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2534:. Bernoulli argued that any sonorous body could vibrate in a series of simple modes with a well-defined frequency of oscillation. As he had earlier indicated, these modes could be superposed to produce more complex vibrations. In his reaction to Bernoulli's memoirs, Euler praised his colleague for having best developed the physical part of the problem of vibrating strings, but denied the generality and superiority of the multi-modes solution.
541:), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.)
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The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources),
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of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system). Mode superposition method uses the natural frequencies and mode shapes to characterize the
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and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if
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In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of
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In as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter is, therefore, a continuation of
Chapter 8 . On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important
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by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields that arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the
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Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems.
2424:, the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses.
2349:{\displaystyle {\begin{aligned}{\dot {x}}_{1}&=Ax_{1}+Bu_{1}+\phi (y_{d}),&&x_{1}(0)=x_{0},\\{\dot {x}}_{2}&=Ax_{2}+Bu_{2}+\phi \left(c^{\mathsf {T}}x_{1}+c^{\mathsf {T}}x_{2}\right)-\phi (y_{d}),&&x_{2}(0)=0\end{aligned}}}
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categories of diffraction relate to the interference that accompanies division of the wavefront, so
Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront.
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418:. Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different
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399:. If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist).
745:." However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself uses ket vector representations of states to decompose or split, for example, a ket vector
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between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in
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whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.
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theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior.
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Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a
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the effect is called diffraction. That is the difference between the two phenomena is of degree only, and basically, they are two limiting cases of superposition effects.
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By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute.
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are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation:
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1820:{\displaystyle {\begin{aligned}{\dot {x}}_{1}&=Ax_{1}+Bu_{1},&&x_{1}(0)=x_{0},\\{\dot {x}}_{2}&=Ax_{2}+Bu_{2},&&x_{2}(0)=0\end{aligned}}}
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green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.
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1126:." Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase, they actually mean phase translation symmetry derived from
99:, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input
426:.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.
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Mechanical
Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192
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if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state
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of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques,
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There are exact correspondences between the superposition presented in the main on this page and the quantum superposition. For example, the
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in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by
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the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory
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motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions:
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The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by
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Waves are usually described by variations in some parameters through space and time—for example, height in a water wave,
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Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example".
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The projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a
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is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into
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holds only approximately in water and only for waves with small amplitudes relative to their wavelengths.
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464:, and direction). As long as the superposition principle holds (which is often but not always; see
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2011:{\displaystyle {\dot {x}}=Ax+B(u_{1}+u_{2})+\phi \left(c^{\mathsf {T}}x\right),\qquad x(0)=x_{0},}
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writes: "The principle of superposition ... has no analogy in classical physics". According to
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phenomena that arise when the superposition principle does not exactly hold, see the articles
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Topics on
Numerics for Wave Propagation, Basque Center for Applied Mathematics, 2012, Spain,
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Lectures in
Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass
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1391:{\displaystyle F(y_{1})=F(y_{2})=\cdots =0\quad \Rightarrow \quad F(y_{1}+y_{2}+\cdots )=0,}
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Fundamental physics principle stating that physical solutions of linear systems are linear
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The
Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London.
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This article is about the superposition principle in linear systems. For other uses, see
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Finite
Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785
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can be applied to both linear and nonlinear systems. Next, consider a nonlinear system
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there is a large number of them, it seems that the word diffraction is more often used.
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A common type of boundary value problem is (to put it abstractly) finding a function
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of those forms. The stimuli and responses could be numbers, functions, vectors,
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because many physical systems can be modeled as linear systems. For example, a
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Wave propagation in
Periodic Structures: Electric Filters and Crystal Lattices
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In any system with waves, the waveform at a given time is a function of the
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to develop analytical elements capable of being combined in a single model.
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Superposition principle is only available for linear systems. However, the
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allows a well-defined meaning to be given to the relative phases of the
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can be modeled as a linear system where the input stimulus is the
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2740:, 4th edition, Oxford, UK: Oxford University Press, p. 14.
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1607:{\displaystyle {\dot {x}}=Ax+B(u_{1}+u_{2}),\qquad x(0)=x_{0}.}
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By superposition principle, the system can be decomposed into
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702:, a principal task is to compute how a certain type of wave
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This decomposition can help to simplify controller design.
