电脑控制弦音计

电脑控制弦音计 PASCO实验 Experiment 7 Sonometer 实验七 电脑控制弦音计 实验原理: Theory of Waves on a Stretched String 1.驻波 1. Standing Waves 一简单的正弦波在拉紧的金属线上传播,可以由方程式 A simple sine wave traveling along a taut =ysin2来描述.若金属线一端固定,波到达该端时将被反射回y,(x/,,t/n)1mstring can be described by the equation y=ysin2. If the string is fixed ,(x/,,t/n)1m来,反射波为: y= ysin2. ,(x/,，t/n)2mat one end, the wave will be reflected back 假设波幅足够小,未超出金属线的弹性限制,则叠加后的波形即为两波形之和: when it strikes that end, The reflected wave can y=y+y=ysin2 ,(x/,,t/n)12mbe described by the equation y= ysin2. Assuming the + ysin2 ,(x/,，t/n),(x/,，t/n)2mm 由恒等式: amplitudes of these waves are small enough so that the elastic limit of the string is not sinA+sinB=2sin(A+B)/2cos(B-A)/2, 上式可改写为: exceeded, the resultant waveform will be just y=2ysin(2. ,x/,)cos(2,t/n)mthe sum of the two waves: y=y+y=ysin2+ ,(x/,,t/n)12m 该方程具有一些特点:对一固定时间t,则金属线的波形为一正弦波,最大波oysin2 幅为2ycos(2.对一固定的x,金属线也现为谐振动,最大振幅为,(x/,，t/n),t/,)mmoo 2ysin(2. x= l/2,l,3l/2,2l,等时,波幅为0. ,x/,)Using the trigonometric identity: moo sinA+sinB=2sin(A+B)/2cos(B-A)/2, 该种波形即为驻波,因为金属线上并没有波形的传播.时间方向的驻波,其表This equation becomes: 现形式如图1. y=2ysin(2. ,x/,)cos(2,t/n)m This equation has some interesting characteristics .At a fixed time t, the shape of o the string is a sine wave with a maximum amplitude of 2ycos(2. At a fixed ,t/,)moFigure 1 position on the string, x, the string is o undergoing simple harmonic motion, with an amplitude of 2ysin(2,x/,). therefore, at mo points of the string where x=l/2,l,3l/2,2l,etc., o the amplitude of the oscillations will be zero. 该模式为驻波波胞.金属线上每一点上下振幅取决于该点波胞。最大振幅处 This waveform is called a standing wave 即为波腹,振幅为0处即为波节. because there is no propagation of the P7—1 waveform along the string .A time exposure of the standing wave would show a pattern something like the one in Figure 1. This pattern is called the envelope of the standing wave. Each point of the string oscillates up and down with its amplitude determined by the envelope. The points of maximum amplitude are called antinodes. The points of zero amplitude are called nodes. 2.共振 2.Resonance 以上建立在假设驻波为原始波与反射波的叠加的基础之上.事实上若 The analysis above assumes that the 金属线两端都固定,每个波在到达固定端时都将被反射.总的来说,叠加在一起的standing wave is formed by the superposition 反射波并非都同相,其波幅也很小.但对于某些振动频率,所有反射波都处于同一of an original wave and one reflected wave. In 相位,产生一振幅很高的驻波.这些频率即为共振频率. fact if the string is fixed at both ends, each wave will be reflected every time if reaches either end of the string. In general, the multiply reflected waves will not all is in phase, and the amplitude of the wave pattern will be small. However at certain frequencies of oscillation, 实验1研究线长与共振频率间的关系.共振产生时,通过对波长与线长的分all the reflected waves are in phase, resulting in 析，很容易得出这样的结论:共振产生时，线长与波长间的关系为线长为半波a very high amplitude standing wave. These 长的整数倍.即; n=1,2,3,4…则驻波的节点一定位于两固定端. ,,2L/nfrequencies are called resonant frequencies. In Experiment 1,the relationship between the length of the string and the frequencies at which resonance occurs is investigated. It is shown that the conditions for resonance are more easily understood in terms of the wavelength of the wave pattern, rather than in terms of the frequency. In general, resonance occurs when the wavelength (,) satisfies the condition: ; n=1,2,3,4… ,,2L/n Another way of stating this same relationship is to say that the length of the P7—2 string to equal to an integral number of half wavelengths. This means that the standing wave is such that a node of the wave pattern exists naturally at each fixed end of the string. 