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Superposition principle
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<h2 id="relation-to-fourier-analysis-and-similar-methods">Relation to Fourier analysis and similar methods</h2> <p>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. </p><p>For example, in <a href="/facts/Fourier_analysis/sUA8JP6q">Fourier analysis</a>, the stimulus is written as the superposition of infinitely many <a href="/facts/Sine_wave/mD1CB09M">sinusoids</a>. 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 <a href="/facts/Amplitude/aEckuXJM">amplitude</a> and <a href="/facts/Phase_(waves)/cpGtuVll">phase</a>.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses. </p><p>As another common example, in <a href="/facts/Green%27s_function/rKxKKPcp">Green's function analysis</a>, the stimulus is written as the superposition of infinitely many <a href="/facts/Impulse_function/V8PjRP8T">impulse functions</a>, and the response is then a superposition of <a href="/facts/Impulse_response/l4vaE5er">impulse responses</a>. </p><p>Fourier analysis is particularly common for <a href="/facts/Wave/fU0HH6St">waves</a>. For example, in electromagnetic theory, ordinary <a href="/facts/Light/jeofpMn4">light</a> is described as a superposition of <a href="/facts/Plane_wave/0aDUbNhQ">plane waves</a> (waves of fixed <a href="/facts/Frequency/QUSbPaW8">frequency</a>, <a href="/facts/Polarization_(waves)/fP33m0eD">polarization</a>, and direction). As long as the superposition principle holds (which is often but not always; see <a href="/facts/Nonlinear_optics/maerngXd">nonlinear optics</a>), the behavior of any light wave can be understood as a superposition of the behavior of these simpler <a href="/facts/Plane_wave/0aDUbNhQ">plane waves</a>. </p> <h2 id="wave-superposition">Wave superposition</h2> <p class="note">Further information: <a href="/facts/Wave/fU0HH6St">Wave</a> and <a href="/facts/Wave_equation/J2ksD2Az">Wave equation</a></p> <p>Waves are usually described by variations in some parameters through space and time—for example, height in a water wave, <a href="/facts/Pressure/eyc37GRG">pressure</a> in a sound wave, or the <a href="/facts/Electromagnetic_field/MacraS8y">electromagnetic field</a> in a light wave. The value of this parameter is called the <a href="/facts/Amplitude/aEckuXJM">amplitude</a> of the wave and the wave itself is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> specifying the amplitude at each point. </p><p>In any system with waves, the waveform at a given time is a function of the <a href="/facts/Wave_equation/J2ksD2Az">sources</a> (i.e., external forces, if any, that create or affect the wave) and <a href="/facts/Initial_condition/sA4ohFP4">initial conditions</a> of the system. In many cases (for example, in the classic <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a>), 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.) </p> <h3>Wave diffraction vs. wave interference</h3> <p>With regard to wave superposition, <a href="/facts/Richard_Feynman/32vGFrtB">Richard Feynman</a> wrote:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <blockquote><p>No-one has ever been able to define the difference between <a href="/facts/Interference_(wave_propagation)/lI5f78I4">interference</a> 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 there is a large number of them, it seems that the word diffraction is more often used.</p></blockquote> <p>Other authors elaborate:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <blockquote><p>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), the effect is called diffraction. That is the difference between the two phenomena is [a matter] of degree only, and basically, they are two limiting cases of superposition effects.</p></blockquote> <p>Yet another source concurs:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <blockquote><p>In as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter [Fraunhofer diffraction] is, therefore, a continuation of Chapter 8 [Interference]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important 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.