Massachusetts Institute of Technology Physics 8 03 Fall 2004 Problem Set 4 Due Friday October 8 2004 at 4 PM Reading Assignment French pages 135 152 201 230 238 243 253 264 Beke Barrett pages 117 139 Problem 4 1 Travelling pulse Do Problem 7 12 from French A P Vibrations and Waves New York N Y W W Norton and Company January 1 1971 ISBN 0393099369 The gure shows a pulse on a string of length 100 m with xed ends The pulse is travelling to the right without any change of shape at a speed of 40 m sec a Make a clear sketch showing how the transverse velocity of the string varies with distance along the string at the instant when the pulse is in the position shown b What is the maximum transverse velocity of the string approximately c If the total mass of the string is 2 kg what is the tension T in it d Write an equation for y x t that numerically describes sinusoidal waves of wavelength 5 m and amplitude 0 2 m travelling in the negative x direction on a very long string made of the same material and under the same tension as above Problem 4 2 Travelling pulse Do Problem 7 13 from French A P Vibrations and Waves New York N Y W W Norton and Company January 1 1971 ISBN 0393099369 A pulse travelling along a stretched string is described by the following equation y x t b2 b3 2x ut 2 a Sketch the graph of y against x for t 0 b What are the speed of the pulse and its direction of travel c The transverse velocity of a given point of the string is de ned by vy y t Calculate vy as a function of x for the instant t 0 and show by means of a sketch what this tells us about the motion of the pulse during a short time t 1 Problem 4 3 Pulse re ection at a boundary Two strings with mass per unit length 1 0 1 kg m and 2 0 3 kg m respectively are jointed seamlessly They are under tension T 20 N A travelling wave of a triangular shape shown in the gure is moving to the right along the lighter string The tick marks set the scale of the pulse width a Find the re ection and transmission coe cients at the interface including the signs b Make a careful sketch of the total deformation of the string when the incident pulse has its peak exactly at the interface Indicate how you arrived at your answer on your sketch c Make a careful sketch of the total deformation of the string when both the re ected and transmitted pulses have moved away from the interface d What is unphysical about the shape of this pulse Be quantitative Problem 4 4 Boundary conditions on a string A very long string of mass density and tension T is attached to a small hoop with negligible mass The hoop slides on a greased vertical rod and experiences a vertical force Fy b y t when it moves a Apply Newton s law to the hoop to nd the boundary condition at the end of the string Express your result in terms of the partial derivatives of y x t at the location of the rod b Show that the boundary condition is satis ed by an incident pulse f x vt and a re ected pulse g x vt Find g in terms of f c Show that your result has the correct behavior in the limits b 0 the string is free to slip and b the string is rmly clamped 2 Problem 4 5 Boundary conditions in a pipe Pressure oscillations in a hollow pipe of length L are described by the wave equation 2p 0 2 p 2 z t2 where p is the over pressure over and above the one atmosphere ambient pressure 0 is the density of the gas in the pipe is the bulk modulus and z is the longitudinal direction along the pipe Assuming a solution of the form p z t A coskz B sinkz cos t nd all the unknowns A B k and for the case where the pipe is open at both ends and p z L 2 t 0 p0 Problem 4 6 Normal modes of discrete vs continuous systems Referring to the diagram below you are given a uniform string of length L and total mass M that is stretched to a tension T You are also given a set of 5 beads each of mass M 5 spaced at equal intervals on a massless string with tension T and total length L a Use boundary conditions to derive a general expression for the frequencies of the normal modes of oscillation of the string Give the frequencies in terms of n T L and M b Write down the frequencies of the ve lowest normal modes of transverses oscillation of the string c Compare the numerical values of these normal mode frequencies with the normal mode frequencies of ve beads on the massless string Hint You do not have to solve the frequencies of the beads You may use French eqns 5 25 and 5 26 d Sketch the ve lowest normal modes you found for the massive string Sketch also the ve normal modes of the massless string with ve beads e In a sentence or two discuss the di erences if any in the normal modes of the two systems considered here 3