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No-one has ever been able to define the difference between
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in a light wave. The value of this parameter is called the
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391:, time-varying signals, or any other object that satisfies
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of the system. In many cases (for example, in the classic
170:. Superposition can be defined by two simpler properties:
52:(diagonal lines) from a distant source and waves from the
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2463:, superposition is used to solve for beam and structure
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imply that the (possibly time-varying) distributions of
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that satisfies the superposition principle is called a
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Later it became accepted, largely through the work of
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Nevertheless, on the topic of quantum superposition,
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2628:, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361.
2475:, the superposition principle is applied to the
403:Relation to Fourier analysis and similar methods
259:{\displaystyle F(x_{1}+x_{2})=F(x_{1})+F(x_{2})}
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1107:representing different types of classical pure
607:than the component variations; this is called
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444:Fourier analysis is particularly common for
2643:, Cambridge University Press, Nov 8, 2012.
345:on the beam and the output response is the
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329:This principle has many applications in
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2468:dynamic response of a linear structure.
371:The superposition principle applies to
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2817:Applied Partial Differential Equations
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1962:
926:{\displaystyle C_{j}\in {\textbf {C}}}
545:Wave diffraction vs. wave interference
1398:while the boundary values superpose:
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603:, the summed variation has a smaller
522:of the wave and the wave itself is a
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2641:Introduction to the Physics of Waves
1245:would be an operator that restricts
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2738:The Principles of Quantum Mechanics
2679:The Principles of Quantum Mechanics
1097:two-level quantum mechanical system
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549:With regard to wave superposition,
452:is described as a superposition of
83:, and rotation without translation.
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1077:{\displaystyle |\psi _{i}\rangle }
961:{\displaystyle |\psi _{i}\rangle }
808:{\displaystyle |\phi _{j}\rangle }
773:{\displaystyle |\psi _{i}\rangle }
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1509:Consider a simple linear system:
1183:with some boundary specification
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2487:. This principle is used in the
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37:For the geologic principle, see
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1979:
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933:. The equivalence class of the
611:. In other cases, such as in a
591:Interference (wave propagation)
2800:, McGraw–Hill, New York, p. 2.