</p></blockquote> <h3>Wave interference</h3> <p class="note">Main article: <a href="/facts/Interference_(wave_propagation)/lI5f78I4">Interference (wave propagation)</a></p> <p>The phenomenon of <a href="/facts/Interference_(wave_propagation)/lI5f78I4">interference</a> 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 <a href="/facts/Noise-canceling_headphones/zpwbXLkc">noise-canceling headphones</a>, the summed variation has a smaller <a href="/facts/Amplitude/aEckuXJM">amplitude</a> than the component variations; this is called <i>destructive interference</i>. In other cases, such as in a <a href="/facts/Line_array/0grMqJa4">line array</a>, the summed variation will have a bigger amplitude than any of the components individually; this is called <i>constructive interference</i>. </p> <table><tbody><tr><td>combined waveform</td><td colspan="2" rowspan="3"></td></tr><tr><td>wave 1</td></tr><tr><td>wave 2</td></tr><tr><td></td><td>Two waves in phase</td><td>Two waves 180° out of phase</td></tr></tbody></table> <h3>Departures from linearity</h3> <p>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 phenomena that arise when the superposition principle does not exactly hold, see the articles <a href="/facts/Nonlinear_optics/maerngXd">nonlinear optics</a> and <a href="/facts/Nonlinear_acoustics/sCN6t1yS">nonlinear acoustics</a>. </p> <h3>Quantum superposition</h3> <p class="note">Main article: <a href="/facts/Quantum_superposition/4z1UJg9o">Quantum superposition</a></p> <p>In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, a principal task is to compute how a certain type of wave <a href="/facts/Wave_propagation/PXhWIt7k">propagates</a> and behaves. The wave is described by a <a href="/facts/Wave_function/KgM5ykjY">wave function</a>, and the equation governing its behavior is called the <a href="/facts/Schr%C3%B6dinger_equation/3csgSIWF">Schrödinger equation</a>. A primary approach to computing the behavior of a wave function is to write it as a superposition (called "<a href="/facts/Quantum_superposition/4z1UJg9o">quantum superposition</a>") of (possibly infinitely many) other wave functions of a certain type—<a href="/facts/Stationary_state/HBQvbb6s">stationary states</a> 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.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>The projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a <i>ray</i> in <a href="/facts/Projective_Hilbert_space/JEA7AWel">projective Hilbert space</a>, not a <i>vector</i>. According to <a href="/facts/Paul_Dirac/6bt5yRdk">Dirac</a>: "<i>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</i> [italics in original]."<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> 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 | ψ i ⟩ {\displaystyle |\psi _{i}\rangle } into superposition of component ket vectors | ϕ j ⟩ {\displaystyle |\phi _{j}\rangle } as: | ψ i ⟩ = ∑ j C j | ϕ j ⟩ , {\displaystyle |\psi _{i}\rangle =\sum _{j}{C_{j}}|\phi _{j}\rangle ,} where the C j ∈ C {\displaystyle C_{j}\in {\textbf {C}}} . The equivalence class of the | ψ i ⟩ {\displaystyle |\psi _{i}\rangle } allows a well-defined meaning to be given to the relative phases of the C j {\displaystyle C_{j}} .,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> but an absolute (same amount for all the C j {\displaystyle C_{j}} ) phase change on the C j {\displaystyle C_{j}} does not affect the equivalence class of the | ψ i ⟩ {\displaystyle |\psi _{i}\rangle } . </p><p>There are exact correspondences between the superposition presented in the main on this page and the quantum superposition. For example, the <a href="/facts/Bloch_sphere/CvgaFMNf">Bloch sphere</a> to represent <a href="/facts/Pure_state/7jbxd7Dx">pure state</a> of a <a href="/facts/Two-level_system/hQrw3gR9">two-level quantum mechanical system</a> (<a href="/facts/Qubit/WveeEubw">qubit</a>) is also known as the <a href="/facts/Bloch_sphere/CvgaFMNf">Poincaré sphere</a> representing different types of classical pure <a href="/facts/Polarization_(waves)/fP33m0eD">polarization</a> states. </p><p>Nevertheless, on the topic of quantum superposition, <a href="/facts/Hans_Kramers/M44sqHjf">Kramers</a> writes: "The principle of [quantum] superposition ... has no analogy in classical physics". According to <a href="/facts/Paul_Dirac/6bt5yRdk">Dirac</a>: "<i>the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory</i> [italics in original]."