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is open, the pressure variations are very small, corresponding to a pressure node (which is an anti-node for molecular motions, because the molecules near the open end can move freely). The exact location of the pressure node at an open end is actually slightly beyond the open end of the pipe, so the effective length of the pipe for the purpose of determining resonant frequencies is slightly longer than its physical length. This difference in length is ignored in this example. In terms of reflections, open ends partially reflect waves back into the pipe, allowing some energy to be released into the outside air. Ideally, closed ends reflect the entire wave back in the other direction.
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along the direction of wave motion. The wave propagates by alternately compressing and expanding air in segments of the pipe, which displaces the air slightly from its rest position and transfers energy to neighboring segments through the forces exerted by the alternating high and low air pressures. Equations resembling those for the wave on a string can be written for the change in pressure Î
20:
3997:-direction to shift the location of the nodes and anti-nodes. For example, the longest wavelength that resonatesâthe fundamental modeâis again twice the length of the pipe, except that the ends of the pipe have pressure anti-nodes instead of pressure nodes. Between the ends there is one pressure node. In the case of two closed ends, the wavelength is again restricted to
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is an example of a standing wave in an enclosed body of water. It is characterised by the oscillatory behaviour of the water level at either end of the body and typically has a nodal point near the middle of the body where very little change in water level is observed. It should be distinguished from
1704:
off the right fixed end and travels back to the left, reflects again off the left fixed end and travels back to the right, and so on. Eventually, a steady state is reached where the string has identical right- and left-traveling waves as in the infinite-length case and the power dissipated by damping
2868:
traveling to the right or left through the pipe. While the transverse waves on the string from the previous examples vary in their displacement perpendicular to the direction of wave motion, the waves traveling through the air in the pipe vary in terms of their pressure and longitudinal displacement
6848:
Standing waves can be mechanically induced into a solid medium using resonance. One easy to understand example is two people shaking either end of a jump rope. If they shake in sync, the rope will form a regular pattern with nodes and antinodes and appear to be stationary, hence the name standing
3317:
If the end of a pipe is closed, the pressure is maximal since the closed end of the pipe exerts a force that restricts the movement of air. This corresponds to a pressure anti-node (which is a node for molecular motions, because the molecules near the closed end cannot move). If the end of the pipe
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for a pipe that is closed at both ends. Considering longitudinal displacement, the molecules at the ends and the molecules in the middle are not displaced by the wave, representing nodes of longitudinal displacement. Halfway between the nodes there are longitudinal displacement anti-nodes where
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A pure standing wave does not transfer energy from the source to the destination. However, the wave is still subject to losses in the medium. Such losses will manifest as a finite SWR, indicating a travelling wave component leaving the source to supply the losses. Even though the SWR is now
3978:
of the air. In terms of longitudinal displacement, closed ends of pipes correspond to nodes since air movement is restricted and open ends correspond to anti-nodes since the air is free to move. A similar, easier to visualize phenomenon occurs in longitudinal waves propagating along a spring.
3303:
3879:
molecules are maximally displaced. Considering pressure, the molecules are maximally compressed and expanded at the ends and in the middle, representing pressure anti-nodes. Halfway between the anti-nodes are pressure nodes where the molecules are neither compressed nor expanded as they move.
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This section considers representative one- and two-dimensional cases of standing waves. First, an example of an infinite length string shows how identical waves traveling in opposite directions interfere to produce standing waves. Next, two finite length string examples with different
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We can also consider a pipe that is closed at both ends. In this case, both ends will be pressure anti-nodes or equivalently both ends will be displacement nodes. This example is analogous to the case where both ends are open, except the standing wave pattern has a
6734:
to be identified. Nodes occur at fixed ends and anti-nodes at open ends. If fixed at only one end, only odd-numbered harmonics are available. At the open end of a pipe the anti-node will not be exactly at the end as it is altered by its contact with the air and so
6651:
If the two oppositely moving traveling waves are not of the same amplitude, they will not cancel completely at the nodes, the points where the waves are 180° out of phase, so the amplitude of the standing wave will not be zero at the nodes, but merely a minimum.
6717:
Standing waves are also observed in physical media such as strings and columns of air. Any waves traveling along the medium will reflect back when they reach the end. This effect is most noticeable in musical instruments where, at various multiples of a
6180:
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demonstrate how the boundary conditions restrict the frequencies that can form standing waves. Next, the example of sound waves in a pipe demonstrates how the same principles can be applied to longitudinal waves with analogous boundary conditions.
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wave. Similarly a cantilever beam can have a standing wave imposed on it by applying a base excitation. In this case the free end moves the greatest distance laterally compared to any location along the beam. Such a device can be used as a
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finite, it may still be the case that no energy reaches the destination because the travelling component is purely supplying the losses. However, in a lossless medium, a finite SWR implies a definite transfer of energy to the destination.
174:
1922:
1237:
1343:. The animation at the beginning of this article depicts what is happening. As the left-traveling blue wave and right-traveling green wave interfere, they form the standing red wave that does not travel and instead oscillates in place.
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6682:. If they shake in sync the rope can form a regular pattern of waves oscillating up and down, with stationary points along the rope where the rope is almost still (nodes) and points where the arc of the rope is maximum (antinodes).
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of two waves: a travelling wave component and a stationary wave component. An SWR of one indicates that the wave does not have a stationary component â it is purely a travelling wave, since the ratio of amplitudes is equal to 1.
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The constants that determine the exact sinusoidal functions depend on the boundary conditions and initial conditions. To see how the boundary conditions apply, consider an example like the sheet that has been pulled taut where
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155:
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is used to place it exactly. The density of a string will affect the frequency at which harmonics will be produced; the greater the density the lower the frequency needs to be to produce a standing wave of the same harmonic.
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501:, illustrated in the animations above, the nodes become nodal lines, lines on the surface at which there is no movement, that separate regions vibrating with opposite phase. These nodal line patterns are called
344:
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at any point in space (but it may undergo a 180° inversion every half cycle). A finite, non-zero SWR indicates a wave that is partially stationary and partially travelling. Such waves can be decomposed into a
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Melde, Franz. Ueber einige krumme FlÀchen, welche von Ebenen, parallel einer bestimmten Ebene, durchschnitten, als
Durchschnittsfigur einen Kegelschnitt liefern: Inaugural-Dissertation... Koch, 1859.
5420:
5200:
6444:
242:: 1.7 â 4.5, surpassing 4.5 results in direct standing wave) and is therefore neither significantly slowed down by the obstacle nor pushed to the side. Many standing river waves are popular
7978:
6910:'s longitudinal period is 73 minutes and its transversal seiche has a period of around 10 minutes, while Lake Huron can be seen to have resonances with periods between 1 and 2 hours. See
7757:
Feng, Z.; Kim, C.-Y.; Elam, J.W.; Ma, Q.; Zhang, Z.; Bedzyk, M.J. (2009). "Direct Atomic-Scale
Observation of Redox-Induced Cation Dynamics in an Oxide-Supported Monolayer Catalyst: WO
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the amplitude is maximal, with a value of twice the amplitude of the right- and left-traveling waves that interfere to produce this standing wave pattern. These locations are called
318:
6656:(SWR) is the ratio of the amplitude at the antinode (maximum) to the amplitude at the node (minimum). A pure standing wave will have an infinite SWR. It will also have a constant
6082:
360:
513:, there are nodal surfaces. This section includes a two-dimensional standing wave example with a rectangular boundary to illustrate how to extend the concept to higher dimensions.
457:, which is a superposition of a standing wave and a traveling wave. The degree to which the wave resembles either a pure standing wave or a pure traveling wave is measured by the
1009:{\displaystyle y(x,t)=y_{\text{R}}+y_{\text{L}}=y_{\text{max}}\sin \left({2\pi x \over \lambda }-\omega t\right)+y_{\text{max}}\sin \left({2\pi x \over \lambda }+\omega t\right).}
7806:
Bauza, Marcin B.; Hocken, Robert J.; Smith, Stuart T.; Woody, Shane C. (2005). "Development of a virtual probe tip with an application to high aspect ratio microscale features".
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formed by waves with the same wave period moving in opposite directions. These may form near storm centres, or from reflection of a swell at the shore, and are the source of
5817:
3383:
154:
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is a non-linear standing wave at the air-liquid interface induced by hydrodynamic instability. It can be used as a liquid-based template to assemble microscale materials.
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and time. Alternatively, the wave can be written in terms of its longitudinal displacement of air, where air in a segment of the pipe moves back and forth slightly in the
4719:
8042:
673:
562:
1814:
1820:
4178:-direction. Examples of this type of wave are water waves in a pool or waves on a rectangular sheet that has been pulled taut. The waves displace the surface in the
1646:{\displaystyle x=\ldots ,-{5\lambda \over 4},\;-{3\lambda \over 4},\;-{\lambda \over 4},\;{\lambda \over 4},\;{3\lambda \over 4},\;{5\lambda \over 4},\ldots }
1138:
105:
This phenomenon can occur because the medium is moving in the direction opposite to the movement of the wave, or it can arise in a stationary medium as a result of
4887:{\displaystyle {\frac {1}{X(x)}}{\frac {\partial ^{2}X}{\partial x^{2}}}+{\frac {1}{Y(y)}}{\frac {\partial ^{2}Y}{\partial y^{2}}}+{\frac {\omega ^{2}}{c^{2}}}=0.}
1025:
7952:
4342:{\displaystyle {\frac {\partial ^{2}z}{\partial t^{2}}}\;=\;c^{2}\left({\frac {\partial ^{2}z}{\partial x^{2}}}+{\frac {\partial ^{2}z}{\partial y^{2}}}\right),}
4660:
problem where the frequencies correspond to eigenvalues that then correspond to frequency-specific modes or eigenfunctions. Specifically, this is a form of the
6800:(XSW) field. Because of the short wavelength of X-rays (less than 1 nanometer), this phenomenon can be exploited for measuring atomic-scale events at material
453:
In practice, losses in the transmission line and other components mean that a perfect reflection and a pure standing wave are never achieved. The result is a
306:
272:
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178:
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423:) at fixed points along the transmission line. Such a standing wave may be formed when a wave is transmitted into one end of a transmission line and is
4391:
3298:{\displaystyle \Delta p(x,t)=\Delta p_{\text{R}}(x,t)+\Delta p_{\text{L}}(x,t)=2p_{\text{max}}\sin \left({2\pi x \over \lambda }\right)\cos(\omega t).}
8552:
6824:
yield from the atoms near the surface. This shift can be analyzed to pinpoint the location of a particular atomic species relative to the underlying
6366:{\displaystyle z(x,y,t)=z_{\text{max}}\sin \left({\frac {n\pi x}{L_{x}}}\right)\sin \left({\frac {m\pi y}{L_{y}}}\right)\cos \left(\omega t\right).}
5590:
8438:
1484:{\displaystyle x=\ldots ,-{3\lambda \over 2},\;-\lambda ,\;-{\lambda \over 2},\;0,\;{\lambda \over 2},\;\lambda ,\;{3\lambda \over 2},\ldots }
71:
of the wave oscillations at any point in space is constant with respect to time, and the oscillations at different points throughout the wave are
6698:
7990:
7716:
4646:{\displaystyle {\frac {\partial ^{2}Z}{\partial x^{2}}}+{\frac {\partial ^{2}Z}{\partial y^{2}}}=-{\frac {\omega ^{2}}{c^{2}}}Z(x,y,\omega ).}
1284:
7937:
7625:
2663:
This leads to a different set of wavelengths than in the two-fixed-ends example. Here, the wavelength of the standing waves is restricted to
3905:
6633:
determines the eigenvalue in the
Helmholtz equation above, the number of modes corresponding to each frequency relates to the frequency's
6449:
So, standing waves inside this fixed rectangular boundary oscillate in time at certain resonant frequencies parameterized by the integers
8165:
5775:. As in the one dimensional example of the string fixed at both ends, the sinusoidal function that satisfies this boundary condition is
7560:
5837:
3713:. This pipe has boundary conditions analogous to the string with only one fixed end. Its standing waves have wavelengths restricted to
1696:
fixed end a sinusoidal force is applied that drives the string up and down in the y-direction with a small amplitude at some frequency
5435:
5029:
6761:
1689:. The string will have some damping as it is stretched by traveling waves, but assume the damping is very small. Suppose that at the
8708:
8095:
8076:
7190:
3329:. Given that the pressure must be zero at both open ends, the boundary conditions are analogous to the string with two fixed ends,
6906:
where no oscillation is present. In sizeable lakes, the period of such oscillations may be between minutes and hours, for example
2819:
multiple of a quarter wavelength. Thus the fundamental mode in this example only has one quarter of a complete sine cycleâzero at
2608:
6574:
2490:
to a ring that can slide freely up and down a pole. The string again has small damping and is driven by a small driving force at
1950:
3709:
is that the pressure of the closed end will follow that of the point to its left. Examples of this setup include a bottle and a
7162:
7091:
Melde, Franz. "Ueber die
Erregung stehender Wellen eines fadenförmigen Körpers." Annalen der Physik 185, no. 2 (1860): 193â215.
2129:
106:
8106:
8034:
7156:
5339:{\displaystyle {\frac {1}{Y(y)}}{\frac {\partial ^{2}Y}{\partial y^{2}}}=(ik_{y})^{2}=k_{x}^{2}-{\frac {\omega ^{2}}{c^{2}}},}
7986:
7545:
7524:
7070:
7049:
6780:, 10 m) so the standing waves are microscopic in size. One use for standing light waves is to measure small distances, using
5359:
8012:
2506:) still describes the standing wave pattern that can form on the string, and the string has the same boundary condition of
8545:
6994:
3326:
3133:
If identical right- and left-traveling waves travel through the pipe, the resulting superposition is described by the sum
367:
351:
6634:
6377:
4657:
2222:
To compare this example's nodes to the description of nodes for standing waves in the infinite length string, Equation (
7956:
7469:
8916:
7497:. Animations for Physics and Astronomy. Pennsylvania State University. Also available as YouTube Video ID: NMlys8A0_4s
4156:
Next, consider transverse waves that can move along a two dimensional surface within a rectangular boundary of length
200:
As an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the
8277:
823:
For identical right- and left-traveling waves on the same string, the total displacement of the string is the sum of
294:
A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
8579:
7617:
1346:
Because the string is of infinite length, it has no boundary condition for its displacement at any point along the
7920:
Lemmin, Ulrich (2012), "Surface
Seiches", in Bengtsson, Lars; Herschy, Reginald W.; Fairbridge, Rhodes W. (eds.),
7100:
Melde, Franz. Die Lehre von den
Schwingungscurven...: mit einem Atlas von 11 Tafeln in Steindruck. JA Barth, 1864.
