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Y d gdydushxu r hdudusjfh fee jsjjdhrhzhu Dr hsjudh d jzjj fee j DE h DE jej s jz?
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Y d gdydushxu r hdudusjfh fee jsjjdhrhzhu Dr hsjudh d jzjj fee j DE h DE jej s jz?
Consider a flexible, elastic genous string or thread of length l,which undergoes relatively small transverse vibrations. For instance, itcould be a guitar string or a plucked violin string. At a given instantt, the string might look as shown in Figure 2. Assume that it remainsin a plane. Let u(x, t) be its displacement from equilibrium position attime t and position x. Because the string is perfectly flexible, the tension(force) is directed tangentially along the string (Figure 3). Let T(x, t) bethe magnitude of this tension vector. Let ρ be the density (mass per unitlength) of the string. It is a constant because the string is geneous.We shall write down Newton’s law for the part of the string betweenany two points at x = x0 and x = x1. The slope of the string at x1 isFigure 212 CHAPTER 1 WHERE PDEs COME FROMFigure 3ux(x1, t). Newton’s law F = ma in its longitudinal (x) and transverse (u)components isT1 + u2xx1x0= 0 (longitudinal)T ux 1 + u2xx1x0= x1x0ρutt dx (transverse)The right sides are the components of the mass times the accelerationintegrated over the piece of string. Since we have assumed that themotion is purely transverse, there is no longitudinal motion.Now we also assume that the motion is small—more specifically,that |ux | is quite small. Then 1 + u2x may be replaced by 1. This isjustified by the Taylor expansion, actually the binomial expansion, 1 + u2x = 1 + 12 u2x +···where the dots represent higher powers of ux. If ux is small, it makessense to drop the even smaller quan y u2x and its higher powers. Withthe square roots replaced by 1, the first equation then says that T isconstant along the string. Let us assume that T is independent of t aswell as x. The second equation, differentiated, says that(Tux )x = ρutt .That is,utt = c2ux x where c =Tρ . (2)This is the wave equation. At this point it is not clear why c is definedin this manner, but shortly we’ll see that c is the wave speed. 1.3 FLOWS, VIBRATIONS, AND DIFFUSIO
whoops, mods plz delete last post....that was supposed to be a pm to fabbs but i clicked the wrong button
Is this a thread about the triangle offense?
I'm the only 3 time best poster in ST history in case everyone forgot
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