<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase, they actually mean phase translation symmetry derived from <a href="/facts/Time_translation_symmetry/bFCNU4UD">time translation symmetry</a>, which is also applicable to classical states, as shown above with classical polarization states. </p> <h2 id="boundary-value-problems">Boundary-value problems</h2> <p class="note">Further information: <a href="/facts/Boundary-value_problem/QEfmUqmP">Boundary-value problem</a></p> <p>A common type of boundary value problem is (to put it abstractly) finding a function <i>y</i> that satisfies some equation F ( y ) = 0 {\displaystyle F(y)=0} with some boundary specification G ( y ) = z . {\displaystyle G(y)=z.} For example, in <a href="/facts/Laplace%27s_equation/8n1YkmFS">Laplace's equation</a> with <a href="/facts/Dirichlet_problem/uHErgNCb">Dirichlet boundary conditions</a>, <i>F</i> would be the <a href="/facts/Laplacian/kov9u3PT">Laplacian</a> operator in a region <i>R</i>, <i>G</i> would be an operator that restricts <i>y</i> to the boundary of <i>R</i>, and <i>z</i> would be the function that <i>y</i> is required to equal on the boundary of <i>R</i>. </p><p>In the case that <i>F</i> and <i>G</i> 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: F ( y 1 ) = F ( y 2 ) = ⋯ = 0 ⇒ F ( y 1 + y 2 + ⋯ ) = 0 , {\displaystyle F(y_{1})=F(y_{2})=\cdots =0\quad \Rightarrow \quad F(y_{1}+y_{2}+\cdots )=0,} while the boundary values superpose: G ( y 1 ) + G ( y 2 ) = G ( y 1 + y 2 ) . {\displaystyle G(y_{1})+G(y_{2})=G(y_{1}+y_{2}).} 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. </p> <h2 id="additive-state-decomposition">Additive state decomposition</h2> <p class="note">Main article: <a href="/facts/Additive_state_decomposition/fSEWDeHG">Additive state decomposition</a></p> <p>Consider a simple linear system: x ˙ = A x + B ( u 1 + u 2 ) , x ( 0 ) = x 0 . {\displaystyle {\dot {x}}=Ax+B(u_{1}+u_{2}),\qquad x(0)=x_{0}.} </p><p>By superposition principle, the system can be decomposed into x ˙ 1 = A x 1 + B u 1 , x 1 ( 0 ) = x 0 , x ˙ 2 = A x 2 + B u 2 , x 2 ( 0 ) = 0 {\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}}} with x = x 1 + x 2 . {\displaystyle x=x_{1}+x_{2}.} </p><p>Superposition principle is only available for linear systems. However, the <a href="/facts/Additive_state_decomposition/fSEWDeHG">additive state decomposition</a> can be applied to both linear and nonlinear systems. Next, consider a nonlinear system x ˙ = A x + B ( u 1 + u 2 ) + ϕ ( c T x ) , x ( 0 ) = x 0 , {\displaystyle {\dot {x}}=Ax+B(u_{1}+u_{2})+\phi \left(c^{\mathsf {T}}x\right),\qquad x(0)=x_{0},} where ϕ {\displaystyle \phi } is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into x ˙ 1 = A x 1 + B u 1 + ϕ ( y d ) , x 1 ( 0 ) = x 0 , x ˙ 2 = A x 2 + B u 2 + ϕ ( c T x 1 + c T x 2 ) − ϕ ( y d ) , x 2 ( 0 ) = 0 {\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}}} with x = x 1 + x 2 . {\displaystyle x=x_{1}+x_{2}.} </p><p>This decomposition can help to simplify controller design. </p> <h2 id="other-example-applications">Other example applications</h2> <ul><li>In <a href="/facts/Electrical_engineering/YYXW1KLW">electrical engineering</a>, in a <a href="/facts/Linear_circuit/eDxQYWo2">linear circuit</a>, 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.</li> <li>In <a href="/facts/Physics/a7aH2y08">physics</a>, <a href="/facts/Maxwell%27s_equations/LfJ1drHS">Maxwell's equations</a> imply that the (possibly time-varying) distributions of <a href="/facts/Electric_charge/xxsBfmlB">charges</a> and <a href="/facts/Electric_current/Dc1xdnUX">currents</a> are related to the <a href="/facts/Electric_field/tt4MooBs">electric</a> and <a href="/facts/Magnetic_field/Ta8Ev588">magnetic fields</a> 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 <a href="/facts/Heat_equation/2JPPr5GG">heat equation</a>.</li> <li>In <a href="/facts/Engineering/RFVg8H3I">engineering</a>, superposition is used to solve for beam and structure <a href="/facts/Deflection_(engineering)/xroy2lX2">deflections</a> 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).<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></li> <li>In <a href="/facts/Hydrogeology/4HEanFa0">hydrogeology</a>, the superposition principle is applied to the <a href="/facts/Drawdown_(hydrology)/L4PNdlde">drawdown</a> of two or more <a href="/facts/Water_well/aPFvzk3V">water wells</a> pumping in an ideal <a href="/facts/Aquifer/UwG6ksdG">aquifer</a>. This principle is used in the <a href="/facts/Analytic_element_method/GxJAoxvm">analytic element method</a> to develop analytical elements capable of being combined in a single model.</li> <li>In <a href="/facts/Process_control/tmjQhchm">process control</a>, the superposition principle is used in <a href="/facts/Model_predictive_control/FBDVEmjk">model predictive control</a>.