2115:
Waves can only form standing waves on this string if they have a wavelength that satisfies this relationship with
8926:
8687:
8398:
8137:
6816:. By tuning the crystal geometry or X-ray wavelength, the XSW can be translated in space, causing a shift in the
189:
109:
between two waves traveling in opposite directions. The most common cause of standing waves is the phenomenon of
8931:
8921:
8538:
8453:
7005:
6175:{\displaystyle \omega =c\pi {\sqrt {\left({\frac {n}{L_{x}}}\right)^{2}+\left({\frac {m}{L_{y}}}\right)^{2}}}.}
4003:
3719:
3502:
2669:
2338:
2234:
2014:
2839:
This example also demonstrates a type of resonance and the frequencies that produce standing waves are called
8936:
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541:
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4148:
provides a way to visualize the pressure variations of the standing waves in a tube with two closed ends.
3078:{\displaystyle \Delta p_{\text{L}}(x,t)=p_{\text{max}}\sin \left({2\pi x \over \lambda }+\omega t\right),}
2973:{\displaystyle \Delta p_{\text{R}}(x,t)=p_{\text{max}}\sin \left({2\pi x \over \lambda }-\omega t\right),}
1020:
486:
447:
6776:, exciting standing waves of light in the cavity. The wavelength of light is very short (in the range of
4098:
3891:-direction as the pressure varies and waves travel in either or both directions. The change in pressure Î
3814:
3597:
2761:
8642:
8413:
8311:
8228:
8215:
7492:
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gives complicated but predictable two-dimensional patterns of nodes and antinodes inside the rectangle.
2196:
1932:
420:
412:
388:
325:
8671:
7849:
3671:(and therefore has a pressure anti-node). The closed "free end" boundary condition for the pressure at
3486:{\displaystyle \Delta p(L,t)=2p_{\text{max}}\sin \left({2\pi L \over \lambda }\right)\cos(\omega t)=0,}
7676:
Batterman, Boris W. (1969). "Detection of
Foreign Atom Sites by Their X-Ray Fluorescence Scattering".
6490:
From the dispersion relation, in certain situations different modesâmeaning different combinations of
6457:. As they oscillate in time, they do not travel and their spatial variation is sinusoidal in both the
4045:
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2380:
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2071:
1701:
428:
7641:
Batterman, Boris W.; Cole, Henderson (1964). "Dynamical
Diffraction of X Rays by Perfect Crystals".
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in the string equals the power supplied by the driving force so the waves have constant amplitude.
764:{\displaystyle y_{\text{L}}(x,t)=y_{\text{max}}\sin \left({2\pi x \over \lambda }+\omega t\right),}
653:{\displaystyle y_{\text{R}}(x,t)=y_{\text{max}}\sin \left({2\pi x \over \lambda }-\omega t\right).}
458:
446:. The failure of the line to transfer power at the standing wave frequency will usually result in
7419:
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8365:
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1917:{\displaystyle y(L,t)=2y_{\text{max}}\sin \left({2\pi L \over \lambda }\right)\cos(\omega t)=0.}
102:) around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings.
8778:
7298:
3098:
is the pressure amplitude or the maximum increase or decrease in air pressure due to each wave,
1775:
79:, and the locations where the absolute value of the amplitude is maximum are called antinodes.
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1232:{\displaystyle y(x,t)=2y_{\text{max}}\sin \left({2\pi x \over \lambda }\right)\cos(\omega t).}
795:
384:
226:. A requirement for this in river currents is a flowing water with shallow depth in which the
6076:
Restricting the wave numbers to these values also restricts the frequencies that resonate to
1123:{\displaystyle \sin a+\sin b=2\sin \left({a+b \over 2}\right)\cos \left({a-b \over 2}\right)}
8723:
8703:
8499:
8418:
8382:
8350:
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7884:
7823:
7780:
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7693:
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7000:
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1927:
1714:) still describes the standing wave pattern that can form on this string, but now Equation (
510:
165:
67:
that oscillates in time but whose peak amplitude profile does not move in space. The peak
8728:
8654:
8530:
8494:
8448:
8289:
8247:
7564:
6801:
526:
83:
6975:
5572:) is a superposition of modes where each mode is the product of sinusoidal functions for
2123:
along the string, then equivalently the frequency of the standing waves is restricted to
7819:
7735:
7689:
7654:
7590:
2530:
where the string can move freely there should be an anti-node with maximal amplitude of
8766:
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8617:
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8428:
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6828:
or mirror surface. The XSW method has been used to clarify the atomic-scale details of
6809:
6753:
6736:
5005:{\displaystyle {\frac {1}{X(x)}}{\frac {\partial ^{2}X}{\partial x^{2}}}=(ik_{x})^{2}.}
3648:
2448:
1495:
502:
408:
396:
279:
212:
76:
3314:), so a stationary pressure wave forms that is fixed in space and oscillates in time.
2199:
and has a wavelength that is twice the length of the string. Higher integer values of
2006:
is given, so the boundary condition restricts the wavelength of the standing waves to
8910:
8875:
8813:
8622:
8408:
8338:
8293:
8255:
8205:
8200:
7577:
Barbosa Aguiar, Ana C. (2010). "A laboratory model of Saturn's North Polar
Hexagon".
6980:
6821:
6657:
6498:âmay resonate at the same frequency even though they have different shapes for their
6066:{\displaystyle \sin {k_{y}y},\quad k_{y}={\frac {m\pi }{L_{y}}},\quad m=1,2,3,\dots }
4193:
3883:
So far, the wave has been written in terms of its pressure as a function of position
3689:
2552:
1944:
537:
443:
243:
239:
75:. The locations at which the absolute value of the amplitude is minimum are called
6678:
One easy example to understand standing waves is two people shaking either end of a
6213:-dependence is a cosine function, then standing waves for this system take the form
1660:. The distance between two consecutive nodes or anti-nodes is half the wavelength,
439:
8895:
8403:
8112:. Department of Physics. PHYS122 Fundamentals of Physics II. University of Maryland
7166:
6950:
6886:
6874:
6781:
220:
87:
8004:
7929:
6465:-directions such that they satisfy the boundary conditions. The fundamental mode,
4495:{\displaystyle Z(x,y,\omega )=\int _{-\infty }^{\infty }z(x,y,t)e^{-i\omega t}dt.}
1700:. In this situation, the driving force produces a right-traveling wave. That wave
7924:, Encyclopedia of Earth Sciences Series, Springer Netherlands, pp. 751â753,
7598:
6760:
use optical cavities in the form of a pair of facing mirrors, which constitute a
5556:
Multiplying these functions together and applying the inverse Fourier transform,
3864:
only takes odd values just like in the case of the string fixed at only one end.
8870:
8828:
8783:
8748:
8632:
8612:
8595:
8423:
8235:
7020:
6907:
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is that the motion of the "free end" will follow that of the point to its left.
354:, an example of standing waves in two dimensions. This is the fundamental mode.
91:
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8803:
8798:
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of the resonance of the fiber. One application is as a measurement device for
6833:
6723:
3322:
3126:
2865:
816:
473:
469:
117:
due to interference between waves reflected back and forth at the resonator's
19:
7063:
Nonlinear Instability of Nonparallel Flows: IUTAM Symposium Potsdam, New York
5698:{\displaystyle z(x,y,t)\sim e^{\pm ik_{x}x}e^{\pm ik_{y}y}e^{\pm i\omega t}.}
8850:
8788:
8743:
8681:
8265:
8195:
8174:
6777:
6731:
6679:
3113:
2836:âthe first harmonic has three quarters of a complete sine cycle, and so on.
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223:
201:
125:
114:
110:
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7567:, M. Allison, D. A. Godfrey, R. F. Beebe, Science vol. 247, pg. 1061 (1990)
6804:. The XSW is generated in the region where an X-ray beam interferes with a
2534:. Equivalently, this boundary condition of the "free end" can be stated as
312:
Electric force vector (E) and magnetic force vector (H) of a standing wave.
192:- Standing waves are formed where the Carmel River meets the Pacific Ocean.
8132:
8865:
8818:
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8360:
8328:
7717:"Solution to the Surface Registration Problem Using X-Ray Standing Waves"
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1936:
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329:
72:
7424:. Flipping Physics. Event occurs at 4:11. YouTube video ID: 3QbmvunlQR0
7266:. Digital Sound & Music. May 21, 2014. YouTube Video ID: oZ38Y0K8e-Y
1336:{\displaystyle 2y_{\text{max}}\sin \left({2\pi x \over \lambda }\right)}
8885:
8860:
8843:
8838:
8771:
8756:
8306:
7338:
7336:
7334:
7332:
7330:
7328:
7220:
7218:
7216:
7203:
7201:
6970:
6769:
3975:
392:
231:
227:
205:
129:
52:
7827:
7784:
3867:
2428:
and the frequencies that produce standing waves can be referred to as
8855:
8833:
8823:
8793:
7061:
Theodore Y. Wu, "Stability of nonlinear waves resonantly sustained",
6965:
6911:
6898:
6850:
3960:{\displaystyle \Delta p=-\rho v^{2}{\frac {\partial s}{\partial x}},}
3321:
First consider a pipe that is open at both ends, for example an open
216:
278:
Standing wave in stationary medium. The red dots represent the wave
7842:"Precision Engineering and Manufacturing Solutions â IST Precision"
667:-direction for an identical harmonic wave traveling to the left is
8880:
8479:
6793:
6757:
6695:
2439:
465:
88:
standing waves on the surface of a liquid in a vibrating container
18:
6709:
at the north pole of Saturn was initially thought to be standing
5910:{\displaystyle k_{x}={\frac {n\pi }{L_{x}}},\quad n=1,2,3,\dots }
3808:
or equivalently the frequency of standing waves is restricted to
1277:) simply oscillates in time with an amplitude that varies in the
8637:
8469:
6479:, has a single antinode in the middle of the rectangle. Varying
5546:{\displaystyle Y(y)=C_{k_{y}}e^{ik_{y}y}+D_{k_{y}}e^{-ik_{y}y}.}
5140:{\displaystyle X(x)=A_{k_{x}}e^{ik_{x}x}+B_{k_{x}}e^{-ik_{x}x}.}
133:
64:
8534:
8147:
7711:
Golovchenko, J. A.; Patel, J. R.; Kaplan, D. R.; Cowan, P. L.;
7449:. YouTube. Event occurs at 6:04. YouTube video ID: dihQuwrf9yQ
4897:
This leads to two coupled ordinary differential equations. The
4381:
To solve this differential equation, let's first solve for its
3308:
This formula for the pressure is of the same form as Equation (
1350:-axis. As a result, a standing wave can form at any frequency.
505:. In three-dimensional resonators, such as musical instrument
407:
propagating in opposite directions. The effect is a series of
7365:
7363:
8143:
7979:"Bonanza for Lake Superior: Seiches Do More Than Move Water"
5725:) must be zero all around the rectangular boundary. For the
2653:{\displaystyle \cos \left({2\pi L \over \lambda }\right)=0.}
497:. With standing waves on two-dimensional membranes such as
7841:
6619:{\displaystyle \omega ={\frac {c\pi }{L_{x}}}{\sqrt {50}}.}
3496:
which only occurs when the wavelength of standing waves is
1995:{\displaystyle \sin \left({2\pi L \over \lambda }\right)=0}
493:
Standing waves can also occur in two- or three-dimensional
6748:
Standing waves are also observed in optical media such as
2178:{\displaystyle f={\frac {v}{\lambda }}={\frac {nv}{2L}}.}
1494:
the amplitude is always zero. These locations are called
521:
To begin, consider a string of infinite length along the
8067:
Halliday, David; Resnick, Robert; Walker, Jearl (2005).
7558:
A Wave Dynamical Interpretation of Saturn's Polar Region
7494:
2D Standing Wave Patterns (rectangular fixed boundaries)
7470:"Helmholtz Differential Equation--Cartesian Coordinates"
2320:
In this variation of the expression for the wavelength,
253:
7873:"Microscale Assembly Directed by Liquid-Based Template"
6713:. However, this explanation has recently been disputed.
5415:{\displaystyle \omega =c{\sqrt {k_{x}^{2}+k_{y}^{2}}}.}
4189:
defined as the height of the surface when it is still.
204:
of mountain ranges. Such waves are often exploited by
6836:
on surfaces, and chemical transformations involved in
1261:) does not describe a traveling wave. At any position
540:
traveling to the right along the string, the string's
82:
Standing waves were first described scientifically by
7393:
7381:
7342:
7284:
7224:
7207:
6577:
6506:-dependence. For example, if the boundary is square,
6380:
6222:
6085:
5971:
5840:
5784:
5593:
5438:
5362:
5203:
5186:
determined by the boundary conditions. Likewise, the
5032:
4914:
4737:
4677:
4514:
4394:
4205:
4101:
4048:
4006:
3908:
3817:
3764:
3722:
3600:
3547:
3505:
3386:
3338:
3142:
2987:
2882:
2764:
2714:
2672:
2611:
2383:
2341:
2279:
2237:
2132:
2074:
2017:
1953:
1823:
1778:
1758:
and because we assume the driving force at the fixed
1515:
1370:
1287:
1141:
1028:
846:
676:
565:
8035:"Japanese earthquake literally made waves in Norway"
6873:
Standing surface waves on the Earth are observed as
2873:
due to a right- or left-traveling wave in the pipe.
2853:
Acoustic resonance § Resonance of a tube of air
2480:
direction. For example, the string might be tied at
2324:
must be even. Cross multiplying we see that because
8742:
8462:
8391:
8288:
8246:
8181:
4505:Taking the Fourier transform of the wave equation,
2861:. The air inside the pipe serves as the medium for
6618:
6439:{\displaystyle n=1,2,3,\dots \quad m=1,2,3,\dots }
6438:
6365:
6174:
6065:
5909:
5811:
5697:
5545:
5414:
5338:
5139:
5004:
4886:
4713:
4645:
4494:
4341:
4133:
4081:
4033:
3959:
3849:
3797:
3749:
3661:(and therefore has a pressure node) and closed at
3632:
3580:
3532:
3485:
3371:
3297:
3077:
2972:
2796:
2744:
2699:
2652:
2413:
2368:
2309:
2264:
2177:
2104:
2044:
1994:
1947:. The latter boundary condition is satisfied when
1916:
1808:
1645:
1483:
1335:
1231:
1122:
1008:
763:
652:
5745:) must vary in a way that it can be zero at both
4192:In two dimensions and Cartesian coordinates, the
3871:Molecular representation of a standing wave with
31:created by the superposition of a left traveling
1765:end has small amplitude. Checking the values of
788:of the displacement of the string for each wave,
8107:"Chapter 16 â Superposition and Standing Waves"
6641:Standing wave ratio, phase, and energy transfer
7612:Pedrotti, Frank L.; Pedrotti, Leno M. (2017).
464:Another example is standing waves in the open
128:traveling in opposing directions, there is on
8546:
8159:
8090:(3rd ed.). Saunders College Publishing.
8086:Serway, Raymond A.; Faughn, Jerry S. (1992).
2857:Consider a standing wave in a pipe of length
2755:Equivalently, the frequency is restricted to
1668:Standing wave on a string with two fixed ends
8:
7418:Thomas-Palmer, Jonathan (October 16, 2019).
