P2 - Longitudinal Mechanical Waves (10pts]: The wave equation is a pervasive aspect of the dynamics of physical systems.

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P2 Longitudinal Mechanical Waves 10pts The Wave Equation Is A Pervasive Aspect Of The Dynamics Of Physical Systems 1 (442.36 KiB) Viewed 35 times

P2 - Longitudinal Mechanical Waves (10pts]: The wave equation is a pervasive aspect of the dynamics of physical systems. Indeed, it governs the linear regime of most of the waves you are familiar with: light, sound, water waves, earthquakes, and, in special conditions, even entropy waves in superfluid helium (a.k.a. temperature waves, or second sound) and concentration waves in solutions (stationary waves of the latter kind are responsible for the emergence of segments in animal embryos: vertebrae in vertebrates, segments in insects, rings in annelids, etc.). In this problem you, will show that the wave equation applies to the propagation of longitudinal waves in a system composed by several identical masses, each with mass m, connected by identical massless springs with rest length lo and elastic constant k, when consecutive masses are separated by a distance l. Imagine that the spring chain is held in tension because it is stretched between two fixed constraints at its opposite ends. Let's assume that the masses are along the z axis, and consider three arbitrary consecutive masses, indexed as n -1, n, and n + 1. Let's call An-1(t), An(t), and An+1(t) their longitudinal displacement with respect to their equilibrium positions, An(t) = zn(t) – 27,0, etc., as shown in the figure below m k Anti n-1 נגנונות n - 1 n n+1 (a) [3pts] Using Newton's second equation (in 1D), Fn(t) = man(t) = mdIn(t)/dt, the fact that the force acting on a particle, Fn, is given by the sum of the spring forces acting on that particle, and Hooke's law for the elastic force exerted by a spring, f = -k(l – lo), demonstrate that the displacement An(t) is governed by the following differential equation d'An(t) m =k(An+1(t) – 2An(t) + An-1(t)] dt2 (b) [2pts) Assume now that all the displacements are much smaller than the separation between consecutive masses, An(t) <l. Then, instead of identifying A by the index n, we can also identify the corresponding mass by its equilibrium position, z = nl, thus replacing An(t) with A(z,t), and An +1(t) with A(z+l,t). Using the lowest-order finite-difference approximation to the derivative, F'(2) = [f(z + 1/2) – f(2 - 1/2)]/l, show that it is possible to convert the Eq. (1) into the wave equation d'A(z,t) m & A(z,t) 0. (2) dz2 k12 dt2 (C) 3pts Show that, for any sufficiently regular) function F(x), the two functions F-(2,t) and F+(z,t), defined as F+(2,t) = F(z – cit), F-(2,t) = F(z+at), (3) are solutions to Eq. (2), provided that c = {vk/m. Demonstrate that the barycenter of these two waves, (Z+)(t) = / dzz |F+(2,t)?, moves with velocity c in the positive/negative z direction, respectively. (d) [2pts] Consider the special case of F(2) = A cos(kz), where k is a real positive parameter. Demonstrate that the corresponding wave F+(2,t) is periodic in space, i.e., that there is a non-zero smallest real quantity such that F+(2+1, t) = F+(2,t). Demonstrate that F+(z,t) is periodic in time, i.e., that there is a non-zero smallest real quantity T such that F+(z,t+T) = F+(2,t). Find the expression of and T in terms of k and cl.