Vlny nekonečnej struny - PDF

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Vlny nekonečnej struny Fázová rýchlosť Hustota struny na jednotku dĺžky Sila napnutia struny v základnom stave Rovinná vlna 1 2 ), ( ), ( η ξ η η ξ ξ η ξ q t t q t + = Hľadanie riešenia vlnovej rovnice

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Vlny nekonečnej struny Fázová rýchlosť Hustota struny na jednotku dĺžky Sila napnutia struny v základnom stave Rovinná vlna 1 2 ), ( ), ( η ξ η η ξ ξ η ξ q t t q t + = Hľadanie riešenia vlnovej rovnice nekonečnej struny v -v ), ( ), ( η ξ η η ξ ξ η ξ q x x q x + = Rýchlosť modulácie Struna budená v bode x=0 signálom Z(t) Riešenie vlnovej rovnice Modulácia 4 Grupová rýchlosť Ak k 1 -k 2 0 Grupová rýchlosť je vždy menšia ako rýchlosť svetla. Tento výrok nie je pravdivý. 5 Fourierov rozvoj inpulzu 0 t/2 T δω = ω 1 = 2π T t f ( t) = j=0 Aj δω cos δω jδωt δω dω 7 Fourierov integrál δ Pre impulz na obrázku t=1 0.6 A(f)/A(0) f=1/ t Akú šírku v priestore bude mať rozruch vyvolaný impulzom? f t 8 10 Grupová rýchlosť vlnového balíka 0 ) ) ( ) cos(( 2 ) ) ( ) cos(( 2 ), ( 0 0 t x k k A t x k k A t x q j n j n j n j n ω ω ω ω ω ω = Disperzne prostredia 11 Putovanie vlnového balíka 12 Príklad šírenie EM vĺn v ionosfére Pre nedispezné vlny (t.j. ω=vk pre všetky ω) sa tvar vlnového balíka nemení a grupová rýchlosť v g =v. Vo všeobecnosti ω=ω (k) V ionosfére Ionosféra 80 km nad povrchom zeme ω p ~10 MHz 13 Grupová a fázová rýchlosť, disperzia H.J. Pain, THE PHYSICS OF VIBRATIONS AND WAVES, str Rýchlosť signálu This means, therefore, that neither the group nor the phase velocity can be used to describe the speed at which the information in the pulse travels, and we need to define it by another velocity - the signal velocity . This is defined as the speed at which the front of the pulse travels. According to relativity, this speed can never exceed the speed of light in a vacuum because, if it did, it would be equivalent to sending the signal backwards in time, which would violate causality. Grupová rýchlosť zodpovedá rýchlosti signálu len v neabsorbujúcich prostrediach. L.J. Wang, A. Kuzmich and A. Dogariu, Gain-assisted superluminal light propagation, Nature 406, (2000). Their website, contains additional material, including an animation much like Fig. 4 of the present paper. (táto web stránka v súčasnosti neexistuje) 15 Záporná grupová rýchlosť a grupová rýchlosť väčšia ako rýchlosť svetla L.J. Wang, A. Kuzmich and A. Dogariu, Gain-assisted superluminal light propagation, Nature 406, (2000). Nicolas Brunner et al., Direct Measurement of Superluminal GroupVelocity and Signal Velocity in an Optical Fiber, Phys. Rev Lett. 93, (2004) 16 Max Planck Light str. 193 We shall now fix our attention on the special case where the dispersive body, which we shall suppose stretches from x = 0 to x = is perfectly unexcited at first and that from a certain, moment of time onwards, say t = 0, a singly periodic wave of frequency ω is continuously incident on its surface in the normal direction. With what velocity will the disturbance propagate itself into the body? We may be tempted to assume at first sight that the wave simply propagates itself into the body with the velocity c/n, as is the case with a permanently periodic wave. But this is not so. For here we are concerned with a non-periodic wave which becomes deformed as it advances. The complete solution of this problem has been given by Sommerfeld, who resolved the waves into a Fourier integral. But we can find the answer to the above question without making special calculations if we apply the line of reasoning which was fully discussed in 86. The primary wave also advances in the dispersive body