Lista de Exercicios Equacao de Ondas Boyce-Diprima

Lista 14 Equação de ondas extraídos do livro de Boyce e DiPrima Problems Consider an elastic string of length L whose ends are held fixed The string is set in motion with no initial velocity from an initial position ux0 fx In each of Problems 1 through 4 carry out the following steps Let L 10 and a 1 in parts b through d a Find the displacement uxt for the given initial position fx G b Plot uxt versus x for 0 x 10 and for several values of r between t 0 and t 20 G c Plot uxt versus t for 0 t 20 and for several values of x G d Construct an animation of the solution in time for at least one period e Describe the motion of the string in a few sentences 1 fx 2xL 0 x L2 2L xL L2 x L 2 fx 4xL 0 x L4 1 L4 x 3L4 4L xL 3L4 x L 3 fx 8xLx2L3 4 fx 0 0 x L2 1 1 L2 1 x L2 1 assume L 2 0 L2 1 x 1 Consider an elastic string of length L whose ends are held fixed The string is set in motion from its equilibrium position with an initial velocity utx 0 gx In each of Problems 5 through 8 carry out the following steps Let L 10 and a 1 in parts b through d a Find the displacement urt for the given gx G b Plot urt versus r for 0 r 10 and for several values of r between r 0 and r 20 G c Plot uxt versus t for 0 t 20 and for several values of x G d Construct an animation of the solution in time for at least one period e Describe the motion of the string in a few sentences 5 gx 2xL 0 x L 2 2LxL L2 x L 6 gx 4xL 0 x L4 1 L4 x 3L4 4LxL 3L4 x L 7 gx 8xLx2L3 8 gx 0 0 x L2 1 1 L2 1 x L2 1 assume L 2 0 L2 1 x 1 9 If an elastic string is free at one end the boundary condition to be satisfied there is that ut 0 Find the displacement uxr in an elastic string of length L fixed at x 0 and free at x L set in motion with no initial velocity from the initial position ux 0 fx where f is a given function Hint Show that the fundamental solutions for this problem satisfying all conditions except the nonhomogeneous initial condition are ultx r sinλn x cosλn at where λn 2n 1π2L n 1 2 Compare this problem with Problem 15 of Section 106 pay particular attention to the extension of the initial data out of the original interval 0 L 10 Consider an elastic string of length L The end x 0 is held fixed while the end x L is free thus the boundary conditions are ut0 r 0 and uxL t 0 The string is set in motion with no initial velocity from the initial position ux 0 fx where fx 0 0 x L2 1 1 L2 1 x L2 1 assume L 2 0 L2 1 x L a Find the displacement ux r G b With L 10 and a 1 plot u versus x for 0 x 10 and for several values of r Pay particular attention to values of t between 3 and 7 Observe how the initial disturbance is reflected at each end of the string G c With L 10 and a 1 plot u versus t for several values of x G d Construct an animation of the solution in time for at least one period e Describe the motion of the string in a few sentences G 11 Suppose that the string in Problem 10 is started instead from the initial position fx 8xL x2L3 Follow the instructions in Problem 10 for this new problem 12 Dimensionless variables can be introduced into the wave equation a2 uxx utt in the following manner a Let s xL and show that the wave equation becomes a2 uxx L2 utt b Show that La has the dimensions of time and therefore can be used as the unit on the time scale Let τ atL and show that the wave equation then reduces to uxx utt Problems 13 and 14 indicate the form of the general solution of the wave equation and the physical significance of the constant a 13 a Show that the wave equation a2 uxx utt can be reduced to the form utt 0 by the change of variables ξ x at η x at b Show that ux t can be written as ux t ϕx at ψx at where ϕ and ψ are arbitrary functions 14 G a Plot the value of ϕx at for t 0 1a 2a and 5oa if ϕc sin c Note that for any t 0 the graph of y ϕx at is the same as that of y ϕx when τ 0 but displaced a distance at in the positive x direction Thus a represents the velocity at which a disturbance moves along the string b What is the interpretation of ϕx at 15 A steel wire 5 ft in length is stretched by a tensile force of 50 lb The wire has a weight per unit length of 0026 lbft a Find the velocity of propagation of transverse waves in the wire b Find the natural frequencies of vibration c If the tension in the wire is increased how are the natural frequencies changed Are the natural modes also changed edisciplinasuspbr