7405:
7369:
7248:
7236:
4152:2D standing wave with a rectangular boundary
3692:. The intuition for this boundary condition
2555:. The intuition for this boundary condition
2436:Standing wave on a string with one fixed end
2211:. Any standing wave on the string will have
370:with two nodal lines crossing at the center.
7540:, p. 251, Cambridge University Press, 2010
1672:Next, consider a string with fixed ends at
8553:
8539:
8531:
8166:
8152:
8144:
5425:Solving the differential equation for the
4244:
4240:
2203:correspond to modes of oscillation called
1943:This boundary condition is in the form of
1621:
1602:
1588:
1571:
1549:
1459:
1452:
1438:
1431:
1414:
1404:
517:Standing wave on an infinite length string
7896:
7421:Longitudinal Standing Waves Demonstration
6606:
6598:
6584:
6576:
6379:
6326:
6309:
6287:
6270:
6254:
6221:
6161:
6149:
6140:
6126:
6114:
6105:
6098:
6084:
6024:
6010:
6001:
5983:
5978:
5970:
5868:
5854:
5845:
5839:
5796:
5791:
5783:
5677:
5662:
5651:
5636:
5625:
5592:
5529:
5518:
5506:
5501:
5483:
5475:
5463:
5458:
5437:
5401:
5396:
5383:
5378:
5372:
5361:
5325:
5315:
5309:
5300:
5295:
5282:
5272:
5250:
5232:
5225:
5204:
5202:
5123:
5112:
5100:
5095:
5077:
5069:
5057:
5052:
5031:
4993:
4983:
4961:
4943:
4936:
4915:
4913:
4870:
4860:
4854:
4842:
4824:
4817:
4796:
4784:
4766:
4759:
4738:
4736:
4676:
4608:
4598:
4592:
4577:
4559:
4552:
4540:
4522:
4515:
4513:
4468:
4434:
4426:
4393:
4322:
4304:
4297:
4285:
4267:
4260:
4249:
4231:
4213:
4206:
4204:
4108:
4100:
4092:and the frequency is again restricted to
4047:
4034:{\displaystyle \lambda ={\frac {2L}{n}},}
4013:
4005:
3934:
3928:
3907:
3824:
3816:
3763:
3750:{\displaystyle \lambda ={\frac {4L}{n}},}
3729:
3721:
3607:
3599:
3546:
3533:{\displaystyle \lambda ={\frac {2L}{n}},}
3512:
3504:
3434:
3418:
3385:
3337:
3252:
3236:
3205:
3174:
3141:
3040:
3023:
2995:
2986:
2935:
2918:
2890:
2881:
2771:
2763:
2713:
2700:{\displaystyle \lambda ={\frac {4L}{n}},}
2679:
2671:
2622:
2610:
2455:Next, consider the same string of length
2382:
2369:{\displaystyle L={\frac {n\lambda }{4}},}
2348:
2340:
2278:
2265:{\displaystyle \lambda ={\frac {4L}{n}},}
2244:
2236:
2152:
2139:
2131:
2073:
2045:{\displaystyle \lambda ={\frac {2L}{n}},}
2024:
2016:
1964:
1952:
1868:
1852:
1822:
1777:
1622:
1603:
1589:
1575:
1553:
1531:
1514:
1460:
1439:
1418:
1386:
1369:
1311:
1295:
1286:
1186:
1170:
1140:
1098:
1066:
1027:
971:
954:
915:
898:
885:
872:
845:
726:
709:
681:
675:
615:
598:
570:
564:
113:, in which standing waves occur inside a
94:coined the term "standing wave" (German:
7491:Gallis, Michael R. (February 15, 2008).
7301:. HyperPhysics. Georgia State University
6832:in semiconductors, atomic and molecular
3866:
1926:
16:Wave that remains in a constant position
8071:(7th ed.). John Wiley & Sons.
7354:
7319:
7033:
6730:, a standing wave is created, allowing
5190:term equals a constant with respect to
4901:term equals a constant with respect to
2215:+ 1 nodes including the fixed ends and
387:is a wave in which the distribution of
340:
302:
268:
146:
7953:"Lake Huron Storm Surge July 13, 1995"
3860:For the case where one end is closed,
3654:Next, consider a pipe that is open at
3591:or equivalently when the frequency is
1931:Standing waves in a string â the
324:Standing waves in a string â the
4371:) is the displacement of the surface,
1021:trigonometric sum-to-product identity
7:
7922:Encyclopedia of Lakes and Reservoirs
7121:"Surfbare WechselsprĂŒnge | Espazium"
5154:-dependence is sinusoidalârecalling
2476:, the string is free to move in the
2459:, but this time it is only fixed at
2424:This example demonstrates a type of
2008:
1132:
548:direction as a function of position
379:As an example of the second type, a
352:standing wave on a circular membrane
164:Kayakers surfing a standing wave in
7472:. MathWorld--A Wolfram Web Resource
7394:Halliday, Resnick & Walker 2005
7382:Halliday, Resnick & Walker 2005
7343:Halliday, Resnick & Walker 2005
7285:Halliday, Resnick & Walker 2005
7225:Halliday, Resnick & Walker 2005
7208:Halliday, Resnick & Walker 2005
4724:Dividing the Helmholtz equation by
4134:{\displaystyle f={\frac {nv}{2L}}.}
3850:{\displaystyle f={\frac {nv}{4L}}.}
3633:{\displaystyle f={\frac {nv}{2L}},}
2797:{\displaystyle f={\frac {nv}{4L}}.}
1361:multiples of a quarter wavelength,
525:-axis that is free to be stretched
8033:Johnson, Scott K. (30 June 2013).
7519:, pp. 153â154, PHI Learning, 2015
7183:Fundamentals of Physical Acoustics
5243:
5229:
4954:
4940:
4835:
4821:
4777:
4763:
4570:
4556:
4533:
4519:
4435:
4430:
4315:
4301:
4278:
4264:
4224:
4210:
3945:
3937:
3909:
3387:
3339:
3198:
3167:
3143:
2988:
2883:
2332:multiple of a quarter wavelength,
1506:multiples of a quarter wavelength
14:
8045:from the original on 30 July 2022
7852:from the original on 31 July 2016
7446:A better description of resonance
7042:Encyclopedia of Nonlinear Science
8515:
8514:
8227:
8131:
7808:Review of Scientific Instruments
7150: This article incorporates
7145:
7119:Dietsche, Daniela (2014-12-31).
6812:surface or a reflection from an
4082:{\displaystyle n=1,2,3,\ldots ,}
3798:{\displaystyle n=1,3,5,\ldots ,}
3581:{\displaystyle n=1,2,3,\ldots ,}
3372:{\displaystyle \Delta p(0,t)=0,}
359:
343:
317:
305:
287:
271:
172:
149:
8015:from the original on 2019-01-26
7443:Mould, Steve (April 13, 2017).
7163:General Services Administration
6408:
6035:
5996:
5879:
3690:the SturmâLiouville formulation
2811:only takes odd values. Because
2745:{\displaystyle n=1,3,5,\ldots }
2553:the SturmâLiouville formulation
2414:{\displaystyle n=2,4,6,\ldots }
2310:{\displaystyle n=2,4,6,\ldots }
2105:{\displaystyle n=1,2,3,\ldots }
1945:the SturmâLiouville formulation
1748:because the string is fixed at
219:and tidal currents such as the
7987:University of Minnesota Duluth
6875:free oscillations of the Earth
6244:
6226:
6185:If the initial conditions for
5812:{\displaystyle \sin {k_{x}x},}
5615:
5597:
5448:
5442:
5279:
5262:
5219:
5213:
5042:
5036:
4990:
4973:
4930:
4924:
4811:
4805:
4753:
4747:
4705:
4699:
4693:
4687:
4637:
4619:
4461:
4443:
4416:
4398:
3895:and longitudinal displacement
3471:
3462:
3405:
3393:
3357:
3345:
3289:
3280:
3223:
3211:
3192:
3180:
3161:
3149:
3013:
3001:
2908:
2896:
1905:
1896:
1839:
1827:
1794:
1782:
1223:
1214:
1157:
1145:
862:
850:
699:
687:
588:
576:
1:
7977:Korgen, Ben (February 2000).
7930:10.1007/978-1-4020-4410-6_226
7181:Blackstock, David T. (2000),
6995:Index of electronics articles
2119:. If waves travel with speed
23:Animation of a standing wave
7599:10.1016/j.icarus.2009.10.022
6197:,0) and its time derivative
438:, discontinuity, such as an
7185:, WileyâIEEE, p. 141,
6808:beam from a nearly perfect
6707:The hexagonal cloud feature
5353:for this wave is therefore
4714:{\displaystyle Z=X(x)Y(y).}
4664:and it can be solved using
3310:
2577:
2502:
2224:
1716:
1710:
1257:
230:of the water overcomes its
86:in 1831. Faraday observed
8953:
7744:10.1103/PhysRevLett.49.560
7698:10.1103/PhysRevLett.22.703
7618:Cambridge University Press
7538:Physics for the IB Diploma
7044:, p. 683, Routledge, 2006
6762:FabryâPĂ©rot interferometer
6689:
6644:
3688:, which is in the form of
2850:
2815:is an anti-node, it is an
2551:, which is in the form of
215:also form on fast flowing
8688:Music On A Long Thin Wire
8591:
8577:
8510:
8222:
7663:10.1103/RevModPhys.36.681
7643:Reviews of Modern Physics
7065:, p. 368, Springer, 2012
6768:in the cavity (such as a
4377:is the speed of the wave.
2591:the largest amplitude of
1809:{\displaystyle y(0,t)=0,}
427:from the other end by an
403:of two waves of the same
8584:HornbostelâSachs numbers
7406:Serway & Faughn 1992
7370:Serway & Faughn 1992
7249:Serway & Faughn 1992
7237:Serway & Faughn 1992
7006:Characteristic impedance
5962:, which is satisfied by
2500:In this case, Equation (
2451:reflecting at a boundary
663:The displacement in the
480:Mathematical description
8191:Architectural acoustics
8069:Fundamentals of Physics
7724:Physical Review Letters
7678:Physical Review Letters
5940:) must be zero at both
4666:separation of variables
2847:Standing wave in a pipe
2188:The standing wave with
368:standing wave on a disk
8665:Long-string instrument
8278:FletcherâMunson curves
8273:Equal-loudness contour
8183:Acoustical engineering
7955:. NOAA. Archived from
7889:10.1002/adma.201402079
7814:(9): 095112â095112â8.
7614:Introduction to Optics
7158:Federal Standard 1037C
7152:public domain material
6930:Index of wave articles
6714:
6620:
6440:
6367:
6209:,0) are chosen so the
6176:
6067:
5911:
5813:
5699:
5547:
5416:
5340:
5194:that we can define as
5141:
5006:
4905:that we can define as
4888:
4715:
4647:
4496:
4343:
4167:-direction and length
4135:
4083:
4035:
3993:phase shift along the
3961:
3880:
3851:
3799:
3751:
3634:
3582:
3534:
3487:
3373:
3299:
3079:
2974:
2826:and the first peak at
2798:
2746:
2701:
2654:
2452:
2415:
2370:
2311:
2266:
2228:) can be rewritten as
2179:
2106:
2046:
1996:
1940:
1918:
1810:
1647:
1498:. At locations on the
1485:
1337:
1233:
1124:
1010:
765:
654:
448:attenuation distortion
48:
8414:Hermann von Helmholtz
8312:Fundamental frequency
8216:Sympathetic resonance
7517:Microwave Engineering
6863:dimensional metrology
6792:Interference between
6705:
6621:
6441:
6368:
6177:
6068:
5912:
5814:
5700:
5548:
5417:
5341:
5142:
5007:
4889:
4716:
4648:
4497:
4344:
4136:
4084:
4036:
3962:
3870:
3852:
3800:
3752:
3635:
3583:
3535:
3488:
3374:
3300:
3080:
2975:
2799:
2747:
2702:
2655:
2447:analysis of a damped
2443:
2416:
2371:
2312:
2267:
2197:fundamental frequency
2180:
2107:
2047:
1997:
1935:mode and the first 5
1930:
1919:
1811:
1648:
1486:
1338:
1234:
1125:
1011:
766:
655:
455:partial standing wave
328:mode and the first 5
134:propagation of energy
39:and right traveling
22:
8140:at Wikimedia Commons
8105:Streets, J. (2010).
8009:www.soest.hawaii.edu
7297:Nave, C. R. (2016).
6853:to track changes in
6575:
6378:
6220:
6083:
5969:
5838:
5782:
5591:
5436:
5360:
5201:
5030:
4912:
4735:
4675:
4512:
4392:
4203:
4099:
4046:
4004:
3906:
3815:
3762:
3720:
3598:
3545:
3503:
3384:
3336:
3140:
2985:
2880:
2841:resonant frequencies
2762:
2712:
2670:
2609:
2575:Reviewing Equation (
2430:resonant frequencies
2381:
2339:
2328:is a node, it is an
2277:
2235:
2130:
2072:
2015:
1951:
1821:
1776:
1513:
1368:
1353:At locations on the
1285:
1139:
1026:
844:
674:
563:
8434:Werner Meyer-Eppler
8344:Missing fundamental
7820:2005RScI...76i5112B
7779:(51): 18200â18201.
7736:1982PhRvL..49..560G
7690:1969PhRvL..22..703B
7655:1964RvMP...36..681B
7591:2010Icar..206..755B
7468:Weisstein, Eric W.
6961:Resonant room modes
6798:X-ray standing wave
6654:Standing wave ratio
6647:Standing wave ratio
5406:
5388:
5351:dispersion relation
5305:
4439:
1722:boundary conditions
487:boundary conditions
459:standing wave ratio
211:Standing waves and
124:For waves of equal
8917:1831 introductions
8672:Melde's experiment
8317:Frequency spectrum
7877:Advanced Materials
7563:2011-10-21 at the
7251:, p. 475-476.
7040:Alwyn Scott (ed),
6956:Metachronal rhythm
6818:X-ray fluorescence
6796:beams can form an
6750:optical waveguides
6715:
6692:Acoustic resonance
6686:Acoustic resonance
6637:as an eigenvalue.
6616:
6436:
6363:
6172:
6063:
5907:
5809:
5767:for all values of
5695:
5543:
5412:
5392:
5374:
5336:
5291:
5137:
5002:
4884:
4711:
4662:Helmholtz equation
4643:
4492:
4422:
4339:
4131:
4079:
4031:
3957:
3881:
3847:
3795:
3747:
3630:
3578:
3530:
3483:
3369:
3295:
3075:
2970:
2794:
2742:
2697:
2650:
2453:
2411:
2366:
2307:
2262:
2195:oscillates at the
2175:
2102:
2042:
1992:
1941:
1914:
1806:
1643:
1481:
1333:
1229:
1120:
1006:
761:
650:
366:A higher harmonic
236:supercritical flow
119:resonant frequency
59:, also known as a
49:
8904:
8903:
8660:Longitudinal wave
8528:
8527:
8490:Musical acoustics
8322:harmonic spectrum
8136:Media related to
7939:978-1-4020-4410-6
7883:(34): 5936â5941.