3.波传播速度 3. Velocity of Wave propagation 对一柔韧有弹性的金属线,波在金属线上的传播速度(V)由两个变量决定: Assuming a perfectly flexible, perfectly ),及金属线所受张力(T).关系式为: 金属线的线密度(,elastic string, the velocity of wave propagation T(V) on a stretched string depends on two V,,variables: the mass per unit length or linear density of the sting () and the tension of the , string (T). The relationship is given by the equation: T V,, 该公式与牛顿第二定律相似:描述了力,惯量,及线密度间的关系.但金属线 Without going into the derivation of this 的振动与只受一个力的简单刚体运动并不相同.(不论速度，加速度都是物体运equation, its basic form can be appreciated. The 动时所关注的量。但金属线上的波并没有加速度，这也许算一个合理的解释)。 equation is analogous to Newton’s Second law, providing a relationship between a measure of force and a measure of inertia, and linear density of the string. That the form of the two equations is not exactly the same to be expected. The motion of the string is considerably different than the motion of a simple rigid body acted on by a single force. (It could be asked whether velocity, rather than acceleration, is the right measure of motion to focus on. Since the waves on the string do not accelerate, this is at least a reasonable assumption.) P7—3 如果允许这种类推,则可以假设波速只由张力和线密度决定.空间分析可知 If the analogy with Newton’s Law is )来得出波速。 该方程是正确的。没有其他方法可由张力(T)及线密度(,accepted, and it is assumed that the wave velocity depends only on tension and linear density, dimensional analysis shows that the form of the equation must be as it is. There is no other way to combine tension (with units of -2-1MLT) with linear density (ML) to get -1velocity (LT). 该方程应通过实验来验证.实验2中不同的金属线有不同的线密度.力的大 Of course, the equation must be verified 小可通过改变悬挂物的质量或位置来改变.波长可通过共振模式时的频率来确experimentally. This is done in Experiment 2, ,此时我们就可以验证波速与线密度及线所受定.则波速可由下式得出: V=,,in which the linear density of the string is 张力的关系. varied by using different strings. The tension is varied using hanging weights on a lever arm .The wavelength is then measured by adjusting the frequency until a resonance pattern develops. The velocity can then be calculated using the relationship V=, ,, And the effects of tension and linear density on velocity can be determined. Experiment 1 Resonance Modes of a Stretched String 实验1 一段金属线的共振模式 实验目的: Objective: 了解驻波产生的原因。 Determine the relation between the 通过实验收集的数据，确定共振发生时金属线长与波长间的关系. wavelength of the wave and the string length when resonance occurs. 实验仪器及简介(图2): Equipment and the brief introduction: ,WA-9611弦音计 -WA-9611 Sonometer -砝码及其挂钩 -Mass and mass hanger -WA-9613 驱动/探测器 -WA-9613Driver/Detector Coils -示波器或计算机 -Dual trace oscilloscope -频率发生器 -Function generator capable of delivering 0.5 amp 本实验提供10根金属线,每两个一组,其线密度为: 10 wires (guitar strings), 2each of the following P7—4 diameters (liner densities): -0.010”(0.39gm/m) -0.010”(0.39gm/m) -0.014”(0.78gm/m) -0.014”(0.78gm/m) -0.017”(1.12gm/m) -0.017”(1.12gm/m) -0.020”(1.50gm/m) -0.020”(1.50gm/m) -0.022”(1.84gm/m) -0.022”(1.84gm/m) Figure 2 Figure 3 频率发生器的应用(图3): Operation of the Function Generator(see figure 3): 1(电源开关打到OFF 1.Flip the power ON/OFF switch to OFF. 2(连线，接通电源(所用电源为.110V，60Hz交流电，或220V,50Hz2.Plug the power cord into a well 交流电) grounded outlet of the appropriate voltage (110 VAC,60Hz,or 220V,50Hz) P7—5 3(将振幅调节逆时针旋转到底. 