with the velocity c. The secondary waves due to the vibrating dipoles do the same. But the result of this joint action is not the same as in the case of a permanently periodic wave. 17 Planck Light str. 193 For the dipoles do not vibrate periodically; they were at rest initially and, on account of their inertia, they will only gradually be made to vibrate, the later the greater their coordinate x, and these vibrations, strictly speaking, will acquire their constant character only after an infinitely long period of time and so give rise to the singly periodic wave-motion whose velocity of propagation is c/n. So long as this is not the case the primary wave will not be neutralized by the secondary waves and so it manifests itself as a disturbance which advances in the dispersive body with the velocity c. Since on the other hand the secondary waves, no matter what their form may be, also propagate themselves with the velocity c it follows that the desired velocity with which the head of the wave advances into the body, the front velocity, is always equal to c. In contrast c with the front velocity is the phase-velocity, -, which a permanently singly periodic wave has; in future we shall denote it by u : u=c/n. We must not regard it as an inherent contradiction to the theory that the phase-velocity u may also be greater than the front velocity c (n l). For the wave is not singly periodic and the head of the wave that penetrates into the body has no constant phase. Hence there can be no question of the head of the wave which advances with the front velocity being passed by the hinder part of the wave which advances with the phase-velocity. It further follows from this that the velocity of propagation of a light-signal emitted into the body, the signal velocity, can at most be equal to c. This maximum value is attained in an ideal detector, that is, a receiver which reacts to even the smallest disturbance. 18 Fourierov integrál v reálnom tvare 19 F Fourierov integrál v komplexnom tvare 1 f ( t) F( ) e i ω t d 2 = ω ω π ( ω ) 1 ( ) 2 iωt = f t e dt π Pozor na iné definície Fourierovho integrálu a Fourierových obrazov!!!! 20 Impedancia prostredia 0 21 Impedancia prostredia m=ρ x x zdroj 0 3. Newtonov zákon 22 Výkon dodávaný zo zdroja, tok energie vlnením Bežiace vlny Analógia: Ak q je náboj P=ZI 2 =UI 23 Rozhranie dvoch prostredí l i l i+1 q q q i+1 i-1 i x 0 Z + q+ ( x, t) t = Z q ( x, t) t 24 25 Odraz vlnenia opísaného vlnovou funkciou Ψ na rozhraní dvoch prostredí v 1D R T Spojitosť vlnovej funkcie 26 28 Prispôsobovanie impedancii Ψ = Ψ 1 Ψ 12 = 23 = R i Ae R T ( k x ω ) 1 t i Ae e ( k R x ω ) 1 t T ik2l ik2l i( k1x ωt ) i2k2l 12 + R23e = Ae Odrazená vlna má byť rovná nule Ψ12 + Ψ23 = 0 e 0 Z 1 Z 2 Z 3 x=0 L x Reálna časť=0 Imaginárna časť=0 Z Z 1 1 Z + Z 2 2 = Z Z 2 2 Z + Z 3 3 L = λ 30 Pozdĺžne a priečne vlny - zvuk 31 Zvuk Stredná voľná dráha musí byť oveľa menšia ako vlnová dĺžka ( l λ.) inak by sa minimá a maximá v hustote zvuku takmer okamžite vyhladili. Silný zvuk zodpovedá zmenám tlaku iba 10-5 atm, t.j. P e P 0 32 0 x x 33 Pohybová rovnica x 34 Newtonove odvodenie rýchlosti zvuku izotermické šírenie zvuku Newtonov model Experiment 35 Laplaceov adiabatický model šírenia sa zvuku 36 Intenzita zvuku Intenzita zvuku energia prechádzajúca cez jednotkovú plochu za jednotku času Reč zodpovedá asi Jednotka intenzity zvuku Prach bolesti I z0 Prach počuteľnosti I z0 Ucho pracuje v logaritmickej mierke Ako citlivé sú naše uši? Sú schopné registrovať výchylky bubienka na úrovni rozmeru atómu 37 Pozdĺžne a priečne vlny 38 39 40 Lineárny kvadrupólový zdroj Laterálny kvadrupólový zdroj Kmitanie dipólu a ladičky 42
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