7871:Chen, Pu (2014).
7846:www.insitutec.com
7828:10.1063/1.2052027
7785:10.1021/ja906816y
7627:978-1-108-42826-2
6946:Longitudinal mode
6936:Amphidromic point
6826:crystal structure
6728:natural frequency
6703:
6611:
6604:
6332:
6293:
6257:
6167:
6155:
6120:
6030:
5874:
5407:
5331:
5257:
5223:
4968:
4934:
4876:
4849:
4815:
4791:
4757:
4614:
4584:
4547:
4383:Fourier transform
4329:
4292:
4238:
4182:-direction, with
4126:
4026:
3952:
3842:
3742:
3681:can be stated as
3625:
3525:
3450:
3421:
3268:
3239:
3208:
3177:
3106:angular frequency
3056:
3026:
2998:
2951:
2921:
2893:
2789:
2692:
2638:
2361:
2257:
2170:
2147:
2066:
2065:
2037:
1980:
1884:
1855:
1769:at the two ends,
1635:
1616:
1597:
1583:
1566:
1544:
1473:
1447:
1426:
1399:
1327:
1298:
1253:
1252:
1202:
1173:
1114:
1082:
987:
957:
931:
901:
888:
875:
796:angular frequency
742:
712:
684:
631:
601:
573:
511:cavity resonators
399:is formed by the
385:transmission line
377:
376:
183:
159:
8944:
8927:1860s neologisms
8724:String vibration
8555:
8548:
8541:
8532:
8518:
8517:
8419:Carleen Hutchins
8351:Combination tone
8238:
8231:
8211:String vibration
8168:
8161:
8154:
8145:
8135:
8121:
8119:
8117:
8111:
8101:
8082:
8054:
8053:
8051:
8050:
8030:
8024:
8023:
8021:
8020:
8001:
7995:
7994:
7989:. Archived from
7983:seagrant.umn.edu
7974:
7968:
7967:
7965:
7964:
7949:
7943:
7942:
7917:
7911:
7910:
7900:
7868:
7862:
7861:
7859:
7857:
7838:
7832:
7831:
7803:
7797:
7796:
7773:J. Am. Chem. Soc
7754:
7748:
7747:
7721:
7708:
7702:
7701:
7673:
7667:
7666:
7638:
7632:
7631:
7609:
7603:
7602:
7574:
7568:
7555:
7549:
7534:
7528:
7513:
7507:
7506:
7504:
7502:
7488:
7482:
7481:
7479:
7477:
7465:
7459:
7458:
7456:
7454:
7440:
7434:
7433:
7431:
7429:
7415:
7409:
7403:
7397:
7391:
7385:
7379:
7373:
7367:
7358:
7352:
7346:
7340:
7323:
7317:
7311:
7310:
7308:
7306:
7299:"Standing Waves"
7294:
7288:
7282:
7276:
7275:
7273:
7271:
7263:String Resonance
7258:
7252:
7246:
7240:
7234:
7228:
7222:
7211:
7205:
7196:
7195:
7178:
7172:
7171:
7170:
7165:. Archived from
7149:
7148:
7142:
7136:
7135:
7133:
7132:
7116:
7110:
7107:
7101:
7098:
7092:
7089:
7083:
7080:
7074:
7059:
7053:
7038:
7001:Cavity resonator
6844:Mechanical waves
6754:optical cavities
6720:vibrating string
6704:
6625:
6623:
6622:
6617:
6612:
6607:
6605:
6603:
6602:
6593:
6585:
6568:all resonate at
6567:
6560:
6553:
6546:
6539:
6532:
6525:
6478:
6471:
6445:
6443:
6442:
6437:
6372:
6370:
6369:
6364:
6359:
6355:
6337:
6333:
6331:
6330:
6321:
6310:
6298:
6294:
6292:
6291:
6282:
6271:
6259:
6258:
6255:
6181:
6179:
6178:
6173:
6168:
6166:
6165:
6160:
6156:
6154:
6153:
6141:
6131:
6130:
6125:
6121:
6119:
6118:
6106:
6099:
6072:
6070:
6069:
6064:
6031:
6029:
6028:
6019:
6011:
6006:
6005:
5992:
5988:
5987:
5961:
5946:
5916:
5914:
5913:
5908:
5875:
5873:
5872:
5863:
5855:
5850:
5849:
5818:
5816:
5815:
5810:
5805:
5801:
5800:
5766:
5751:
5704:
5702:
5701:
5696:
5691:
5690:
5672:
5671:
5667:
5666:
5646:
5645:
5641:
5640:
5552:
5550:
5549:
5544:
5539:
5538:
5534:
5533:
5513:
5512:
5511:
5510:
5493:
5492:
5488:
5487:
5470:
5469:
5468:
5467:
5421:
5419:
5418:
5413:
5408:
5405:
5400:
5387:
5382:
5373:
5345:
5343:
5342:
5337:
5332:
5330:
5329:
5320:
5319:
5310:
5304:
5299:
5287:
5286:
5277:
5276:
5258:
5256:
5255:
5254:
5241:
5237:
5236:
5226:
5224:
5222:
5205:
5158:âwith constants
5146:
5144:
5143:
5138:
5133:
5132:
5128:
5127:
5107:
5106:
5105:
5104:
5087:
5086:
5082:
5081:
5064:
5063:
5062:
5061:
5011:
5009:
5008:
5003:
4998:
4997:
4988:
4987:
4969:
4967:
4966:
4965:
4952:
4948:
4947:
4937:
4935:
4933:
4916:
4893:
4891:
4890:
4885:
4877:
4875:
4874:
4865:
4864:
4855:
4850:
4848:
4847:
4846:
4833:
4829:
4828:
4818:
4816:
4814:
4797:
4792:
4790:
4789:
4788:
4775:
4771:
4770:
4760:
4758:
4756:
4739:
4720:
4718:
4717:
4712:
4652:
4650:
4649:
4644:
4615:
4613:
4612:
4603:
4602:
4593:
4585:
4583:
4582:
4581:
4568:
4564:
4563:
4553:
4548:
4546:
4545:
4544:
4531:
4527:
4526:
4516:
4501:
4499:
4498:
4493:
4482:
4481:
4438:
4433:
4348:
4346:
4345:
4340:
4335:
4331:
4330:
4328:
4327:
4326:
4313:
4309:
4308:
4298:
4293:
4291:
4290:
4289:
4276:
4272:
4271:
4261:
4254:
4253:
4239:
4237:
4236:
4235:
4222:
4218:
4217:
4207:
4188:
4140:
4138:
4137:
4132:
4127:
4125:
4117:
4109:
4088:
4086:
4085:
4080:
4040:
4038:
4037:
4032:
4027:
4022:
4014:
3992:
3991:
3987:
3966:
3964:
3963:
3958:
3953:
3951:
3943:
3935:
3933:
3932:
3877:
3856:
3854:
3853:
3848:
3843:
3841:
3833:
3825:
3804:
3802:
3801:
3796:
3756:
3754:
3753:
3748:
3743:
3738:
3730:
3708:
3698:
3687:
3680:
3670:
3660:
3639:
3637:
3636:
3631:
3626:
3624:
3616:
3608:
3587:
3585:
3584:
3579:
3539:
3537:
3536:
3531:
3526:
3521:
3513:
3492:
3490:
3489:
3484:
3455:
3451:
3446:
3435:
3423:
3422:
3419:
3378:
3376:
3375:
3370:
3304:
3302:
3301:
3296:
3273:
3269:
3264:
3253:
3241:
3240:
3237:
3210:
3209:
3206:
3179:
3178:
3175:
3108:or equivalently
3084:
3082:
3081:
3076:
3071:
3067:
3057:
3052:
3041:
3028:
3027:
3024:
3000:
2999:
2996:
2979:
2977:
2976:
2971:
2966:
2962:
2952:
2947:
2936:
2923:
2922:
2919:
2895:
2894:
2891:
2835:
2825:
2807:In this example
2803:
2801:
2800:
2795:
2790:
2788:
2780:
2772:
2751:
2749:
2748:
2743:
2706:
2704:
2703:
2698:
2693:
2688:
2680:
2659:
2657:
2656:
2651:
2643:
2639:
2634:
2623:
2601:
2590:
2571:
2561:
2550:
2540:
2529:
2519:
2512:
2496:
2489:
2475:
2465:
2420:
2418:
2417:
2412:
2375:
2373:
2372:
2367:
2362:
2357:
2349:
2316:
2314:
2313:
2308:
2271:
2269:
2268:
2263:
2258:
2253:
2245:
2194:
2184:
2182:
2181:
2176:
2171:
2169:
2161:
2153:
2148:
2140:
2111:
2109:
2108:
2103:
2060:
2051:
2049:
2048:
2043:
2038:
2033:
2025:
2009:
2001:
1999:
1998:
1993:
1985:
1981:
1976:
1965:
1923:
1921:
1920:
1915:
1889:
1885:
1880:
1869:
1857:
1856:
1853:
1815:
1813:
1812:
1807:
1764:
1757:
1747:
1737:
1730:
1720:) is subject to
1695:
1688:
1678:
1652:
1650:
1649:
1644:
1636:
1631:
1623:
1617:
1612:
1604:
1598:
1590:
1584:
1576:
1567:
1562:
1554:
1545:
1540:
1532:
1490:
1488:
1487:
1482:
1474:
1469:
1461:
1448:
1440:
1427:
1419:
1400:
1395:
1387:
1342:
1340:
1339:
1334:
1332:
1328:
1323:
1312:
1300:
1299:
1296:
1247:
1238:
1236:
1235:
1230:
1207:
1203:
1198:
1187:
1175:
1174:
1171:
1133:
1129:
1127:
1126:
1121:
1119:
1115:
1110:
1099:
1087:
1083:
1078:
1067:
1015:
1013:
1012:
1007:
1002:
998:
988:
983:
972:
959:
958:
955:
946:
942:
932:
927:
916:
903:
902:
899:
890:
889:
886:
877:
876:
873:
798:or equivalently
770:
768:
767:
762:
757:
753:
743:
738:
727:
714:
713:
710:
686:
685:
682:
659:
657:
656:
651:
646:
642:
632:
627:
616:
603:
602:
599:
575:
574:
571:
363:
347:
321:
309:
291:
275:
254:
185:
184:
166:Great Falls Park
161:
160:
44:
36:
28:
8952:
8951:
8947:
8946:
8945:
8943:
8942:
8941:
8932:Michael Faraday
8922:1831 in science
8907:
8906:
8905:
8900:
8809:Japanese fiddle
8747:
8738:
8729:Transverse wave
8677:Mersenne's laws
8655:String harmonic
8587:
8573:
8559:
8529:
8524:
8506:
8458:
8449:D. Van Holliday
8387:
8356:Mersenne's laws
8290:Audio frequency
8284:
8248:Psychoacoustics
8242:
8241:
8234:
8220:
8177:
8172:
8128:
8115:
8113:
8109:
8104:
8098:
8088:College Physics
8085:
8079:
8066:
8063:
8058:
8057:
8048:
8046:
8032:
8031:
8027:
8018:
8016:
8003:
8002:
7998:
7976:
7975:
7971:
7962:
7960:
7951:
7950:
7946:
7940:
7919:
7918:
7914:
7870:
7869:
7865:
7855:
7853:
7840:
7839:
7835:
7805:
7804:
7800:
7770:
7766:
7762:
7756:
7755:
7751:
7719:
7710:
7709:
7705:
7684:(14): 703â705.
