Université des Sciences et Technologies de Lille

Niveau: Supérieur, Master, Bac+5
Université des Sciences et Technologies de Lille 1 2011/2012 Master degree in Mathematical Engineering Refresher Course in Physics Semester 3 Waves, Propagation Phenomena Preliminary remark Verify that a complex valued function ? is a solution of ∆? = 1 c2 ∂2? ∂t2 if and only Re(?) and Im(?) are. Exercise 1 Consider a conducting and vibrating string. Denote ? its mass by unit of length. Suppose that the motion takes place only the y and z directions. Suppose that there is a constant density of current j in the string and that there is a constant magnetic field B on the x direction. Recall that the Laplace force on a piece of string d~l is given by ~F = jd~l ? ~B. (1) Determine the propagation equation. (2) Let (y(x, t), z(x, t)) = (y0eı(kx??t), z0eı(kx??t)) be a solution of this equation. Deter- mine the dispersion relation. (3) Express y0 in terms of z0. Exercise 2 Consider the membrane of a drum that is initially in the plane (xOy) and that it is slightly perturbed on the z axis (and not on the x and y axis). Denote µ the mass per unit of surface.

• ∂t ∂x


• propagation phenomena


• group velocity


• dispersion relation


• persion relation between


• constant magnetic field


• klein- gordon equation


• t0f ?


• tension force


• denote ?