3.Turn the amplitude to zero by rotating the Amplitude knob counterclockwise as far as it will go. 4(将插头插入”output”,使频率发生器接入整个线路，输出所需信4.Using banana plug connectors or 号. hookup wire, connect the output of the generator to the circuit or device to which you want to apply the signal. 5(选择所需波形(正弦波或方波). 5.Set the Wave Shape switch to select a sine or square wave. 6(打开电源开关.指示灯会亮. 6.Flip the power switch on .The power switch will light up. 7(选择”Frequency Multiplier”旋钮到所需倍数.旋转大的刻度盘改变频7.Set the Frequency Multiplier knob to 率. the desired decade. Turn the large Frequency dial to vary the frequency within the selected decade. 8(测量输出频率: 8.To measure the output frequency to within 1%. (1)用数字万用表测量”DVM Output”电压. (1) Plug a digital voltmeter into the DVM (2)选择适当的量程.使测量范围为0到10V。 Output jacks using banana plugs. (3)万用表读数与选择量程相乘为输出频率.如”Frequency Multiplier” (2) Set the voltage range of the voltmeter 置为100Hz,万用表读数为5.75伏,则输出频率为575Hz。 so it will measure voltages from 0 to 10 volts. (3) To determine the output frequency, just multiply the reading on the voltmeter by the selected range. For example, if the multiplier knob is set to 100 Hz, and the voltmeter reads 5.75 volts, the output frequency is 575Hz. 软件的应用: Using the Science Workshop program 1(界面及仪器准备.在实验安装窗口,点击并拖动”analog sensor” 图像1. Prepare the interface and apparatus. 插头到”Analog Channels A or B”.由传感器菜单中选择”Sound Sensor”.其灵Start Science Workshop. In the Experiment 敏度将自动为X100.点击并拖动”Scope “数据显示图标至”Sound Sensor” Setup window, click and drag the analog sensor plug icon to Analog Channels A or B. Select “Sound Sensor” from the list of sensors. This will automatically set the Sensitivity to P7—6 X100.click and drag the Scope data display icon to the Sound Sensor icon. 2(调节”Scope”.其灵敏度及扫描速度由实验装置决定.例如:用金2. Adjust the Scope so it can display the 属线(017)作传质的弦音计,两桥间距60cm,水平张力杆上第二凹槽处悬挂signal from the sensor .The sensitivity and 1kg重量,驱动频率为130Hz,5V,应将电压分度调节到0.010或0.020v/div,扫sweep speed will depend on the experimental 描速度为1.00或2.00ms/div. setup. For example, if you are using the Sonometer with the medium gauge wire (0170,60cm between the bridges, a one kilogram mass on the second notch of the tensioning lever, and a driving frequency of 130Hz and 5 Volts, you may need to increase the volts/division (sensitivity) to about 0.010 or 0.020v/div and the sweep speed to 1.00 or 2.00 ms/div. 实验步骤: Procedure 1(按如下要求安装弦音计:两桥间距60cm,张力杆悬挂大约1kg质量.1.Set up the Sonometer as following 调节线调节旋钮使张力杆水平.将驱动器安旨在距其中一壳5cm处,探测器(Figure 4):Start with the bridges 60 cm apart. 安置在金属线中间(如图4).将线长,力(mg),线密度在表格1. Use any of the included strings and hang a mass of approximately 1 kg from the tensioning lever. Adjust the string adjustment knob so that the tensioning lever is horizontal. Position the driver coil approximately 5 cm from one of the bridges and position the detector near the center of the wire. Record the length, tension (mg), and linear density of the string in Table 1.1. 2(信号发生器产生一正弦波,调到约5mv/cm. 2.Set the signal generator to produce a sine wave and set the gain of the oscilloscope to approximately 5 mV/cm. 3(缓慢增加信号频率.收听弦音计的声音增强信号,或观察示波器的探3.Slowly increase the frequency of the 测信号的增强(第一次约在20,30Hz).引起弦线最大振动的频率为共振频signal to the driver coil. starting at 率.