7675:
7674:
7670:
7640:
7639:
7635:
7628:
7611:
7610:
7606:
7576:
7575:
7571:
7565:Wayback Machine
7556:
7552:
7535:
7531:
7514:
7510:
7500:
7498:
7490:
7489:
7485:
7475:
7473:
7467:
7466:
7462:
7452:
7450:
7442:
7441:
7437:
7427:
7425:
7417:
7416:
7412:
7404:
7400:
7392:
7388:
7380:
7376:
7368:
7361:
7353:
7349:
7341:
7326:
7318:
7314:
7304:
7302:
7296:
7295:
7291:
7283:
7279:
7269:
7267:
7260:
7259:
7255:
7247:
7243:
7235:
7231:
7223:
7214:
7206:
7199:
7193:
7180:
7179:
7175:
7155:
7146:
7144:
7143:
7139:
7130:
7128:
7125:www.espazium.ch
7118:
7117:
7113:
7108:
7104:
7099:
7095:
7090:
7086:
7081:
7077:
7060:
7056:
7039:
7035:
7030:
7025:
6990:
6985:
6925:
6920:
6895:
6883:
6871:
6846:
6790:
6746:
6696:
6694:
6688:
6676:
6649:
6643:
6629:Recalling that
6594:
6586:
6573:
6572:
6562:
6555:
6548:
6541:
6534:
6527:
6524:
6515:
6507:
6473:
6466:
6376:
6375:
6348:
6344:
6322:
6311:
6305:
6283:
6272:
6266:
6250:
6218:
6217:
6145:
6136:
6135:
6110:
6101:
6100:
6081:
6080:
6020:
6012:
5997:
5979:
5967:
5966:
5960:
5948:
5941:
5864:
5856:
5841:
5836:
5835:
5830:
5792:
5780:
5779:
5765:
5753:
5746:
5673:
5658:
5647:
5632:
5621:
5589:
5588:
5525:
5514:
5502:
5497:
5479:
5471:
5459:
5454:
5434:
5433:
5358:
5357:
5321:
5311:
5278:
5268:
5246:
5242:
5228:
5227:
5209:
5199:
5198:
5185:
5184:
5171:
5170:
5156:Euler's formula
5119:
5108:
5096:
5091:
5073:
5065:
5053:
5048:
5028:
5027:
4989:
4979:
4957:
4953:
4939:
4938:
4920:
4910:
4909:
4866:
4856:
4838:
4834:
4820:
4819:
4801:
4780:
4776:
4762:
4761:
4743:
4733:
4732:
4673:
4672:
4604:
4594:
4573:
4569:
4555:
4554:
4536:
4532:
4518:
4517:
4510:
4509:
4464:
4390:
4389:
4318:
4314:
4300:
4299:
4281:
4277:
4263:
4262:
4259:
4255:
4245:
4227:
4223:
4209:
4208:
4201:
4200:
4183:
4172:
4161:
4154:
4118:
4110:
4097:
4096:
4044:
4043:
4015:
4002:
4001:
3989:
3985:
3984:
3944:
3936:
3924:
3904:
3903:
3899:are related as
3872:
3834:
3826:
3813:
3812:
3760:
3759:
3731:
3718:
3717:
3700:
3693:
3682:
3672:
3662:
3655:
3617:
3609:
3596:
3595:
3543:
3542:
3514:
3501:
3500:
3436:
3430:
3414:
3382:
3381:
3334:
3333:
3254:
3248:
3232:
3201:
3170:
3138:
3137:
3097:
3042:
3039:
3035:
3019:
2991:
2983:
2982:
2937:
2934:
2930:
2914:
2886:
2878:
2877:
2855:
2849:
2827:
2820:
2781:
2773:
2760:
2759:
2710:
2709:
2681:
2668:
2667:
2624:
2618:
2607:
2606:
2596:
2582:
2563:
2556:
2542:
2535:
2521:
2514:
2507:
2491:
2481:
2467:
2460:
2438:
2379:
2378:
2350:
2337:
2336:
2275:
2274:
2246:
2233:
2232:
2189:
2162:
2154:
2128:
2127:
2070:
2069:
2058:
2026:
2013:
2012:
1966:
1960:
1949:
1948:
1870:
1864:
1848:
1819:
1818:
1774:
1773:
1759:
1749:
1739:
1732:
1725:
1690:
1680:
1673:
1670:
1624:
1605:
1555:
1533:
1511:
1510:
1502:-axis that are
1462:
1388:
1366:
1365:
1357:-axis that are
1313:
1307:
1291:
1283:
1282:
1245:
1188:
1182:
1166:
1137:
1136:
1100:
1094:
1068:
1062:
1024:
1023:
973:
970:
966:
950:
917:
914:
910:
894:
881:
868:
842:
841:
836:
829:
783:
728:
725:
721:
705:
677:
672:
671:
617:
614:
610:
594:
566:
561:
560:
519:
503:Chladni figures
482:
371:
364:
355:
348:
333:
322:
313:
310:
295:
292:
283:
276:
263:
252:
213:hydraulic jumps
198:
197:
196:
193:
186:
173:
168:
162:
150:
142:
84:Michael Faraday
61:stationary wave
42:
34:
26:
17:
12:
11:
5:
8950:
8948:
8940:
8939:
8937:Wave mechanics
8934:
8929:
8924:
8919:
8909:
8908:
8902:
8901:
8899:
8898:
8893:
8888:
8883:
8878:
8873:
8868:
8863:
8858:
8853:
8848:
8847:
8846:
8841:
8836:
8831:
8826:
8821:
8811:
8806:
8801:
8796:
8791:
8786:
8781:
8776:
8775:
8774:
8767:Bladder fiddle
8764:
8759:
8753:
8751:
8740:
8739:
8737:
8736:
8731:
8726:
8721:
8716:
8711:
8706:
8701:
8696:
8691:
8684:
8679:
8674:
8669:
8668:
8667:
8657:
8640:
8635:
8630:
8625:
8620:
8615:
8610:
8609:
8608:
8598:
8592:
8589:
8588:
8578:
8575:
8574:
8560:
8558:
8557:
8550:
8543:
8535:
8526:
8525:
8523:
8522:
8511:
8508:
8507:
8505:
8504:
8503:
8502:
8497:
8487:
8482:
8477:
8472:
8466:
8464:
8463:Related topics
8460:
8459:
8457:
8456:
8451:
8446:
8444:Joseph Sauveur
8441:
8436:
8431:
8429:Marin Mersenne
8426:
8421:
8416:
8411:
8406:
8401:
8395:
8393:
8389:
8388:
8386:
8385:
8380:
8379:
8378:
8368:
8363:
8358:
8353:
8348:
8347:
8346:
8341:
8336:
8326:
8325:
8324:
8314:
8309:
8304:
8298:
8296:
8286:
8285:
8283:
8282:
8281:
8280:
8270:
8269:
8268:
8263:
8252:
8250:
8244:
8243:
8240:
8239:
8232:
8224:
8223:
8221:
8219:
8218:
8213:
8208:
8203:
8198:
8193:
8187:
8185:
8179:
8178:
8173:
8171:
8170:
8163:
8156:
8148:
8142:
8141:
8138:Standing waves
8127:
8126:External links
8124:
8123:
8122:
8102:
8096:
8083:
8077:
8062:
8059:
8056:
8055:
8025:
7996:
7993:on 2007-12-27.
7969:
7944:
7938:
7912:
7863:
7833:
7798:
7768:
7764:
7758:
7749:
7730:(8): 560â563.
7703:
7668:
7649:(3): 681â717.
7633:
7626:
7616:(3 ed.).
7604:
7585:(2): 755â763.
7569:
7550:
7529:
7508:
7483:
7460:
7435:
7410:
7408:, p. 477.
7398:
7396:, p. 451.
7386:
7384:, p. 458.
7374:
7372:, p. 478.
7359:
7347:
7345:, p. 457.
7324:
7312:
7289:
7287:, p. 450.
7277:
7253:
7241:
7239:, p. 472.
7229:
7227:, p. 434.
7212:
7210:, p. 432.
7197:
7191:
7173:
7169:on 2022-01-22.
7137:
7111:
7102:
7093:
7084:
7075:
7054:
7032:
7031:
7029:
7026:
7024:
7023:
7018:
7013:
7008:
7003:
6998:
6991:
6989:
6986:
6984:
6983:
6978:
6973:
6968:
6963:
6958:
6953:
6948:
6943:
6938:
6933:
6926:
6924:
6921:
6919:
6916:
6894:
6891:
6882:
6879:
6870:
6867:
6845:
6842:
6810:single crystal
6789:
6786:
6772:) emits light
6745:
6742:
6737:end correction
6690:Main article:
6687:
6684:
6675:
6672:
6645:Main article:
6642:
6639:
6627:
6626:
6615:
6610:
6601:
6597:
6592:
6589:
6583:
6580:
6520:
6511:
6447:
6446:
6435:
6432:
6429:
6426:
6423:
6420:
6417:
6414:
6411:
6407:
6404:
6401:
6398:
6395:
6392:
6389:
6386:
6383:
6373:
6362:
6358:
6354:
6351:
6347:
6343:
6340:
6336:
6329:
6325:
6320:
6317:
6314:
6308:
6304:
6301:
6297:
6290:
6286:
6281:
6278:
6275:
6269:
6265:
6262:
6253:
6249:
6246:
6243:
6240:
6237:
6234:
6231:
6228:
6225:
6183:
6182:
6171:
6164:
6159:
6152:
6148:
6144:
6139:
6134:
6129:
6124:
6117:
6113:
6109:
6104:
6097:
6094:
6091:
6088:
6074:
6073:
6062:
6059:
6056:
6053:
6050:
6047:
6044:
6041:
6038:
6034:
6027:
6023:
6018:
6015:
6009:
6004:
6000:
5995:
5991:
5986:
5982:
5977:
5974:
5956:
5924:dependence of
5920:Likewise, the
5918:
5917:
5906:
5903:
5900:
5897:
5894:
5891:
5888:
5885:
5882:
5878:
5871:
5867:
5862:
5859:
5853:
5848:
5844:
5831:restricted to
5826:
5820:
5819:
5808:
5804:
5799:
5795:
5790:
5787:
5761:
5706:
5705:
5694:
5689:
5686:
5683:
5680:
5676:
5670:
5665:
5661:
5657:
5654:
5650:
5644:
5639:
5635:
5631:
5628:
5624:
5620:
5617:
5614:
5611:
5608:
5605:
5602:
5599:
5596:
5554:
5553:
5542:
5537:
5532:
5528:
5524:
5521:
5517:
5509:
5505:
5500:
5496:
5491:
5486:
5482:
5478:
5474:
5466:
5462:
5457:
5453:
5450:
5447:
5444:
5441:
5423:
5422:
5411:
5404:
5399:
5395:
5391:
5386:
5381:
5377:
5371:
5368:
5365:
5347:
5346:
5335:
5328:
5324:
5318:
5314:
5308:
5303:
5298:
5294:
5290:
5285:
5281:
5275:
5271:
5267:
5264:
5261:
5253:
5249:
5245:
5240:
5235:
5231:
5221:
5218:
5215:
5212:
5208:
5180:
5176:
5166:
5162:
5148:
5147:
5136:
5131:
5126:
5122:
5118:
5115:
5111:
5103:
5099:
5094:
5090:
5085:
5080:
5076:
5072:
5068:
5060:
5056:
5051:
5047:
5044:
5041:
5038:
5035:
5013:
5012:
5001:
4996:
4992:
4986:
4982:
4978:
4975:
4972:
4964:
4960:
4956:
4951:
4946:
4942:
4932:
4929:
4926:
4923:
4919:
4895:
4894:
4883:
4880:
4873:
4869:
4863:
4859:
4853:
4845:
4841:
4837:
4832:
4827:
4823:
4813:
4810:
4807:
4804:
4800:
4795:
4787:
4783:
4779:
4774:
4769:
4765:
4755:
4752:
4749:
4746:
4742:
4722:
4721:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4654:
4653:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4611:
4607:
4601:
4597:
4591:
4588:
4580:
4576:
4572:
4567:
4562:
4558:
4551:
4543:
4539:
4535:
4530:
4525:
4521:
4503:
4502:
4491:
4488:
4485:
4480:
4477:
4474:
4471:
4467:
4463:
4460:
4457:
4454:
4451:
4448:
4445:
4442:
4437:
4432:
4429:
4425:
4421:
4418:
4415:
4412:
4409:
4406:
4403:
4400:
4397:
4379:
4378:
4372:
4350:
4349:
4338:
4334:
4325:
4321:
4317:
4312:
4307:
4303:
4296:
4288:
4284:
4280:
4275:
4270:
4266:
4258:
4252:
4248:
4243:
4234:
4230:
4226:
4221:
4216:
4212:
4170:
4159:
4153:
4150:
4142:
4141:
4130:
4124:
4121:
4116:
4113:
4107:
4104:
4090:
4089:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4041:
4030:
4025:
4021:
4018:
4012:
4009:
3968:
3967:
3956:
3950:
3947:
3942:
3939:
3931:
3927:
3923:
3920:
3917:
3914:
3911:
3858:
3857:
3846:
3840:
3837:
3832:
3829:
3823:
3820:
3806:
3805:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3757:
3746:
3741:
3737:
3734:
3728:
3725:
3649:speed of sound
3641:
3640:
3629:
3623:
3620:
3615:
3612:
3606:
3603:
3589:
3588:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3540:
3529:
3524:
3520:
3517:
3511:
3508:
3494:
3493:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3454:
3449:
3445:
3442:
3439:
3433:
3429:
3426:
3417:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3379:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3306:
3305:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3272:
3267:
3263:
3260:
3257:
3251:
3247:
3244:
3235:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3204:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3173:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3131:
3130:
3120:
3099:
3095:
3086:
3085:
3074:
3070:
3066:
3063:
3060:
3055:
3051:
3048:
3045:
3038:
3034:
3031:
3022:
3018:
3015:
3012:
3009:
3006:
3003:
2994:
2990:
2980:
2969:
2965:
2961:
2958:
2955:
2950:
2946:
2943:
2940:
2933:
2929:
2926:
2917:
2913:
2910:
2907:
2904:
2901:
2898:
2889:
2885:
2848:
2845:
2805:
2804:
2793:
2787:
2784:
2779:
2776:
2770:
2767:
2753:
2752:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2707:
2696:
2691:
2687:
2684:
2678:
2675:
2661:
2660:
2649:
2646:
2642:
2637:
2633:
2630:
2627:
2621:
2617:
2614:
2520:. However, at
2449:traveling wave
2437:
2434:
2422:
2421:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2376:
2365:
2360:
2356:
2353:
2347:
2344:
2318:
2317:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2272:
2261:
2256:
2252:
2249:
2243:
2240:
2186:
2185:
2174:
2168:
2165:
2160:
2157:
2151:
2146:
2143:
2138:
2135:
2113:
2112:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2064:
2063:
2054:
2052:
2041:
2036:
2032:
2029:
2023:
2020:
1991:
1988:
1984:
1979:
1975:
1972:
1969:
1963:
1959:
1956:
1925:
1924:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1888:
1883:
1879:
1876:
1873:
1867:
1863:
1860:
1851:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1816:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1669:
1666:
1654:
1653:
1642:
1639:
1634:
1630:
1627:
1620:
1615:
1611:
1608:
1601:
1596:
1593:
1587:
1582:
1579:
1574:
1570:
1565:
1561:
1558:
1552:
1548:
1543:
1539:
1536:
1530:
1527:
1524:
1521:
1518:
1492:
1491:
1480:
1477:
1472:
1468:
1465:
1458:
1455:
1451:
1446:
1443:
1437:
1434:
1430:
1425:
1422:
1417:
1413:
1410:
1407:
1403:
1398:
1394:
1391:
1385:
1382:
1379:
1376:
1373:
1331:
1326:
1322:
1319:
1316:
1310:
1306:
1303:
1294:
1290:
1281:-direction as
1251:
1250:
1241:
1239:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1206:
1201:
1197:
1194:
1191:
1185:
1181:
1178:
1169:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1118:
1113:
1109:
1106:
1103:
1097:
1093:
1090:
1086:
1081:
1077:
1074:
1071:
1065:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1017:
1016:
1005:
1001:
997:
994:
991:
986:
982:
979:
976:
969:
965:
962:
953:
949:
945:
941:
938:
935:
930:
926:
923:
920:
913:
909:
906:
897:
893:
884:
880:
871:
867:
864:
861:
858:
855:
852:
849:
834:
827:
821:
820:
810:
789:
781:
772:
771:
760:
756:
752:
749:
746:
741:
737:
734:
731:
724:
720:
717:
708:
704:
701:
698:
695:
692:
689:
680:
661:
660:
649:
645:
641:
638:
635:
630:
626:
623:
620:
613:
609:
606:
597:
593:
590:
587:
584:
581:
578:
569:
518:
515:
509:and microwave
481:
478:
397:field strength
375:
374:
373:
372:
365:
358:
356:
349:
342:
337:
336:
335:
334:
323:
316:
314:
311:
304:
299:
298:
297:
296:
293:
286:
284:
277:
270:
265:
264:
262:
261:
260:Standing waves
257:
251:
250:Opposing waves
248:
195:
194:
187:
171:
169:
163:
148:
145:
144:
143:
141:
138:
96:stehende Welle
15:
13:
10:
9:
6:
4:
3:
2:
8949:
8938:
8935:
8933:
8930:
8928:
8925:
8923:
8920:
8918:
8915:
8914:
8912:
8897:
8894:
8892:
8889:
8887:
8884:
8882:
8879:
8877:
8876:Tromba marina
8874:
8872:
8869:
8867:
8864:
8862:
8859:
8857:
8854:
8852:
8849:
8845:
8842:
8840:
8837:
8835:
8832:
8830:
8827:
8825:
8822:
8820:
8817:
8816:
8815:
8812:
8810:
8807:
8805:
8802:
8800:
8797:
8795:
8792:
8790:
8787:
8785:
8782:
8780:
8777:
8773:
8770:
8769:
8768:
8765:
8763:
8760:
8758:
8755:
8754:
8752:
8750:
8745:
8741:
8735:
8732:
8730:
8727:
8725:
8722:
8720:
8719:Standing wave
8717:
8715:
8712:
8710:
8707:
8705:
8702:
8700:
8697:
8695:
8692:
8690:
8689:
8685:
8683:
8680:
8678:
8675:
8673:
8670:
8666:
8663:
8662:
8661:
8658:
8656:
8652:
8648:
8644:
8641:
8639:
8636:
8634:
8631:
8629:
8626:
8624:
8621:
8619:
8616:
8614:
8611:
8607:
8604:
8603:
8602:
8599:
8597:
8594:
8593:
8590:
8585:
8581:
8576:
8572:
8568:
8564:
8556:
8551:
8549:
8544:
8542:
8537:
8536:
8533:
8521:
8513:
8512:
8509:
8501:
8498:
8496:
8493:
8492:
8491:
8488:
8486:
8483:
8481:
8478:
8476:
8473:
8471:
8468:
8467:
8465:
8461:
8455:
8452:
8450:
8447:
8445:
8442:
8440:
8439:Lord Rayleigh
8437:
8435:
8432:
8430:
8427:
8425:
8422:
8420:
8417:
8415:
8412:
8410:
8409:Ernst Chladni
8407:
8405:
8402:
8400:
8397:
8396:
8394:
8390:
8384:
8381:
8377:
8374:
8373:
8372:
8371:Standing wave
8369:
8367:
8364:
8362:
8359:
8357:
8354:
8352:
8349:
8345:
8342:
8340:
8339:Inharmonicity
8337:
8335:
8332:
8331:
8330:
8327:
8323:
8320:
8319:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8300:
8299:
8297:
8295:
8291:
8287:
8279:
8276:
8275:
8274:
8271:
8267:
8264:
8262:
8259:
8258:
8257:
8254:
8253:
8251:
8249:
8245:
8237:
8233:
8230:
8226:
8225:
8217:
8214:
8212:
8209:
8207:
8206:Soundproofing
8204:
8202:
8201:Reverberation
8199:
8197:
8194:
8192:
8189:
8188:
8186:
8184:
8180:
8176:
8169:
8164:
8162:
8157:
8155:
8150:
8149:
8146:
8139:
8134:
8130:
8129:
8125:
8108:
8103:
8099:
8097:0-03-076377-0
8093:
8089:
8084:
8080:
8078:0-471-42959-7
8074:
8070:
8065:
8064:
8060:
8044:
8040:
8036:
8029:
8026:
8014:
8010:
8006:
8000:
7997:
7992:
7988:
7984:
7980:
7973:
7970:
7959:on 2008-09-16
7958:
7954:
7948:
7945:
7941:
7935:
7931:
7927:
7923:
7916:
7913:
7908:
7904:
7899:
7894:
7890:
7886:
7882:
7878:
7874:
7867:
7864:
7851:
7847:
7843:
7837:
7834:
7829:
7825:
7821:
7817:
7813:
7809:
7802:
7799:
7794:
7790:
7786:
7782:
7778:
7774:
7761:
7753:
7750:
7745:
7741:
7737:
7733:
7729:
7725:
7718:
7714:
7713:Bedzyk, M. J.