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1148:that satisfies some equation
381:linear differential equations
2834:Superposition of sound waves
2551:Additive state decomposition
1880:additive state decomposition
1505:Additive state decomposition
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575:Yet another source concurs:
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671:Departures from linearity
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375:linear system, including
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731:projective Hilbert space
609:destructive interference
514:in a sound wave, or the
48:Superposition of almost
2853:Superposition principle
2489:analytic element method
1214:{\displaystyle G(y)=z.}
1134:Boundary-value problems
1103:) is also known as the
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89:superposition principle
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207:
195:
194:
165:
163:
162:
157:
91:, also known as
21:
2908:
2907:
2903:
2902:
2901:
2899:
2898:
2897:
2873:
2872:
2843:
2838:
2827:
2810:
2808:Further reading
2805:
2804:
2792:
2788:
2780:
2776:
2764:
2760:
2748:
2744:
2734:Dirac, P. A. M.
2732:
2728:
2689:
2685:
2675:Dirac, P. A. M.
2673:
2669:
2653:
2649:
2639:Tim Freegarde,
2638:
2634:
2623:
2619:
2612:
2608:
2603:
2599:
2594:
2547:
2532:Joseph Lagrange
2516:
2496:process control
2479:of two or more
2449:magnetic fields
2413:
2386:
2382:
2373:
2369:
2361:
2358:
2357:
2343:
2342:
2321:
2317:
2314:
2302:
2298:
2278:
2274:
2267:
2266:
2262:
2253:
2249:
2242:
2241:
2237:
2236:
2232:
2220:
2216:
2204:
2200:
2190:
2184:
2173:
2172:
2171:
2168:
2167:
2158:
2154:
2136:
2132:
2129:
2117:
2113:
2098:
2094:
2082:
2078:
2068:
2062:
2051:
2050:
2049:
2045:
2043:
2040:
2039:
2023:
2020:
2019:
1999:
1995:
1961:
1960:
1956:
1955:
1951:
1936:
1932:
1923:
1919:
1890:
1889:
1887:
1884:
1883:
1857:
1853:
1844:
1840:
1832:
1829:
1828:
1814:
1813:
1792:
1788:
1785:
1776:
1772:
1760:
1756:
1746:
1740:
1729:
1728:
1727:
1724:
1723:
1714:
1710:
1692:
1688:
1685:
1676:
1672:
1660:
1656:
1646:
1640:
1629:
1628:
1627:
1623:
1621:
1618:
1617:
1595:
1591:
1563:
1559:
1550:
1546:
1517:
1516:
1514:
1511:
1510:
1507:
1501:
1472:
1468:
1459:
1455:
1437:
1433:
1415:
1411:
1403:
1400:
1399:
1364:
1360:
1351:
1347:
1315:
1311:
1293:
1289:
1281:
1278:
1277:
1188:
1185:
1184:
1153:
1150:
1149:
1142:
1136:
1105:Poincaré sphere
1065:
1061:
1056:
1054:
1051:
1050:
1033:
1029:
1027:
1024:
1023:
1006:
1002:
1000:
997:
996:
979:
975:
973:
970:
969:
949:
945:
940:
938:
935:
934:
917:
916:
907:
903:
901:
898:
897:
874:
870:
865:
858:
854:
853:
847:
831:
827:
822:
820:
817:
816:
796:
792:
787:
785:
782:
781:
761:
757:
752:
750:
747:
746:
737:. According to
696:
690:
673:
663:
634:
593:
587:
551:Richard Feynman
547:
488:
478:
405:
366:linear operator
323:
274:
271:
270:
247:
243:
225:
221:
203:
199:
190:
186:
178:
175:
174:
168:linear function
142:
139:
138:
42:
35:
28:
23:
22:
15:
12:
11:
5:
2906:
2896:
2895:
2893:Systems theory
2890:
2885:
2871:
2870:
2856:
2842:
2841:External links
2839:
2837:
2836:
2831:
2825:
2811:
2809:
2806:
2803:
2802:
2786:
2774:
2758:
2742:
2726:
2699:(2): 185–195.
2683:
2667:
2647:
2632:
2617:
2606:
2596:
2595:
2593:
2590:
2589:
2588:
2583:
2578:
2573:
2568:
2563:
2558:
2553:
2546:
2543:
2539:Joseph Fourier
2528:Leonhard Euler
2520:LĂ©on Brillouin
2515:
2512:
2511:
2510:
2503:
2492:
2469:
2457:
2425:
2422:linear circuit
2412:
2409:
2394:
2389:
2385:
2381:
2376:
2372:
2368:
2365:
2341:
2338:
2335:
2332:
2329:
2324:
2320:
2316:
2313:
2310:
2305:
2301:
2297:
2294:
2291:
2287:
2281:
2277:
2270:
2265:
2261:
2256:
2252:
2245:
2240:
2235:
2231:
2228:
2223:
2219:
2215:
2212:
