单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,*,BASIC AOUSTICAS(6),Transverse Motion-The Vibrating String,BASIC AOUSTICAS(6)Transver,1,Vibrations of extended systems,In the previous chapter it was assumed that the mass moves as a rigid body so that it could be considered concentrated at a single point.,However,most vibrating bodies are not so simple.A loudspeaker has its mass distributed over its surface so that the cone does move as a unit.A piano sting.,Vibrations of extended systems,2,A flexible string under tension provides the easiest example for visualizing how waves work and developing physical concepts and techniques for their study.,The vibrating string is interesting both for its own sake(as a source of sound on a guitar or violin)and as a model for the motion of other systems.,We study free motion of a string.The procedures we use will apply in our later study of other kinds of waves.,A flexible string under tensio,3,FUNDAMENTALS-OF-ACOUSTICS6-声学基础(英文版教学课件),4,FUNDAMENTALS-OF-ACOUSTICS6-声学基础(英文版教学课件),5,Initial disturbance at t=0,Separate disturbance at t,1,0,Separate disturbance at t,2,t,1,Propagation of a transverse disturbance along a stretched string,Initial disturbance at t=0 Sep,6,It is observed that the speed of propagation of all small displacements is independent of the shape and amplitude of the initial displacement and depends only on the mass per unit length of the string and its tension,Experiment and theory show that this seed is given by,Where c is in m/s,T is the tension in,N,and,p,l,is the mass per unit length of the string in kg/m.,It is observed that the speed,7,The equation of motion,Assume a string of uniform linear density,p,l,and negligible stiffness,stretched to a tension T great enough that the effects of gravity can be neglected.,Also assume that there are no dissipative forces(such as those associated with friction or with the radiation of acoustic energy),The equation of motionAssume a,8,Fig.A isolates an infinitesimal element of the string with equilibrium position x and equilibrium length dx.,When the string is at rest,the tensions at x and at x+dx are precisely equal in magnitude and opposite in direction,making zero total force.,Fig.A,Fig.A isolates an infinitesi,9,If,(the transverse displacement of this element from its equilibrium position)is small,the tension T remains constant along the string and the difference between the,Component of the tension at the two ends of the element is,If (the transverse displ,10,If,is small,We get,Applying the Taylors series expansion,If is small,We getApplying th,11,Since the mass of the element is,p,l,dx and its acceleration in the,direction is,Newtons law gives,Then yields the equation of motion,where the constant c,2,is defined by,Since the mass of the element,12,GENERAL SOLUTION OF THE EQUATIONG OF MOTION,Equation(2-1)is a second-order,partial differential equation.Its complete solution contains two arbitrary functions.The most general solution is,are completely arbitrary functions of arguments(ct-x)and(ct+x),respectively.Possible examples of such arbitrary functions include log(ct+x),(ct+x),2,sinw(t+x/c),et al.,GENERAL SOLUTION OF THE EQUATI,13,We can prove that any function of argument(ct-x)is a solution of the wave equation(2-1).Similarly,it can be shown that f,2,(ct+x)is also a solution.The sum of these two functions is the complete general solution of the equation of motion,.,We can prove that any function,14,Consider the solution f,1,(ct-x).At time t,1,the transverse displacement of the string is given by f,1,(ct-x).As suggested by Fig.B,x,1,x,2,At a later time t,2,the shape of the string will be given by f,1,(ct,2,-x,2,),Consider the solution f1(ct-x),15,The particular transverse displacement f,1,(ct,1,-x,1,)of the string that was found at x,1,when t=t,1,must be found at a position x,2,when t=t,2,where,ct,1,-x,1,=ct,2,-x,2,Thus,this particular displacement has moved a distance,x,2,-x,1,=c(t,2,-t,1,)to the right.,The particular transverse disp,16,Since the particular displacement chosen was arbitrary,any transverse displacement must move to the right with the same speed.,This means that the shape of the disturbance remains unchanged and travels along the string to the right at a constant speed c.,The function f,1,(ct-x)represents a wave traveling in the+x direction,called wave function.,Since the particular displacem,17,STANDING WAVES,Consider now a string of finite length L.Describing all motions of this string in terms of traveling waves remains possible in principle.,Because of repeated reflections between the two ends,that is usually not the most helpful description.,We find it more convenient to study standing waves.,STANDING WAVESConsider now a s,18,FUNDAMENTALS-OF-ACOUSTICS6-声学基础(英文版教学课件),19,We limit to solutions that meet the proper boundary conditions.,Suppose specifically that both ends of the string are fixed,that is:,Substitute the initial conditions,and obtain,We limit to solutions that mee,20,The only way to