驻波与克拉尼图形

驻波与克拉尼图形 实验8 驻波与克拉尼图形 驻波是由两列传播方向相反而振幅与频率都相同的波叠加而成的。驻波有一维驻波、二维驻波等。按某些频率激发弦乐器的弦线振动，弦线就会形成一维驻波。对于话筒的膜片、锣鼓鼓面，它们形成的驻波分布在平面或曲面上，是二维驻波。把细沙撒在薄板上，当薄板振动时，薄板上的细沙就会显示各种各样的图形，图形随振动频率而变，这种图形是由德国物理学家克拉尼(Chladni)发现的，因而命名为克拉尼图形。用来显示克拉尼图形的薄板就称为克拉尼板。早期是用弓弦摩擦克拉尼板的边缘方法使板振动。现在使板振动的方法很多，用压电陶瓷片使克拉尼板振动的方法就是其中的一种。通过本实验，要求对一维驻波、二维驻波有所了解。 实验原理 当弦线上的前进波遇到障碍物后反射，反射波与前进波叠加就形成了驻波，如图1所示。图中实线代前进波，虚线代表反射波，粗线代表叠加波——驻波。驻波的特点是从波形上看不出波在前进，在弦线上的某些点始终不动，这些不动点称为波节;在相邻两个波节中间的点只作上下振动，振动最大处称为波腹。弦线上产生的驻波是一维驻波。 图1 驻波的形成 construction budget and feedback to departments, for inclusion in the settlement, to avoid losses, in order to better control costs. 7, strengthen construction equipment management, eliminate man-made damage, improve the utilization of machines, to shorten the duration. 8, actively carry out QC group activities to improve efficiency and ensure quality. Third, labour resources, machinery and equipment, materials, adequate supply of labor resources: give full play to the company's resources, early labor organizations, and concerned about labor, imposed on the project earmarked to protect the interests of migrant workers, not workers ' wages in arrears. Machines and equipment: using a combination of to buy or rent, rental advanced equipment, to ensure normal progress of the duration. Purchase advanced equipment, to provide work. Ensuring construction quality and schedule. Materials: supplier purchases contracts in advance, and study visits to the providers, ensure the quality of supplied materials, bid can quickly supply organizations, to ensure the project quality. 13th chapter, and the owner, general contractor, supervision and design of supporting measures, and owner of fit 1, meet the requirements of construction on contract duration and quality management, take corresponding measures in construction procedure 前进波沿x方向传播，其位移y与时间t的关系为 y = Acos2π(νt , kx) (1) 前进 其中，A为振幅，ν为波的振动频率，k为波数，它等于波长入倒数。波的传播速度为 , u = ν (2) 反射波的相应表示式为 y=,Acos2π(νt + kx) (3) 反射 前进波与反射波相遇，并在空间叠加后，其合振动为 y =2Asin2πkxsin2πνt (4) 上式为驻波的函数，它是时间t的函数f(t )=sin2πνt和位置x的函数(x)=2Asin2πkx的,乘积。 ,在x=?(n+1/2)/2(n=0,1,2，3，…)处，满足2πkx=?(n+1/2)π的条件时，各点的振幅最大，就是波腹的位置。 ,在x=?n/2(n=0，1，2，3，…)处，满足2πkx=?nπ的条件时，各点的振幅为零，就是波节的位置。 上式为一维驻波的波函数。而话筒的膜片等振动可以看成二维驻波，这些驻波分布在平面或曲面上。一块四周固定的矩形板的振动是最简单二维驻波，如图2所示，它的波函 ,数也可以表示为一位置函数 (x，y)与时间函数f(t)这两部分的乘积。从x方向看，膜片的振动可以看成是许多平行于x轴的线条上的驻波联结在一起;从y方向看，可以看成是许多平行于y轴的线条上的驻波联结在一起;边缘各点的振幅则均为零。总的振动位移可写为 , Z(x，y)=(x，y)sin2πνt = Csin2πkxsin2πkysin2πνt (5) 12 ,,式中，C为常数，kl/= n/2L，k= l/=n/2L，n和n为整数。膜也有一系列本征频1=1 11221212 率，但与弦线的情况不同，它们不等于一个基频的整数倍。膜的本征频率与边界条件等许多因素有关，情况很复杂，本实验不再展开论述。图3是克拉尼板显示的一个克拉尼(二维驻波)图形。 图2 矩形膜 图3 一种克拉尼图形 实验仪器 图4为一台XD7型正弦波信号发生器的示意图。该信号发生器能输出频率范围从20Hz到200kHz的正弦波信号。信号的电压幅度可调，并可通过伏特表显示。在使用时还要注 g measations, to ensure the project quality. 13th chapter, and the owner, general contractor, supervision and design of supportinracts in advance, and study visits to the providers, ensure the quality of supplied materials, bid can quickly supply organizs contation. Purchase advanced equipment, to provide work. Ensuring construction quality and schedule. Materials: supplier purchaseachines and equipment: using a combination of to buy or rent, rental advanced equipment, to ensure normal progress of the durrs. Merned about labor, imposed on the project earmarked to protect the interests of migrant workers, not workers ' wages in arreamaterials, adequate supply of labor resources: give full play to the company's resources, early labor organizations, and conc ve efficiency and ensure quality. Third, labour resources, machinery and equipment,made damage, improve the utilization of machines, to shorten the duration. 