P3 - Transverse Mechanical Waves (10pts]: The mechanical waves considered in the previous problem are called longitudina

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P3 Transverse Mechanical Waves 10pts The Mechanical Waves Considered In The Previous Problem Are Called Longitudina 1 (312.03 KiB) Viewed 32 times

P3 - Transverse Mechanical Waves (10pts]: The mechanical waves considered in the previous problem are called longitudinal because the masses move back and forth in the same direction along which the wave propagates. However, it is also possible to displace the masses away from their equilibrium axis. For example, if we call z the equilibrium axis, we can displace the masses along the x direction (say, in the vertical direction in the plane of the figure), along the y direction (outgoing or incoming, with respect to the page), or along any other transverse diection (i.e., perpendicular to z). In this problem, you will show that also the dynamics of transverse displacements is governed by the wave equation. m k An- lu n+1 n-1 Anti n l m { (a) (3pts] Consider a displacement along x, Ac,n(t). As in the previous problem, show that, when the transverse displacement from the equilibrium position is small, Ar,n(t) < l obeys the differential equation d´An(t) = An+1(t) – 2An(t) + An-i(t) (4) dt2 where T = kl-lo) is the tension along the springs, i.e., the force exerted by the springs when the system is at rest. (b) (3pts) Following the same approximation and passages in points 2.b-c, show that transverse displacements also obey the wave equation dA (z,t) 1 d'Az(z,t) = 0, (5) d22 c? dt2 find the expression for ci in terms of the problem parameters, T, m, and l (notice that, in general, ci #ci), and show that any function of the form A (2, t) = F(2-cit) propagates in the positive z direction with speed ci. (c) (2 pts) Let's write the combination of two transverse displacements, one along r and one along y, as A(z, t) = ĉAr(2,t) + ġAy(z,t). (6) Of course, A. = 0 (transverse motion). Now, let's consider one such specific displacement, moving towards positive z, whose z and y components are sinusoidal and in quadrature, A+(2,t) = Ao [î cos(wt – kz) ġ sin(wt – kz)). (7) Compute the time derivative of this displacement dĀ/dt. Taking as reference the origin of coordi- nates (or any point along the axis, for that matter), compute the angular momentum per unit of length of the mass chain. Consider that the mass per unit of length is m/l, and the angular momentum for an infinitesimal element of the chain, with mass dm, is dỈ = dm Ảx dĀ(F,t) (8) dt 3 (d) [2pts] Draw the trajectory of A+(0,t) and Ă_(0,t) in the xy plane, indicating with an arrow the direction of motion.