7707:
7704:
7699:
7695:
7691:
7687:
7683:
7679:
7672:
7669:
7664:
7660:
7656:
7652:
7648:
7644:
7637:
7634:
7629:
7623:
7619:
7615:
7608:
7605:
7600:
7596:
7592:
7588:
7584:
7580:
7573:
7570:
7566:
7562:
7559:
7554:
7551:
7547:
7543:
7539:
7533:
7530:
7526:
7522:
7518:
7512:
7509:
7496:
7495:
7487:
7484:
7471:
7464:
7461:
7448:
7447:
7439:
7436:
7423:
7422:
7414:
7411:
7407:
7402:
7399:
7395:
7390:
7387:
7383:
7378:
7375:
7371:
7366:
7364:
7360:
7357:, p. 15.
7356:
7351:
7348:
7344:
7339:
7337:
7335:
7333:
7331:
7329:
7325:
7321:
7316:
7313:
7300:
7293:
7290:
7286:
7281:
7278:
7265:
7264:
7257:
7254:
7250:
7245:
7242:
7238:
7233:
7230:
7226:
7221:
7219:
7217:
7213:
7209:
7204:
7202:
7198:
7194:
7192:0-471-31979-1
7188:
7184:
7177:
7174:
7168:
7164:
7160:
7159:
7153:
7141:
7138:
7126:
7122:
7115:
7112:
7106:
7103:
7097:
7094:
7088:
7085:
7079:
7076:
7072:
7068:
7064:
7058:
7055:
7051:
7047:
7043:
7037:
7034:
7027:
7022:
7019:
7017:
7014:
7012:
7009:
7007:
7004:
7002:
6999:
6996:
6993:
6992:
6987:
6982:
6981:Wave Equation
6979:
6977:
6974:
6972:
6969:
6967:
6964:
6962:
6959:
6957:
6954:
6952:
6949:
6947:
6944:
6942:
6939:
6937:
6934:
6931:
6928:
6927:
6922:
6917:
6915:
6913:
6909:
6905:
6900:
6892:
6890:
6888:
6881:Faraday waves
6880:
6878:
6876:
6869:Seismic waves
6868:
6866:
6864:
6860:
6856:
6852:
6843:
6841:
6839:
6835:
6831:
6827:
6823:
6822:photoelectron
6819:
6815:
6811:
6807:
6803:
6799:
6795:
6787:
6785:
6783:
6782:optical flats
6779:
6775:
6771:
6767:
6763:
6759:
6755:
6751:
6744:Visible light
6743:
6741:
6738:
6733:
6729:
6725:
6721:
6712:
6708:
6693:
6685:
6683:
6681:
6673:
6671:
6667:
6664:
6663:superposition
6659:
6655:
6648:
6640:
6638:
6636:
6632:
6613:
6608:
6599:
6595:
6590:
6587:
6581:
6578:
6571:
6570:
6569:
6565:
6558:
6551:
6544:
6537:
6530:
6523:
6519:
6514:
6510:
6505:
6501:
6497:
6493:
6488:
6486:
6482:
6476:
6469:
6464:
6460:
6456:
6452:
6433:
6430:
6427:
6424:
6421:
6418:
6415:
6412:
6409:
6405:
6402:
6399:
6396:
6393:
6390:
6387:
6384:
6381:
6374:
6360:
6356:
6352:
6349:
6345:
6341:
6338:
6334:
6327:
6323:
6318:
6315:
6312:
6306:
6302:
6299:
6295:
6288:
6284:
6279:
6276:
6273:
6267:
6263:
6260:
6251:
6247:
6241:
6238:
6235:
6232:
6229:
6223:
6216:
6215:
6214:
6212:
6208:
6204:
6200:
6196:
6192:
6188:
6169:
6162:
6157:
6150:
6146:
6142:
6137:
6132:
6127:
6122:
6115:
6111:
6107:
6102:
6095:
6092:
6089:
6086:
6079:
6078:
6077:
6060:
6057:
6054:
6051:
6048:
6045:
6042:
6039:
6036:
6032:
6025:
6021:
6016:
6013:
6007:
6002:
5998:
5993:
5989:
5984:
5980:
5975:
5972:
5965:
5964:
5963:
5959:
5955:
5951:
5944:
5939:
5935:
5931:
5927:
5923:
5904:
5901:
5898:
5895:
5892:
5889:
5886:
5883:
5880:
5876:
5869:
5865:
5860:
5857:
5851:
5846:
5842:
5834:
5833:
5832:
5829:
5825:
5806:
5802:
5797:
5793:
5788:
5785:
5778:
5777:
5776:
5774:
5770:
5764:
5760:
5756:
5749:
5744:
5740:
5736:
5732:
5728:
5724:
5720:
5716:
5712:
5692:
5687:
5684:
5681:
5678:
5674:
5668:
5663:
5659:
5655:
5652:
5648:
5642:
5637:
5633:
5629:
5626:
5622:
5618:
5612:
5609:
5606:
5603:
5600:
5594:
5587:
5586:
5585:
5583:
5579:
5575:
5571:
5567:
5563:
5559:
5540:
5535:
5530:
5526:
5522:
5519:
5515:
5507:
5503:
5498:
5494:
5489:
5484:
5480:
5476:
5472:
5464:
5460:
5455:
5451:
5445:
5439:
5432:
5431:
5430:
5428:
5409:
5402:
5397:
5393:
5389:
5384:
5379:
5375:
5369:
5366:
5363:
5356:
5355:
5354:
5352:
5333:
5326:
5322:
5316:
5312:
5306:
5301:
5296:
5292:
5288:
5283:
5273:
5269:
5265:
5259:
5251:
5247:
5238:
5233:
5216:
5210:
5206:
5197:
5196:
5195:
5193:
5189:
5183:
5179:
5175:
5169:
5165:
5161:
5157:
5153:
5134:
5129:
5124:
5120:
5116:
5113:
5109:
5101:
5097:
5092:
5088:
5083:
5078:
5074:
5070:
5066:
5058:
5054:
5049:
5045:
5039:
5033:
5026:
5025:
5024:
5022:
5018:
4999:
4994:
4984:
4980:
4976:
4970:
4962:
4958:
4949:
4944:
4927:
4921:
4917:
4908:
4907:
4906:
4904:
4900:
4881:
4878:
4871:
4867:
4861:
4857:
4851:
4843:
4839:
4830:
4825:
4808:
4802:
4798:
4793:
4785:
4781:
4772:
4767:
4750:
4744:
4740:
4731:
4730:
4729:
4727:
4708:
4702:
4696:
4690:
4684:
4681:
4678:
4671:
4670:
4669:
4667:
4663:
4659:
4640:
4634:
4631:
4628:
4625:
4622:
4616:
4609:
4605:
4599:
4595:
4589:
4586:
4578:
4574:
4565:
4560:
4549:
4541:
4537:
4528:
4523:
4508:
4507:
4506:
4489:
4486:
4483:
4478:
4475:
4472:
4469:
4465:
4458:
4455:
4452:
4449:
4446:
4440:
4427:
4423:
4419:
4413:
4410:
4407:
4404:
4401:
4395:
4388:
4387:
4386:
4384:
4376:
4373:
4370:
4366:
4362:
4358:
4355:
4354:
4353:
4336:
4332:
4323:
4319:
4310:
4305:
4294:
4286:
4282:
4273:
4268:
4256:
4250:
4246:
4241:
4232:
4228:
4219:
4214:
4199:
4198:
4197:
4195:
4194:wave equation
4190:
4186:
4181:
4177:
4173:
4166:
4162:
4151:
4149:
4147:
4128:
4122:
4119:
4114:
4111:
4105:
4102:
4095:
4094:
4093:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4042:
4028:
4023:
4019:
4016:
4010:
4007:
4000:
3999:
3998:
3996:
3980:
3977:
3973:
3954:
3948:
3940:
3929:
3925:
3921:
3918:
3915:
3912:
3902:
3901:
3900:
3898:
3894:
3890:
3886:
3875:
3869:
3865:
3863:
3844:
3838:
3835:
3830:
3827:
3821:
3818:
3811:
3810:
3809:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3758:
3744:
3739:
3735:
3732:
3726:
3723:
3716:
3715:
3714:
3712:
3707:
3703:
3696:
3691:
3685:
3679:
3675:
3669:
3665:
3658:
3652:
3650:
3646:
3627:
3621:
3618:
3613:
3610:
3604:
3601:
3594:
3593:
3592:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3541:
3527:
3522:
3518:
3515:
3509:
3506:
3499:
3498:
3497:
3480:
3477:
3474:
3468:
3465:
3459:
3456:
3452:
3447:
3443:
3440:
3437:
3431:
3427:
3424:
3415:
3411:
3408:
3402:
3399:
3396:
3390:
3380:
3366:
3363:
3360:
3354:
3351:
3348:
3342:
3332:
3331:
3330:
3328:
3324:
3319:
3315:
3313:
3312:
3292:
3286:
3283:
3277:
3274:
3270:
3265:
3261:
3258:
3255:
3249:
3245:
3242:
3233:
3229:
3226:
3220:
3217:
3214:
3202:
3195:
3189:
3186:
3183:
3171:
3164:
3158:
3155:
3152:
3146:
3136:
3135:
3134:
3128:
3124:
3121:
3118:
3115:
3111:
3107:
3103:
3100:
3094:
3091:
3090:
3089:
3072:
3068:
3064:
3061:
3058:
3053:
3049:
3046:
3043:
3036:
3032:
3029:
3020:
3016:
3010:
3007:
3004:
2992:
2981:
2967:
2963:
2959:
2956:
2953:
2948:
2944:
2941:
2938:
2931:
2927:
2924:
2915:
2911:
2905:
2902:
2899:
2887:
2876:
2875:
2874:
2872:
2867:
2864:
2860:
2854:
2846:
2844:
2842:
2837:
2834:
2830:
2823:
2818:
2814:
2810:
2791:
2785:
2782:
2777:
2774:
2768:
2765:
2758:
2757:
2756:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2708:
2694:
2689:
2685:
2682:
2676:
2673:
2666:
2665:
2664:
2647:
2644:
2640:
2635:
2631:
2628:
2625:
2619:
2615:
2612:
2605:
2604:
2603:
2599:
2594:
2589:
2585:
2580:
2579:
2573:
2570:
2566:
2559:
2554:
2549:
2545:
2538:
2533:
2528:
2524:
2517:
2510:
2505:
2504:
2498:
2494:
2488:
2484:
2479:
2474:
2470:
2463:
2458:
2450:
2446:
2442:
2435:
2433:
2431:
2427:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2377:
2363:
2358:
2354:
2351:
2345:
2342:
2335:
2334:
2333:
2331:
2327:
2323:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2273:
2259:
2254:
2250:
2247:
2241:
2238:
2231:
2230:
2229:
2227:
2226:
2220:
2218:
2214:
2210:
2206:
2202:
2198:
2192:
2172:
2166:
2163:
2158:
2155:
2149:
2144:
2141:
2136:
2133:
2126:
2125:
2124:
2122:
2118:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2068:
2067:
2062:
2055:
2053:
2039:
2034:
2030:
2027:
2021:
2018:
2011:
2010:
2007:
2005:
1989:
1986:
1982:
1977:
1973:
1970:
1967:
1961:
1957:
1954:
1946:
1938:
1934:
1929:
1911:
1908:
1902:
1899:
1893:
1890:
1886:
1881:
1877:
1874:
1871:
1865:
1861:
1858:
1849:
1845:
1842:
1836:
1833:
1830:
1824:
1817:
1803:
1800:
1797:
1791:
1788:
1785:
1779:
1772:
1771:
1770:
1768:
1762:
1756:
1752:
1746:
1742:
1735:
1728:
1723:
1719:
1718:
1713:
1712:
1706:
1703:
1699:
1693:
1687:
1683:
1676:
1667:
1665:
1663:
1659:
1640:
1637:
1632:
1628:
1625:
1618:
1613:
1609:
1606:
1599:
1594:
1591:
1585:
1580:
1577:
1572:
1568:
1563:
1559:
1556:
1550:
1546:
1541:
1537:
1534:
1528:
1525:
1522:
1519:
1516:
1509:
1508:
1507:
1505:
1501:
1497:
1478:
1475:
1470:
1466:
1463:
1456:
1453:
1449:
1444:
1441:
1435:
1432:
1428:
1423:
1420:
1415:
1411:
1408:
1405:
1401:
1396:
1392:
1389:
1383:
1380:
1377:
1374:
1371:
1364:
1363:
1362:
1360:
1356:
1351:
1349:
1344:
1329:
1324:
1320:
1317:
1314:
1308:
1304:
1301:
1292:
1288:
1280:
1276:
1272:
1268:
1264:
1260:
1259:
1249:
1242:
1240:
1226:
1220:
1217:
1211:
1208:
1204:
1199:
1195:
1192:
1189:
1183:
1179:
1176:
1167:
1163:
1160:
1154:
1151:
1148:
1142:
1135:
1134:
1131:
1116:
1111:
1107:
1104:
1101:
1095:
1091:
1088:
1084:
1079:
1075:
1072:
1069:
1063:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1022:
1003:
999:
995:
992:
989:
984:
980:
977:
974:
967:
963:
960:
951:
947:
943:
939:
936:
933:
928:
924:
921:
918:
911:
907:
904:
895:
891:
882:
878:
869:
865:
859:
856:
853:
847:
840:
839:
838:
833:
826:
818:
814:
811:
808:
805:
801:
797:
793:
790:
787:
780:
777:
776:
775:
758:
754:
750:
747:
744:
739:
735:
732:
729:
722:
718:
715:
706:
702:
696:
693:
690:
678:
670:
669:
668:
666:
647:
643:
639:
636:
633:
628:
624:
621:
618:
611:
607:
604:
595:
591:
585:
582:
579:
567:
559:
558:
557:
555:
551:
547:
543:
539:
538:harmonic wave
534:
532:
528:
524:
516:
514:
512:
508:
504:
500:
496:
491:
488:
479:
477:
475:
471:
467:
462:
460:
456:
451:
449:
445:
441:
437:
433:
430:
426:
422:
418:
414:
410:
406:
402:
401:superposition
398:
394:
390:
386:
382:
381:standing wave
369:
362:
357:
353:
346:
341:
339:
338:
331:
327:
320:
315:
308:
303:
301:
300:
290:
285:
281:
274:
269:
267:
266:
259:
258:
256:
255:
249:
247:
245:
244:river surfing
241:
240:Froude number
237:
233:
229:
225:
222:
218:
214:
209:
207:
206:glider pilots
203:
191:
188:Mouth of the
170:
167:
147:
140:Moving medium
139:
137:
135:
131:
127:
122:
120:
116:
112:
108:
103:
101:
97:
93:
89:
85:
80:
78:
74:
70:
66:
62:
58:
57:standing wave
54:
46:
38:
30:
21:
8896:Washtub bass
8749:musical bows
8718:
8709:Scale length
8686:
8606:Third bridge
8454:Thomas Young
8404:Jens Blauert
8392:Acousticians
8370:
8114:. Retrieved
8087:
8068:
8047:. Retrieved
8039:Ars Technica
8038:
8028:
8017:. Retrieved
8008:
7999:
7991:the original
7982:
7972:
7961:. Retrieved
7957:the original
7947:
7921:
7915:
7880:
7876:
7866:
7854:. Retrieved
7845:
7836:
7811:
7807:
7801:
7776:
7772:
7759:
7752:
7727:
7723:
7706:
7681:
7677:
7671:
7646:
7642:
7636:
7613:
7607:
7582:
7578:
7572:
7553:
7537:
7536:K A Tsokos,
7532:
7516:
7511:
7501:December 28,
7499:. Retrieved
7493:
7486:
7474:. Retrieved
7463:
7451:. Retrieved
7445:
7438:
7426:. Retrieved
7420:
7413:
7401:
7389:
7377:
7355:Streets 2010
7350:
7322:, p. 6.