2207:
2203:
2199:
2196:
2193:
2191:
2187:
2180:
2177:
2170:
2169:
2166:
2161:
2157:
2153:
2150:
2147:
2144:
2139:
2135:
2131:
2128:
2125:
2120:
2116:
2112:
2109:
2106:
2101:
2097:
2093:
2090:
2085:
2081:
2077:
2074:
2071:
2069:
2065:
2058:
2055:
2048:
2047:
2027:
2007:
2002:
1998:
1994:
1991:
1988:
1985:
1982:
1978:
1974:
1970:
1964:
1959:
1954:
1950:
1947:
1944:
1939:
1935:
1931:
1926:
1922:
1918:
1915:
1912:
1909:
1906:
1903:
1897:
1894:
1865:
1860:
1856:
1852:
1847:
1843:
1839:
1836:
1812:
1809:
1806:
1803:
1800:
1795:
1791:
1787:
1784:
1779:
1775:
1771:
1768:
1763:
1759:
1755:
1752:
1749:
1747:
1743:
1736:
1733:
1726:
1725:
1722:
1717:
1713:
1709:
1706:
1703:
1700:
1695:
1691:
1687:
1684:
1679:
1675:
1671:
1668:
1663:
1659:
1655:
1652:
1649:
1647:
1643:
1636:
1633:
1626:
1625:
1603:
1598:
1594:
1590:
1587:
1584:
1581:
1578:
1574:
1571:
1566:
1562:
1558:
1553:
1549:
1545:
1542:
1539:
1536:
1533:
1530:
1524:
1521:
1503:Main article:
1500:
1497:
1483:
1480:
1475:
1471:
1467:
1462:
1458:
1454:
1451:
1448:
1445:
1440:
1436:
1432:
1429:
1426:
1423:
1418:
1414:
1410:
1407:
1387:
1384:
1381:
1378:
1375:
1372:
1367:
1363:
1359:
1354:
1350:
1346:
1343:
1339:
1335:
1332:
1329:
1326:
1323:
1318:
1314:
1310:
1307:
1304:
1301:
1296:
1292:
1288:
1285:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1172:
1169:
1166:
1163:
1160:
1157:
1135:
1132:
1073:
1068:
1064:
1059:
1036:
1032:
1009:
1005:
982:
978:
957:
952:
948:
943:
915:
910:
906:
885:
882:
877:
873:
868:
861:
857:
850:
846:
842:
839:
834:
830:
825:
804:
799:
795:
790:
769:
764:
760:
755:
692:Main article:
689:
686:
672:
669:
666:
665:
660:
657:
654:
653:
649:
648:
644:
643:
636:
589:Main article:
586:
583:
546:
543:
477:
474:
404:
401:
393:certain axioms
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
255:
250:
246:
242:
239:
236:
233:
228:
224:
220:
217:
214:
211:
206:
202:
198:
193:
189:
185:
182:
155:
152:
149:
146:
115:, then input (
97:linear systems
26:
9:
6:
4:
3:
2:
2905:
2894:
2891:
2889:
2886:
2884:
2881:
2880:
2878:
2869:at Wiktionary
2868:
2867:
2861:
2857:
2854:
2849:
2845:
2844:
2835:
2832:
2828:
2822:
2818:
2813:
2812:
2799:
2795:
2794:Brillouin, L.
2790:
2784:
2778:
2772:
2771:0-13-301458-4
2768:
2762:
2756:
2755:0-07-252036-1
2752:
2746:
2739:
2735:
2730:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2687:
2680:
2676:
2671:
2665:
2661:
2657:
2656:Kramers, H.A.
2651:
2645:
2642:
2636:
2630:
2627:
2624:N. K. VERMA,
2621:
2615:
2610:
2601:
2597:
2587:
2584:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2548:
2542:
2540:
2535:
2533:
2529:
2525:
2521:
2518:According to
2508:
2507:linearization
2504:
2501:
2497:
2493:
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2458:
2455:
2454:heat equation
2450:
2446:
2442:
2438:
2434:
2430:
2426:
2423:
2419:
2415:
2414:
2408:
2405:
2392:
2387:
2383:
2379:
2374:
2370:
2366:
2363:
2339:
2336:
2330:
2322:
2318:
2311:
2303:
2299:
2292:
2289:
2285:
2279:
2275:
2263:
2259:
2254:
2250:
2238:
2233:
2229:
2226:
2221:
2217:
2213:
2210:
2205:
2201:
2197:
2194:
2192:
2185:
2178:
2175:
2164:
2159:
2155:
2151:
2145:
2137:
2133:
2126:
2118:
2114:
2107:
2104:
2099:
2095:
2091:
2088:
2083:
2079:
2075:
2072:
2070:
2063:
2056:
2053:
2025:
2005:
2000:
1996:
1992:
1986:
1980:
1976:
1972:
1968:
1957:
1952:
1948:
1945:
1937:
1933:
1929:
1924:
1920:
1913:
1910:
1907:
1904:
1901:
1895:
1892:
1881:
1876:
1863:
1858:
1854:
1850:
1845:
1841:
1837:
1834:
1810:
1807:
1801:
1793:
1789:
1782:
1777:
1773:
1769:
1766:
1761:
1757:
1753:
1750:
1748:
1741:
1734:
1731:
1720:
1715:
1711:
1707:
1701:
1693:
1689:
1682:
1677:
1673:
1669:
1666:
1661:
1657:
1653:
1650:
1648:
1641:
1634:
1631:
1614:
1601:
1596:
1592:
1588:
1582:
1576:
1572:
1564:
1560:
1556:
1551:
1547:
1540:
1537:
1534:
1531:
1528:
1522:
1519:
1506:
1496:
1481:
1473:
1469:
1465:
1460:
1456:
1449:
1446:
1438:
1434:
1427:
1424:
1416:
1412:
1405:
1385:
1382:
1379:
1373:
1370:
1365:
1361:
1357:
1352:
1348:
1341:
1333:
1330:
1327:
1324:
1316:
1312:
1305:
1302:
1294:
1290:
1283:
1275:
1271:
1266:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1233:would be the
1232:
1228:
1224:
1208:
1205:
1202:
1196:
1190:
1170:
1167:
1161:
1155:
1147:
1141:
1131:
1129:
1125:
1121:
1117:
1112:
1110:
1106:
1102:
1098:
1094:
1091:to represent
1090:
1085:
1066:
1062:
1034:
1030:
1007:
1003:
980:
976:
950:
946:
913:
908:
904:
883:
875:
871:
859:
855:
848:
844:
840:
832:
828:
797:
793:
762:
758:
744:
740:
736:
732:
728:
723:
721:
717:
713:
709:
708:wave function
705:
701:
695:
685:
683:
679:
661:
658:
656:
655:
651:
650:
646:
645:
641:
632:
631:
624:
620:
618:
614:
610:
606:
602:
598:
592:
581:
576:
572:
567:
563:
560:
554:
552:
542:
540:
539:wave equation
536:
532:
527:
525:
521:
517:
513:
504:
497:
496:standing wave
492:
487:
486:Wave equation
483:
473:
471:
467:
463:
459:
455:
451:
447:
442:
440:
436:
432:
427:
425:
421:
417:
413:
408:
400:
398:
394:
390:
389:vector fields
386:
382:
378:
374:
369:
367:
363:
359:
355:
352:
348:
344:
340:
336:
332:
327:
322:
303:
297:
294:
291:
285:
282:
276:
269:
248:
244:
237:
234:
226:
222:
215:
212:
204:
200:
196:
191:
187:
180:
173:
169:
150:
144:
137:
132:
130:
126:
122:
118:
114:
110:
106:
102:
98:
94:
90:
82:
78:
74:
70:
63:
59:
55:
51:
46:
40:
33:
19:
2866:interference
2865:
2816:
2797:
2789:
2777:
2761:
2745:
2737:
2729:
2696:
2692:
2686:
2678:
2670:
2650:
2640:
2635:
2625:
2620:
2609:
2600:
2581:Interference
2536:
2530:and then by
2517:
2473:hydrogeology
2406:
1877:
1615:
1508:
1273:
1269:
1267:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1230:
1145:
1143:
1123:
1113:
1109:polarization
1089:Bloch sphere
1086:
742:
734:
726:
724:
697:
674:
616:
608:
597:interference
594:
578:
574:
569:
565:
559:interference
556:
548:
528:
509:
462:polarization
443:
428:
409:
406:
372:
370:
328:
133:
128:
124:
120:
116:
112:
108:
107:, and input
104:
100:
92:
88:
86:
2566:Convolution
2481:water wells
2465:deflections
2461:engineering
470:plane waves
454:plane waves
335:engineering
268:homogeneity
77:translation
50:plane waves
2877:Categories
2592:References
1093:pure state
896:where the
704:propagates
613:line array
397:vector sum
347:deflection
172:additivity
2721:121930907
2293:ϕ
2290:−
2230:ϕ
2179:˙
2108:ϕ
2057:˙
2026:ϕ
1949:ϕ
1896:˙
1735:˙
1635:˙
1523:˙
1374:⋯
1338:⇒
1328:⋯
1235:Laplacian
1072:⟩
1063:ψ
956:⟩
947:ψ
914:∈
881:⟩
872:ϕ
845:∑
838:⟩
829:ψ
803:⟩
794:ϕ
768:⟩
759:ψ
664:of phase
635:waveform
605:amplitude
520:amplitude
458:frequency
420:amplitude
416:sinusoids
62:Linearity
2796:(1946).
2736:(1958).
2677:(1958).
2545:See also
2477:drawdown
2445:electric
2441:currents
1111:states.
733:, not a
633:combined
524:function
512:pressure
136:function
81:rotation
79:without
2701:Bibcode
2514:History
2485:aquifer
2437:charges
2429:physics
2420:, in a
1116:Kramers
652:wave 2
647:wave 1
553:wrote:
531:sources
362:Laplace
358:Fourier
331:physics
73:Rolling
56:of the
2823:
2769:
2753:
2719:
2662:
2018:where
1253:, and
735:vector
383:, and
321:scalar
2888:Waves
2783:p. 39
2717:S2CID
2356:with
1827:with
1225:with
1120:Dirac
1101:qubit
1095:of a
739:Dirac
450:light
446:waves
424:phase
58:ducks
2821:ISBN
2767:ISBN
2751:ISBN
2660:ISBN
2447:and
2439:and
1272:and
815:as:
680:and
484:and
482:Wave
422:and
360:and
343:load
339:beam
333:and
319:for
266:and
87:The
54:wake
2709:doi
2494:In
2471:In
2459:In
2427:In
2416:In
1122:: "
741:: "
729:in
727:ray
698:In
373:any
131:).