确定最小共振频率.该频率为共振基频.测量并将其纪录到表格1. approximately 25 Hz. Listen for an increase in the volume of the sound from the sonometer and /or an increase in the size of the detector Table 1 线长_____张力_________线密度______ signal on the oscilloscope screen. Frequencies P7—7 that result in maximum string vibration are 方式 共振频率 波腹 波节 resonant frequencies. Determine the lowest frequency at which resonance occurs. This is resonance in the first, or fundamental, mode. Measure this frequency and record it in Table 1. Figure 4 4(将探测器尽可能的靠近其中一桥,在缓慢移动探测器时观察示波4. Start with the detector as close as you 器,确定并在表1中记录下每个节点和波腹. can get it to one of the bridges. Watch the oscilloscope as you slide the detector slowly along the string. Locate and record the locations of each node and antinode. Record your results in Table 1. 5(继续增大频率,找到连续的共振频率(5-6个).与表格1.1中纪录每5. Continue increasing the frequency to 个波节和波腹的频率及位置. find successive resonant frequencies (at least five or six). Record the resonance frequency for each mode, and the locations of nodes and antinodes inTable1. 6(由结果来确定,并记录下每次共振模式下的波长. 6. From your results, determine and record the wavelength of each resonance pattern you discovered. 7(改变两桥间的线长,重新构造表格并重新测量(至少3根不同的7. Change the string length by moving one 线长). or both of the bridges. Construct a new data table and repeat your measurements for at least P7—8 three different string lengths. 注:驱动频率可能并不是金属线的振动频率，在示波器上可以很清楚的Note: The driving frequency of the signal 看到这一点。有可能是金属线振动频率的整数倍。 generator may not be the frequency at which the wire is vibrating .By using a dual trace oscilloscope , you can determine if the two frequencies are the same , or if the vibrating frequency is a multiple of the driving frequency. 实验分析: Analysis 应用实验数据确定频率增加时的共振波形，及波形与线长间的关Using your data, determine the shape of 系。画出任意线长的共振波形。波长与线长的关系是什么,你可以用数the successive resonance waveforms as the 学式子表示之。 frequency is increased. How do the wave 注:该实验中，当共振频率为基频的偶数倍(N)时,其在弦线上的表现shapes depend on the length of the string ? 形式为N/2个波腹，N/2+1个波节。 Sketch the resonance waveforms for an arbitrary string length. What relationship holds between the wavelength of the wave and the string length when resonance occurs? Can you state this relationship mathematically? Experiment 2 Velocity of Wave Propagation 试验2 波速的测量 实验目的: Objective: 通过对不同金属线，及相同金属线不同长度共振时相关量Determine the relation among the speed of 的测量，确定共振发生时波速与线所受张力及线密度间的关系. the wave(V),the wire tension(T)and the linear density(). , 实验装置: (同实验1) Equipment: -WA-9611 Sonometer -WA-9613 Driver/Detector Coils -Dual trace oscilloscope -Mass and mass hanger -Function generator capable of delivering 0.5 amp P7—9 实验步骤: Procedure: 1(按如下要求按装弦音计:两桥间距60cm,张力杆悬挂大约1.Set up the Sonometer as shown in Figure 1kg质量.调节线调节旋钮使张力杆水平.将驱动器安置在距其中4 一桥5cm处,探测器安置在金属线中间.. Set the bridges 60 cm a part .Use any of the included strings and hang a mass of approximately1kg from the tensioning lever. Adjust the string adjustment knot so that the tensioning lever is horizontal. Position the driver coil approximately 5 cm from one of the bridges and position the detector near the center of the wire. 2(信号发生器产生一正弦波,示波器调到约5mv/cm. 2.Set the signal generator to produce a sine wave and set the gain of the oscilloscope to approximately 5cm/cm. 3(缓慢增加信号频率(开始时大约1HZ).