7320:Streets 2010
7315:
7303:. Retrieved
7292:
7280:
7268:. Retrieved
7262:
7256:
7244:
7232:
7182:
7176:
7167:the original
7157:
7140:
7129:. Retrieved
7124:
7114:
7105:
7096:
7087:
7078:
7062:
7057:
7041:
7036:
6976:Kundt's tube
6951:Mode-locking
6912:Lake seiches
6896:
6887:Faraday wave
6884:
6872:
6847:
6814:X-ray mirror
6791:
6747:
6716:
6711:Rossby waves
6677:
6668:
6650:
6635:multiplicity
6630:
6628:
6563:
6556:
6549:
6542:
6535:
6528:
6526:, the modes
6521:
6517:
6512:
6508:
6503:
6499:
6495:
6491:
6489:
6484:
6480:
6474:
6467:
6462:
6458:
6454:
6450:
6448:
6210:
6206:
6202:
6198:
6194:
6190:
6186:
6184:
6075:
5957:
5953:
5949:
5942:
5937:
5933:
5929:
5925:
5921:
5919:
5827:
5823:
5821:
5772:
5768:
5762:
5758:
5754:
5747:
5742:
5738:
5734:
5730:
5729:dependence,
5726:
5722:
5718:
5714:
5710:
5707:
5581:
5577:
5573:
5569:
5565:
5561:
5557:
5555:
5426:
5424:
5348:
5191:
5187:
5181:
5177:
5173:
5167:
5163:
5159:
5151:
5149:
5020:
5016:
5015:Solving for
5014:
4902:
4898:
4896:
4725:
4723:
4655:
4504:
4380:
4374:
4368:
4364:
4360:
4356:
4351:
4191:
4184:
4179:
4175:
4168:
4164:
4157:
4155:
4143:
4091:
3994:
3981:
3971:
3969:
3896:
3892:
3888:
3884:
3882:
3873:
3861:
3859:
3807:
3705:
3701:
3694:
3683:
3677:
3673:
3667:
3663:
3656:
3653:
3644:
3642:
3590:
3495:
3320:
3316:
3309:
3307:
3132:
3129:of the wave.
3122:
3116:
3109:
3101:
3092:
3087:
2870:
2863:longitudinal
2858:
2856:
2840:
2838:
2832:
2828:
2821:
2816:
2812:
2808:
2806:
2754:
2662:
2597:
2595:occurs when
2592:
2587:
2583:
2576:
2574:
2568:
2564:
2557:
2547:
2543:
2536:
2531:
2526:
2522:
2515:
2508:
2501:
2499:
2492:
2486:
2482:
2477:
2472:
2468:
2461:
2456:
2454:
2429:
2423:
2329:
2325:
2321:
2319:
2223:
2221:
2219:anti-nodes.
2216:
2212:
2200:
2190:
2187:
2120:
2116:
2114:
2056:
2003:
1942:
1766:
1760:
1754:
1750:
1744:
1740:
1733:
1726:
1715:
1709:
1707:
1697:
1691:
1685:
1681:
1674:
1671:
1661:
1655:
1503:
1499:
1493:
1358:
1354:
1352:
1347:
1345:
1278:
1274:
1270:
1266:
1262:
1256:
1254:
1243:
1018:
831:
824:
822:
819:of the wave.
812:
806:
799:
791:
778:
773:
664:
662:
553:
549:
545:
542:displacement
535:
530:
527:transversely
522:
520:
492:
483:
463:
454:
452:
440:open circuit
435:
421:displacement
413:displacement
380:
378:
221:Saltstraumen
217:river rapids
210:
199:
190:Carmel River
123:
107:interference
104:
99:
95:
81:
60:
56:
50:
40:
32:
24:
8871:Psalmodicon
8784:Diddley bow
8643:Fundamental
8633:Fingerboard
8613:Chordophone
8571:instruments
8424:Franz Melde
8399:John Backus
8383:Subharmonic
8236:Spectrogram
7127:(in German)
7021:Normal mode
6988:Electronics
6908:Lake Geneva
6904:storm surge
6766:gain medium
4656:This is an
4146:Rubens tube
2866:sound waves
1933:fundamental
533:direction.
507:sound boxes
474:microseisms
470:microbaroms
326:fundamental
234:due to the
92:Franz Melde
8911:Categories
8804:Ichigenkin
8799:Ground bow
8744:Monochords
8734:Tuning peg
8714:Soundboard
8628:Enharmonic
8485:Ultrasound
8475:Infrasound
8261:Bark scale
8116:August 23,
8061:References
8049:2023-01-01
8019:2023-01-01
7963:2023-01-01
7546:0521138213
7525:8120351592
7476:January 2,
7453:August 23,
7428:August 23,
7305:August 23,
7270:August 22,
7131:2022-01-13
7071:3642850847
7050:1135455589
6834:adsorption
6806:diffracted
6778:nanometers
6774:coherently
6724:air column
4658:eigenvalue
3323:organ pipe
3127:wavelength
3112:times the
2851:See also:
1708:Equation (
1658:anti-nodes
1255:Equation (
1019:Using the
817:wavelength
802:times the
495:resonators
417:anti-nodes
8851:Langeleik
8789:Duxianqin
8682:Monochord
8651:Overtones
8647:Harmonics
8366:Resonance
8266:Mel scale
8196:Monochord
8175:Acoustics
7771:(0001)".
7515:R S Rao,
7016:Impedance
6902:a simple
6855:frequency
6838:catalysis
6732:harmonics
6680:jump rope
6591:π
6579:ω
6434:…
6406:…
6350:ω
6342:
6316:π
6303:
6277:π
6264:
6096:π
6087:ω
6061:…
6017:π
5976:
5905:…
5861:π
5789:
5685:ω
5679:±
5653:±
5627:±
5619:∼
5520:−
5364:ω
5313:ω
5307:−
5244:∂
5230:∂
5114:−
4955:∂
4941:∂
4858:ω
4836:∂
4822:∂
4778:∂
4764:∂
4668:. Assume
4635:ω
4596:ω
4590:−
4571:∂
4557:∂
4534:∂
4520:∂
4476:ω
4470:−
4436:∞
4431:∞
4428:−
4424:∫
4414:ω
4316:∂
4302:∂
4279:∂
4265:∂
4225:∂
4211:∂
4074:…
4008:λ
3946:∂
3938:∂
3922:ρ
3919:−
3910:Δ
3790:…
3724:λ
3573:…
3507:λ
3466:ω
3460:
3448:λ
3441:π
3428:
3388:Δ
3340:Δ
3284:ω
3278:
3266:λ
3259:π
3246:
3199:Δ
3168:Δ
3144:Δ
3114:frequency
3062:ω
3054:λ
3047:π
3033:
2989:Δ
2957:ω
2954:−
2949:λ
2942:π
2928:
2884:Δ
2740:…
2674:λ
2636:λ
2629:π
2616:
2445:Transient
2426:resonance
2409:…
2355:λ
2305:…
2239:λ
2209:overtones
2205:harmonics
2145:λ
2100:…
2019:λ
1978:λ
1971:π
1958:
1937:harmonics
1900:ω
1894:
1882:λ
1875:π
1862:
1641:…
1629:λ
1610:λ
1592:λ
1578:λ
1573:−
1560:λ
1551:−
1538:λ
1529:−
1523:…
1479:…
1467:λ
1454:λ
1442:λ
1421:λ
1416:−
1409:λ
1406:−
1393:λ
1384:−
1378:…
1325:λ
1318:π
1305:
1218:ω
1212:
1200:λ
1193:π
1180:
1105:−
1092:
1060:
1045:
1033:
993:ω
985:λ
978:π
964:
937:ω
934:−
929:λ
922:π
908:
804:frequency
786:amplitude
748:ω
740:λ
733:π
719:
637:ω
634:−
629:λ
622:π
608:
552:and time
499:drumheads
429:impedance
425:reflected
419:(maximum
405:frequency
330:harmonics
224:maelstrom
126:amplitude
115:resonator
111:resonance
100:Stehwelle
69:amplitude
8866:Onavillu
8819:Genggong
8814:Jaw harp
8762:Berimbau
8704:Re-entry
8561:Musical
8520:Category
8361:Overtone
8329:Harmonic
8043:Archived
8013:Archived
8005:"Seiche"
7907:24956442
7856:28 April
7850:Archived
7793:20028144
7715:(1982).
7561:Archived
7011:Cymatics
6941:Clapotis
6918:See also
6802:surfaces
6674:Examples
5349:and the
3711:clarinet
3695:â(Îp)/âx
3684:â(Îp)/âx
3327:recorder
1702:reflects
432:mismatch
246:breaks.
73:in phase
8886:Umuduri
8861:Masenqo
8844:Mukkuri
8839:Morsing
8779:ÄĂ n báș§u
8772:Boom-ba
8757:Ahardin
8563:strings
8307:Formant
7898:4159433
7816:Bibcode
7732:Bibcode
7686:Bibcode
7651:Bibcode
7587:Bibcode
6971:Trumpet
6893:Seiches
6830:dopants
6770:crystal
4385:, with
4174:in the
4163:in the
3988:⁄
3976:density
3974:is the
3647:is the
3125:is the
3104:is the
2581:), for
815:is the
794:is the
784:is the
544:in the
529:in the
461:(SWR).
393:voltage
389:current
238:speed (
232:gravity
228:inertia
132:no net
130:average
63:, is a
53:physics
8891:Unitar
8856:Lesiba
8834:Kubing
8829:Khomuz
8824:Gogona
8794:Ektara
8618:Course
8601:Bridge
8569:, and
8500:Violin
8334:Series
8094:
8075:
7936:
7905:
7895:
7791:
7624:
7579:Icarus
7544:
7523:
7189:
7069:
7048:
6966:Seiche
6899:seiche
6851:sensor
6788:X-rays
6764:. The
6758:Lasers
6554:, and
6502:- and
6461:- and
5580:, and
5429:term,
4352:where
3970:where
3643:where
3088:where
1724:where
774:where
536:For a
415:) and
411:(zero
8881:Tumbi
8623:Drone
8567:wires
8495:Piano
8480:Sound
8294:pitch
8256:Pitch
8110:(PDF)
7763:/α-Fe
7720:(PDF)
7154:from
7028:Notes
6923:Waves
6859:phase
6794:X-ray
6658:phase
5822:with
5150:This
3325:or a
2602:, or
2598:ây/âx
2558:ây/âx
2537:ây/âx
2466:. At
1496:nodes
466:ocean
444:short
442:or a
409:nodes
395:, or
383:in a
280:nodes
77:nodes
43:green
8694:Node
8638:Fret
8580:List
8470:Echo
8376:Node
8302:Beat
8292:and
8118:2020
8092:ISBN
8073:ISBN
7934:ISBN
7903:PMID
7858:2018
7789:PMID
7622:ISBN
7542:ISBN
7521:ISBN
7503:2020
7478:2021
7455:2020
7430:2020
7307:2020
7272:2020
7187:ISBN
7067:ISBN
7046:ISBN
6885:The
6752:and
6561:and
6547:and
6533:and
6494:and
6483:and
6472:and
6453:and
5947:and
5771:and
5752:and
5172:and
2330:even
1738:and
1679:and
1664:/2.