2879::
2715:.
2707:.
2697:23
2695:.
2541:.
2431:,
1265:.
1241:,
1229:,
1084:.
684:.
619:.
472:.
460:,
441:.
379:,
326:.
134:A
127:+
119:+
60:.
2829:.
2723:.
2711::
2703::
2509:.
2502:.
2456:.
2393:.
2388:2
2384:x
2380:+
2375:1
2371:x
2367:=
2364:x
2340:0
2337:=
2334:)
2331:0
2328:(
2323:2
2319:x
2312:,
2309:)
2304:d
2300:y
2296:(
2286:)
2280:2
2276:x
2269:T
2264:c
2260:+
2255:1
2251:x
2244:T
2239:c
2234:(
2227:+
2222:2
2218:u
2214:B
2211:+
2206:2
2202:x
2198:A
2195:=
2186:2
2176:x
2165:,
2160:0
2156:x
2152:=
2149:)
2146:0
2143:(
2138:1
2134:x
2127:,
2124:)
2119:d
2115:y
2111:(
2105:+
2100:1
2096:u
2092:B
2089:+
2084:1
2080:x
2076:A
2073:=
2064:1
2054:x
2006:,
2001:0
1997:x
1993:=
1990:)
1987:0
1984:(
1981:x
1977:,
1973:)
1969:x
1963:T
1958:c
1953:(
1946:+
1943:)
1938:2
1934:u
1930:+
1925:1
1921:u
1917:(
1914:B
1911:+
1908:x
1905:A
1902:=
1893:x
1864:.
1859:2
1855:x
1851:+
1846:1
1842:x
1838:=
1835:x
1811:0
1808:=
1805:)
1802:0
1799:(
1794:2
1790:x
1783:,
1778:2
1774:u
1770:B
1767:+
1762:2
1758:x
1754:A
1751:=
1742:2
1732:x
1721:,
1716:0
1712:x
1708:=
1705:)
1702:0
1699:(
1694:1
1690:x
1683:,
1678:1
1674:u
1670:B
1667:+
1662:1
1658:x
1654:A
1651:=
1642:1
1632:x
1602:.
1597:0
1593:x
1589:=
1586:)
1583:0
1580:(
1577:x
1573:,
1570:)
1565:2
1561:u
1557:+
1552:1
1548:u
1544:(
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1538:+
1535:x
1532:A
1529:=
1520:x
1482:.
1479:)
1474:2
1470:y
1466:+
1461:1
1457:y
1453:(
1450:G
1447:=
1444:)
1439:2
1435:y
1431:(
1428:G
1425:+
1422:)
1417:1
1413:y
1409:(
1406:G
1386:,
1383:0
1380:=
1377:)
1371:+
1366:2
1362:y
1358:+
1353:1
1349:y
1345:(
1342:F
1334:0
1331:=
1325:=
1322:)
1317:2
1313:y
1309:(
1306:F
1303:=
1300:)
1295:1
1291:y
1287:(
1284:F
1274:G
1270:F
1263:R
1259:y
1255:z
1251:R
1247:y
1243:G
1239:R
1231:F
1209:.
1206:z
1203:=
1200:)
1197:y
1194:(
1191:G
1171:0
1168:=
1165:)
1162:y
1159:(
1156:F
1146:y
1099:(
1067:i
1058:|
1035:j
1031:C
1008:j
1004:C
981:j
977:C
951:i
942:|
919:C
909:j
905:C
884:,
876:j
867:|
860:j
856:C
849:j
841:=
833:i
824:|
798:j
789:|
763:i
754:|
498:.
324:a
307:)
304:x
301:(
298:F
295:a
292:=
289:)
286:x
283:a
280:(
277:F
254:)
249:2
245:x
241:(
238:F
235:+
232:)
227:1
223:x
219:(
216:F
213:=
210:)
205:2
201:x
197:+
192:1
188:x
184:(
181:F
154:)
151:x
148:(
145:F
129:Y
125:X
121:B
117:A
113:Y
109:B
105:X
101:A
41:.
34:.
20:)
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