FUNDAMENTALS-OF-ACOUSTICS(6)-声学基础(英文版教学ppt课件)

BASIC AOUSTICAS(6)Transverse Motion-The Vibrating String BASIC AOUSTICAS(6)Transver1 1Vibrations of extended systemsIn 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 systems2 2A 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 tensio3 3FUNDAMENTALS-OF-ACOUSTICS(6)-声学基础(英文版教学ppt课件)4 4FUNDAMENTALS-OF-ACOUSTICS(6)-声学基础(英文版教学ppt课件)5 5Initial disturbance at t=0 Separate disturbance at t10 Separate disturbance at t2t1Propagation of a transverse disturbance along a stretched stringInitial disturbance at t=0 Sep6 6It 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 tensionExperiment and theory show that this seed is given by Where c is in m/s,T is the tension in N and pl is the mass per unit length of the string in kg/m.It is observed that the speed 7 7The equation of motionAssume a string of uniform linear density pl 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 a8 8Fig.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.AFig.A isolates an infinitesi9 9If (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 displ1010If If is small,We getApplying the Taylors series expansionIf is small,We getApplying th1111Since the mass of the element is pl dx and its acceleration in the direction is Newtons law givesThen yields the equation of motionwhere the constant c2 is defined bySince the mass of the element 1212GENERAL SOLUTION OF THE EQUATIONG OF MOTIONEquation(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 EQUATI1313We can prove that any function of argument(ct-x)is a solution of the wave equation(2-1).Similarly,it can be shown that f2(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 function1414Consider the solution f1(ct-x).At time t1 the transverse displacement of the string is given by f1(ct-x).As suggested by Fig.Bx1x2At a later time t2 the shape of the string will be given by f1(ct 2-x2)Consider the solution f1(ct-x)1515The particular transverse displacement f1(ct1-x1)of the string that was found at x1when t=t1must be found at a position x2 when t=t2 where ct1-x1=ct2-x2Thus,this particular displacement has moved a distance x2-x1=c(t2-t1)to the right.The particular transverse disp1616Since 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 f1(ct-x)represents a wave traveling in the+x direction,called wave function.Since the particular displacem1717STANDING WAVESConsider 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 s1818FUNDAMENTALS-OF-ACOUSTICS(6)-声学基础(英文版教学ppt课件)1919We 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 obtainWe limit to solutions that mee2020The only way to meet the first condition is to set A=0.Then the second condition allows the amplitude B to be anything as long as we require that ul/c be an integral multiple of.The only way to meet the first2121We use We use 2222Normal-mode frequencies that form a harmonic series as above are a very special feature of one-dimensional systems whose properties are uniform everywhere along their length.Normal-mode frequencies that f2323Any sum of these sinusoidal standing waves is also a solution of both the equation of motion and the boundary conditions.So we can represent very general motions of a string fixed at both ends by Any sum of these sinusoidal st2424HomeworkHow to determine the amplitudes and phase when initial conditions have been specified.(text book P71-72)HomeworkHow to determine the a2525