确定最小共振频率,3.Slowly increase the frequency of the 并纪录于表格2. driver coil, staring with a frequency of around 1 HZ. Determine the lowest frequency at which resonance occurs. Record this value in Table 2. 注:确定最低共振频率，其方法为缓慢滑动探测器，若为Note: To be sure you have found the 最低共振频率，则两桥间仅有一个波腹。 lowest resonant frequency, slide the detector coil the length of the string. The wave pattern should have just a single antinode located midway between the two bridges. . 4(于表格2中记录金属线的张力(T)和线密度(,)4.In Table 2.1,record the string tension (T) and the linear density of the string (. ,)张力由下面式子确定:张力=悬挂物质量x 张力杆凹槽数. 线密度手册中已给出. The tension is determined: multiply the weight of the hanging mass buy one, two, three, Table 2 张力 线密度 基 频 波速 four, or five, depending on which notch of the (,) (T) (V) (v) tensioning lever the mass is hanging from. The linear densities of the strings are given in the front of this manual. 5(通过改变悬挂物不同的位置改变金属线的张力,对五组5.Change the string tension by hanging the 不同的张力,重复步骤3和4. mass from a different notch. Repeat steps 3 and 4 for five different values of the string tension. P7—10 6(将线的张力设为中间值,用五根不同的线重复步骤3和46.Set the string tension to a midrange 的测量. value. Then repeat your measurements of step 3 and 4 using each of the five different strings. 实验分析: Anal age: 1(由你所测数据来确定金属线上波速与张力及线密度间的1.Using your measured string length, the 关系。 fundamental frequency, and the equation V=to determine the velocity of the wave on ,v the string for each value of tension and linear density that you used. 2(确定波速与张力间的关系，可由以下3种方法实现: 2.Determine the functional relationship between the speed of the wave and the wire tension. This can be accomplished using either of the following three methods. If you are not familiar with these procedures, you might want to try all three. 2,，A(画一V-T图，V为x轴。若该图并非一直线，可画TA. Plot a graph of V versus T, with Von 1/2T等，直到得到一直线为止。 the y-axis. If the graph is not a straight line ,try plotting V versus some power of T(such as 21/2T,T,etc)，until you get a straight line . p,则有lnV=plnT+lnk. p,kB(假设有这样的关系式:V=kTB. Assume that the functional relationship p为未知量。lnv与lnT成正比。我们就可以得到一斜率为is of the form V=kT. Then lnV=plnT+lnk, p=lnV/lnT的直线。Lnk为y轴截距。 where p and k are unknown constants. Then if lnV is plotted against the independent variable lnT, a straight line will be obtained having a slope p, where p is lnV/lnT and lnk is the y-intercept. C(其他许多计算器都有对V,T做对数衰减直线的方法，其C. Many calculators have the ability to do 实质与B相同。 power regressions or linear regressions on the logarithms of Vand T. This will accomplish essentially what the graph of method Bdid. )间3(用上面所提到的方法，确定波速(V)与线密度(,3.Using one of the methods above, 的关系。 determine the functional relationship of the speed of the wave (V)to the linear density of the string(,). P7—11 结论: Conclusions 金属线上振动模式的特点: Characterize the resonant modes of a vibrating wire. That is: 1。确定金属线长与波长之间的数学关系式.(实验1) 1. Determine a mathematical relationship that describes the wavelengths of the waves that from standing wave patterns in a wire of 来确定线长为L的金2(用问题1的答案及表达式V=,,length L (See Experiment 1). 属线的共振频率. 2.Use your answer to question 1,and the expression V=, to determine the resonant ,, 3(利用实验结果写出量T, 及L的表达式 ,,frequencies of a wire of length L. 3.Use your experimental results to write an expression for the frequencies of a vibrating wire in terms of T, and L. ,, P7—12