:2
1532:)
1528:k
1525:2
1519:m
1514:2
1506:(
1486:ω
1480:2
1477:x
1473:1
1470:x
1466:2
1463:A
1459:1
1456:A
1454:(
1438:0
1435:=
1430:)
1422:2
1418:A
1408:1
1404:A
1397:(
1390:]
1384:k
1381:2
1375:m
1370:2
1360:k
1353:k
1348:k
1345:2
1339:m
1334:2
1323:[
1289:0
1286:=
1277:2
1273:A
1269:)
1266:k
1263:2
1257:m
1252:2
1244:(
1241:+
1236:1
1232:A
1228:k
1221:0
1218:=
1209:2
1205:A
1201:k
1198:+
1193:1
1189:A
1185:)
1182:k
1179:2
1173:m
1168:2
1160:(
1126:t
1120:i
1116:e
1110:2
1106:A
1102:k
1099:2
1091:t
1085:i
1081:e
1075:1
1071:A
1067:k
1064:=
1055:t
1049:i
1045:e
1039:2
1035:A
1031:m
1026:2
1009:t
1003:i
999:e
993:2
989:A
985:k
982:+
977:t
971:i
967:e
961:1
957:A
953:k
950:2
944:=
935:t
929:i
925:e
919:1
915:A
911:m
906:2
864:t
858:i
854:e
848:2
844:A
840:=
833:)
830:t
827:(
822:2
818:x
808:t
802:i
798:e
792:1
788:A
784:=
777:)
774:t
771:(
766:1
762:x
745:ω
723:1
719:x
715:k
712:+
707:2
703:x
699:k
696:2
690:=
687:)
682:2
678:x
669:1
665:x
661:(
658:k
655:+
650:2
646:x
642:k
636:=
627:2
617:x
610:m
601:2
597:x
593:k
590:+
585:1
581:x
577:k
574:2
568:=
565:)
560:1
556:x
547:2
543:x
539:(
536:k
533:+
528:1
524:x
520:k
514:=
505:1
495:x
488:m
468:,
450:x
436:)
434:t
432:(
430:x
418:)
416:t
414:(
412:2
409:x
400:)
398:t
396:(
394:1
391:x
376:k
369:m
144:.
91:)
85:(
80:)
76:(
62:.
20:)
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