</li> <li>The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by <a href="/facts/Linearization/ohYSeQR3">linearization</a>.</li></ul> <h2 id="history">History</h2> <p>According to <a href="/facts/L%C3%A9on_Brillouin/8AAtqyeH">Léon Brillouin</a>, the principle of superposition was first stated by <a href="/facts/Daniel_Bernoulli/NRs5Ne5r">Daniel Bernoulli</a> in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> and then by <a href="/facts/Joseph-Louis_Lagrange/bTQUaSTv">Joseph Lagrange</a>. 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.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p><p>Later it became accepted, largely through the work of <a href="/facts/Joseph_Fourier/sKvYF2fz">Joseph Fourier</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Additive_state_decomposition/fSEWDeHG">Additive state decomposition</a></li> <li><a href="/facts/Beat_(acoustics)/zKhaKnMy">Beat (acoustics)</a></li> <li><a href="/facts/Coherence_(physics)/NF1Rgdoa">Coherence (physics)</a></li> <li><a href="/facts/Convolution/PVPrdz9J">Convolution</a></li> <li><a href="/facts/Green%27s_function/rKxKKPcp">Green's function</a></li> <li><a href="/facts/Impulse_response/l4vaE5er">Impulse response</a></li> <li><a href="/facts/Interference_(wave_propagation)/lI5f78I4">Interference</a></li> <li><a href="/facts/Quantum_superposition/4z1UJg9o">Quantum superposition</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Haberman, Richard (2004). <i>Applied Partial Differential Equations</i>. Prentice Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-065243-0.</li> <li><a href="http://www.acoustics.salford.ac.uk/feschools/waves/super.htm">Superposition of sound waves</a></li></ul> <h2 id="external-links">External links</h2> <ul><li> Media related to Superposition principle at Wikimedia Commons</li> <li> The dictionary definition of interference at Wiktionary</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [1] <a href="https://books.google.com/books?id=S-JFAgAAQBAJ&dq=feynman+interference+and+diffraction&pg=SA30-PA1" target="_blank">https://books.google.com/books?id=S-JFAgAAQBAJ&dq=feynman+interference+and+diffraction&pg=SA30-PA1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>N. K. VERMA, Physics for Engineers, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [2] <a href="https://books.google.com/books?id=kY-7AQAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA361" target="_blank">https://books.google.com/books?id=kY-7AQAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA361</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Tim Freegarde, Introduction to the Physics of Waves, Cambridge University Press, Nov 8, 2012. [3] <a href="https://books.google.com/books?id=eMMgAwAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA106" target="_blank">https://books.google.com/books?id=eMMgAwAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA106</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Quantum Mechanics, Kramers, H.A. publisher Dover, 1957, p. 62 ISBN 978-0-486-66772-0 <a href="/wiki/Hendrik_Anthony_Kramers" target="_blank">/wiki/Hendrik_Anthony_Kramers</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford, UK: Oxford University Press, p. 17. <a href="/wiki/Paul_Adrien_Maurice_Dirac" target="_blank">/wiki/Paul_Adrien_Maurice_Dirac</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195. Bibcode:1993FoPh...23..185S. doi:10.1007/BF01883623. S2CID 121930907. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford, UK: Oxford University Press, p. 14. <a href="/wiki/Paul_Adrien_Maurice_Dirac" target="_blank">/wiki/Paul_Adrien_Maurice_Dirac</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192 ISBN 0-07-252036-1 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785 ISBN 0-13-301458-4 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Topics on Numerics for Wave Propagation, Basque Center for Applied Mathematics, 2012, Spain, p. 39 <a href="http://www.bcamath.org/documentos_public/courses/1_Course2012Chapter1WavesHistoryApplications.pdf" target="_blank">http://www.bcamath.org/documentos_public/courses/1_Course2012Chapter1WavesHistoryApplications.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Brillouin, L. (1946). Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York, p. 2. <a href="/wiki/L%C3%A9on_Brillouin" target="_blank">/wiki/L%C3%A9on_Brillouin</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> </ol>