1359:even
830:and
472:and
436:i.e.
65:wave
55:, a
47:wave
35:blue
8746:and
8699:Nut
8596:Bow
7926:doi
7893:PMC
7885:doi
7824:doi
7781:doi
7777:131
7740:doi
7694:doi
7659:doi
7595:doi
7583:206
6857:or
6820:or
6726:'s
6722:or
6566:= 5
6559:= 5
6552:= 1
6545:= 7
6538:= 7
6531:= 1
6477:= 1
6470:= 1
6339:cos
6300:sin
6261:sin
6256:max
5973:sin
5945:= 0
5786:sin
5750:= 0
5023:),
4196:is
4187:= 0
3876:= 2
3699:at
3697:= 0
3686:= 0
3659:= 0
3457:cos
3425:sin
3420:max
3275:cos
3243:sin
3238:max
3096:max
3030:sin
3025:max
2925:sin
2920:max
2824:= 0
2817:odd
2613:cos
2600:= 0
2562:at
2560:= 0
2541:at
2539:= 0
2518:= 0
2513:at
2511:= 0
2495:= 0
2464:= 0
2207:or
2193:= 1
1955:sin
1891:cos
1859:sin
1854:max
1763:= 0
1736:= 0
1731:at
1729:= 0
1694:= 0
1677:= 0
1504:odd
1302:sin
1297:max
1209:cos
1177:sin
1172:max
1089:cos
1057:sin
1042:sin
1030:sin
961:sin
956:max
905:sin
900:max
782:max
716:sin
711:max
605:sin
600:max
556:is
202:lee
98:or
90:.
51:In
27:red
8913::
8565:,
8041:.
8037:.
8011:.
8007:.
7985:.
7981:.
7932:,
7901:.
7891:.
7881:26
7879:.
7875:.
7848:.
7844:.
7822:.
7812:76
7810:.
7787:.
7775:.
7738:.
7728:49
7726:.
7722:.
7692:.
7682:22
7680:.
7657:.
7647:36
7645:.
7620:.
7593:.
7581:.
7362:^
7327:^
7215:^
7200:^
7161:.
7123:.
6914:.
6897:A
6877:.
6865:.
6840:.
6784:.
6756:.
6609:50
6540:,
6516:=
5952:=
5757:=
5584:,
5576:,
4882:0.
4728:,
4144:A
3704:=
3676:=
3666:=
3651:.
3110:2Ï
2843:.
2831:=
2648:0.
2586:=
2567:=
2546:=
2525:=
2497:.
2485:=
2471:=
2432:.
2002:.
1912:0.
1753:=
1743:=
1684:=
1265:,
1130:,
837:,
800:2Ï
476:.
450:.
434:,
391:,
350:A
208:.
136:.
121:.
8653:/
8649:/
8645:/
8586:)
8582:(
8554:e
8547:t
8540:v
8167:e
8160:t
8153:v
8120:.
8100:.
8081:.
8052:.
8022:.
7966:.
7928::
7909:.
7887::
7860:.
7830:.
7826::
7818::
7795:.
7783::
7769:3
7767:O
7765:2
7760:x
7746:.
7742::
7734::
7700:.
7696::
7688::
7665:.
7661::
7653::
7630:.
7601:.
7597::
7589::
7548:.
7527:.
7505:.
7480:.
7457:.
7432:.
7309:.
7274:.
7134:.
7073:.
7052:.
6997::
6932::
6631:Ï
6614:.
6600:x
6596:L
6588:c
6582:=
6564:m
6557:n
6550:m
6543:n
6536:m
6529:n
6522:y
6518:L
6513:x
6509:L
6504:y
6500:x
6496:m
6492:n
6485:m
6481:n
6475:m
6468:n
6463:y
6459:x
6455:m
6451:n
6431:,
6428:3
6425:,
6422:2
6419:,
6416:1
6413:=
6410:m
6403:,
6400:3
6397:,
6394:2
6391:,
6388:1
6385:=
6382:n
6361:.
6357:)
6353:t
6346:(
6335:)
6328:y
6324:L
6319:y
6313:m
6307:(
6296:)
6289:x
6285:L
6280:x
6274:n
6268:(
6252:z
6248:=
6245:)
6242:t
6239:,
6236:y
6233:,
6230:x
6227:(
6224:z
6211:t
6207:y
6205:,
6203:x
6201:(
6199:ĆŒ
6195:y
6193:,
6191:x
6189:(
6187:z
6170:.
6163:2
6158:)
6151:y
6147:L
6143:m
6138:(
6133:+
6128:2
6123:)
6116:x
6112:L
6108:n
6103:(
6093:c
6090:=
6058:,
6055:3
6052:,
6049:2
6046:,
6043:1
6040:=
6037:m
6033:,
6026:y
6022:L
6014:m
6008:=
6003:y
5999:k
5994:,
5990:y
5985:y
5981:k
5958:y
5954:L
5950:y
5943:y
5938:t
5936:,
5934:y
5932:,
5930:x
5928:(
5926:z
5922:y
5902:,
5899:3
5896:,
5893:2
5890:,
5887:1
5884:=
5881:n
5877:,
5870:x
5866:L
5858:n
5852:=
5847:x
5843:k
5828:x
5824:k
5807:,
5803:x
5798:x
5794:k
5773:t
5769:y
5763:x
5759:L
5755:x
5748:x
5743:t
5741:,
5739:y
5737:,
5735:x
5733:(
5731:z
5727:x
5723:t
5721:,
5719:y
5717:,
5715:x
5713:(
5711:z
5693:.
5688:t
5682:i
5675:e
5669:y
5664:y
5660:k
5656:i
5649:e
5643:x
5638:x
5634:k
5630:i
5623:e
5616:)
5613:t
5610:,
5607:y
5604:,
5601:x
5598:(
5595:z
5582:t
5578:y
5574:x
5570:t
5568:,
5566:y
5564:,
5562:x
5560:(
5558:z
5541:.
5536:y
5531:y
5527:k
5523:i
5516:e
5508:y
5504:k
5499:D
5495:+
5490:y
5485:y
5481:k
5477:i
5473:e
5465:y
5461:k
5456:C
5452:=
5449:)
5446:y
5443:(
5440:Y
5427:y
5410:.
5403:2
5398:y
5394:k
5390:+
5385:2
5380:x
5376:k
5370:c
5367:=
5334:,
5327:2
5323:c
5317:2
5302:2
5297:x
5293:k
5289:=
5284:2
5280:)
5274:y
5270:k
5266:i
5263:(
5260:=
5252:2
5248:y
5239:Y
5234:2
5220:)
5217:y
5214:(
5211:Y
5207:1
5192:y
5188:y
5182:x
5178:k
5174:B
5168:x
5164:k
5160:A
5152:x
5135:.
5130:x
5125:x
5121:k
5117:i
5110:e
5102:x
5098:k
5093:B
5089:+
5084:x
5079:x
5075:k
5071:i
5067:e
5059:x
5055:k
5050:A
5046:=
5043:)
5040:x
5037:(
5034:X
5021:x
5019:(
5017:X
5000:.
4995:2
4991:)
4985:x
4981:k
4977:i
4974:(
4971:=
4963:2
4959:x
4950:X
4945:2
4931:)
4928:x
4925:(
4922:X
4918:1
4903:x
4899:x
4879:=
4872:2
4868:c
4862:2
4852:+
4844:2
4840:y
4831:Y
4826:2
4812:)
4809:y
4806:(
4803:Y
4799:1
4794:+
4786:2
4782:x
4773:X
4768:2
4754:)
4751:x
4748:(
4745:X
4741:1
4726:Z
4709:.
4706:)
4703:y
4700:(
4697:Y
4694:)
4691:x
4688:(
4685:X
4682:=
4679:Z
4641:.
4638:)
4632:,
4629:y
4626:,
4623:x
4620:(
4617:Z
4610:2
4606:c
4600:2
4587:=
4579:2
4575:y
4566:Z
4561:2
4550:+
4542:2
4538:x
4529:Z
4524:2
4490:.
4487:t
4484:d
4479:t
4473:i
4466:e
4462:)
4459:t
4456:,
4453:y
4450:,
4447:x
4444:(
4441:z
4420:=
4417:)
4411:,
4408:y
4405:,
4402:x
4399:(
4396:Z
4375:c
4369:t
4367:,
4365:y
4363:,
4361:x
4359:(
4357:z
4337:,
4333:)
4324:2
4320:y
4311:z
4306:2
4295:+
4287:2
4283:x
4274:z
4269:2
4257:(
4251:2
4247:c
4242:=
4233:2
4229:t
4220:z
4215:2
4185:z
4180:z
4176:y
4171:y
4169:L
4165:x
4160:x
4158:L
4129:.
4123:L
4120:2
4115:v
4112:n
4106:=
4103:f
4077:,
4071:,
4068:3
4065:,
4062:2
4059:,
4056:1
4053:=
4050:n
4029:,
4024:n
4020:L
4017:2
4011:=
3995:x
3990:2
3986:Ï
3972:Ï
3955:,
3949:x
3941:s
3930:2
3926:v
3916:=
3913:p
3897:s
3893:p
3889:x
3885:x
3874:n
3862:n
3845:.
3839:L
3836:4
3831:v
3828:n
3822:=
3819:f
3793:,
3787:,
3784:5
3781:,
3778:3
3775:,
3772:1
3769:=
3766:n
3745:,
3740:n
3736:L
3733:4
3727:=
3706:L
3702:x
3678:L
3674:x
3668:L
3664:x
3657:x
3645:v
3628:,
3622:L
3619:2
3614:v
3611:n
3605:=
3602:f
3576:,
3570:,
3567:3
3564:,
3561:2
3558:,
3555:1
3552:=
3549:n
3528:,
3523:n
3519:L
3516:2
3510:=
3481:,
3478:0
3475:=
3472:)
3469:t
3463:(
3453:)
3444:L
3438:2
3432:(
3416:p
3412:2
3409:=
3406:)
3403:t
3400:,
3397:L
3394:(
3391:p
3367:,
3364:0
3361:=
3358:)
3355:t
3352:,
3349:0
3346:(
3343:p
3311:1
3293:.
3290:)
3287:t
3281:(
3271:)
3262:x
3256:2
3250:(
3234:p
3230:2
3227:=
3224:)
3221:t
3218:,
3215:x
3212:(
3207:L
3203:p
3196:+
3193:)
3190:t
3187:,
3184:x
3181:(
3176:R
3172:p
3165:=
3162:)
3159:t
3156:,
3153:x
3150:(
3147:p
3123:λ
3119:,
3117:f
3102:Ï
3093:p
3073:,
3069:)
3065:t
3059:+
3050:x
3044:2
3037:(
3021:p
3017:=
3014:)
3011:t
3008:,
3005:x
3002:(
2997:L
2993:p
2968:,
2964:)
2960:t
2945:x
2939:2
2932:(
2916:p
2912:=
2909:)
2906:t
2903:,
2900:x
2897:(
2892:R
2888:p
2871:p
2859:L
2833:L
2829:x
2822:x
2813:L
2809:n
2792:.
2786:L
2783:4
2778:v
2775:n
2769:=
2766:f
2737:,
2734:5
2731:,
2728:3
2725:,
2722:1
2719:=
2716:n
2695:,
2690:n
2686:L
2683:4
2677:=
2645:=
2641:)
2632:L
2626:2
2620:(
2593:y
2588:L
2584:x
2578:1
2569:L
2565:x
2548:L
2544:x
2532:y
2527:L
2523:x
2516:x
2509:y
2503:1
2493:x
2487:L
2483:x
2478:y
2473:L
2469:x
2462:x
2457:L
2406:,
2403:6
2400:,
2397:4
2394:,
2391:2
2388:=
2385:n
2364:,
2359:4
2352:n
2346:=
2343:L
2326:L
2322:n
2302:,
2299:6
2296:,
2293:4
2290:,
2287:2
2284:=
2281:n
2260:,
2255:n
2251:L
2248:4
2242:=
2225:2
2217:n
2213:n
2201:n
2191:n
2173:.
2167:L
2164:2
2159:v
2156:n
2150:=
2142:v
2137:=
2134:f
2121:v
2117:L
2097:,
2094:3
2091:,
2088:2
2085:,
2082:1
2079:=
2076:n
2061:)
2059:2
2057:(
2040:,
2035:n
2031:L
2028:2
2022:=
2004:L
1990:0
1987:=
1983:)
1974:L
1968:2
1962:(
1939:.
1909:=
1906:)
1903:t
1897:(
1887:)
1878:L
1872:2
1866:(
1850:y
1846:2
1843:=
1840:)
1837:t
1834:,
1831:L
1828:(
1825:y
1804:,
1801:0
1798:=
1795:)
1792:t
1789:,
1786:0
1783:(
1780:y
1767:y
1761:x
1755:L
1751:x
1745:L
1741:x
1734:x
1727:y
1717:1
1711:1
1698:f
1692:x
1686:L
1682:x
1675:x
1662:λ
1638:,
1633:4
1626:5
1619:,
1614:4
1607:3
1600:,
1595:4
1586:,
1581:4
1569:,
1564:4
1557:3
1547:,
1542:4
1535:5
1526:,
1520:=
1517:x
1500:x
1476:,
1471:2
1464:3
1457:,
1450:,
1445:2
1436:,
1433:0
1429:,
1424:2
1412:,
1402:,
1397:2
1390:3
1381:,
1375:=
1372:x
1355:x
1348:x
1330:)
1321:x
1315:2
1309:(
1293:y
1289:2
1279:x
1275:t
1273:,
1271:x
1269:(
1267:y
1263:x
1258:1
1248:)
1246:1
1244:(
1227:.
1224:)
1221:t
1215:(
1205:)
1196:x
1190:2
1184:(
1168:y
1164:2
1161:=
1158:)
1155:t
1152:,
1149:x
1146:(
1143:y
1117:)
1112:2
1108:b
1102:a
1096:(
1085:)
1080:2
1076:b
1073:+
1070:a
1064:(
1054:2
1051:=
1048:b
1039:+
1036:a
1004:.
1000:)
996:t
990:+
981:x
975:2
968:(
952:y
948:+
944:)
940:t
925:x
919:2
912:(
896:y
892:=
887:L
883:y
879:+
874:R
870:y
866:=
863:)
860:t
857:,
854:x
851:(
848:y
835:L
832:y
828:R
825:y
813:λ
809:,
807:f
792:Ï
779:y
759:,
755:)
751:t
745:+
736:x
730:2
723:(
707:y
703:=
700:)
697:t
694:,
691:x
688:(
683:L
679:y
665:y
648:.
644:)
640:t
625:x
619:2
612:(
596:y
592:=
589:)
586:t
583:,
580:x
577:(
572:R
568:y
554:t
550:x
546:y
531:y
523:x
332:.
282:.
45:)
41:(
37:)
33:(
29:)
25:(
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