8, actively carry out QC group activities to impro-onstruction equipment management, eliminate mancosts. 7, strengthen c construction budget and feedback to departments, for inclusion in the settlement, to avoid losses, in order to better controlng measures in construction procedureures, and owner of fit 1, meet the requirements of construction on contract duration and quality management, take correspondi68 意输出阻抗和外接阻抗匹配。具体的旋钮指示见图示。 图4 XD7型正弦信号发生器 图5为显示弦线上驻波的实验装置。左边是一台XD7型正弦波信号发生器，正弦波信号通过两根信号线送到喇叭上。一根弦线两端固定，其中一头和喇叭相连，另一头连着砝码钩。喇叭的振动带动弦线振动，由于弦线两端固定满足了驻波形成条件，故弦线振动时会出现驻波。 图5 弦上驻波实验装置图 图6 克拉尼板装置 图6为克拉尼板装置。把压电陶瓷片贴在板下，压电陶瓷片与低频正弦波信号发生器 tion on contract duration and quality management, take corresponding measures in construction procedurener, general contractor, supervision and design of supporting measures, and owner of fit 1, meet the requirements of construche owthe quality of supplied materials, bid can quickly supply organizations, to ensure the project quality. 13th chapter, and t struction quality and schedule. Materials: supplier purchases contracts in advance, and study visits to the providers, ensureng cont, rental advanced equipment, to ensure normal progress of the duration. Purchase advanced equipment, to provide work. Ensurihe interests of migrant workers, not workers ' wages in arrears. Machines and equipment: using a combination of to buy or renect tay to the company's resources, early labor organizations, and concerned about labor, imposed on the project earmarked to protensure quality. Third, labour resources, machinery and equipment, materials, adequate supply of labor resources: give full pl ve efficiency andmade damage, improve the utilization of machines, to shorten the duration. 8, actively carry out QC group activities to impro-to avoid losses, in order to better control costs. 7, strengthen construction equipment management, eliminate manconstruction budget and feedback to departments, for inclusion in the settlement, 69 连接，在板上放上细砂。通电后，压电陶瓷片使板振动，在波腹上细砂上下振动，所以在这个地方不可能有细砂存在，而都聚集在没有振动的波节上，形成克拉尼图形。 实验内容 ? 一维振动 1(观察弦线上的一维驻波:把弦线放长至80cm，在和弦线一端相连的砝码钩上套上3个砝码。打开信号发生器，频段旋钮至1档，输出衰减旋钮至8Ω档，使振源振动。从低到高调节信号发生器输出频率，当出现驻波时，观察弦线上的驻波的波节数，并观察驻波的波节数增减。 ,2(使弦线上出现2个完整波腹(弦线两端均为波节)，测出驻波半波长/2 (相邻波节之间距离)、振幅、频率ν，写出该驻波的表达式以及叠加这一驻波的两列行波的表达1 式，并计算出波的传播速度。 ,3(改变信号发生器输出频率，当出现5个完整波腹时，测出此时的半波长/2，根据以上实验得出的波速计算出电压信号的频率ν，并与信号发生器发出的信号频率ν相比220较(完成内容。 4(把弦线放长、拉长或缩短，根据驻波半波长、电压信号频率计算出传播速度，并讨论波的弦线上的传播速度与弦线长度及其松紧程度的关系。 5. 完成表格内容。 二维振动 ? 1、观察克拉尼图形:用压电陶瓷片贴在板下，并固定。在板上均匀地撒上细砂，将信号发生器的输出衰减旋钮打至5KΩ档，并打开信号发生器，由低到高调节信号发生器输出频率。当出现稳定克拉尼图形时，记下此时的频率。 2、用刷子刷去板上细砂的图形。继续增加信号发生器的输出频率，测出出现各种克拉尼图形时所对应的频率，在纸上画出几个典型的克拉尼图形，完成表格内容。 3(写出出现克拉尼图形的频率范围。 观察思考 1(弦线的松紧程度对驻波的波节数及传播速度有何影响, 2(在弹奏弦线乐器时，发出声音的音调与弦线的长度、粗细、松紧程度有什么关系,为什么, 3(克拉尼图形与所使用的压电陶瓷片数量以及贴在板上位置是否有关, 4(是否可以用食盐代替细砂进行克拉尼图形实验, 5(克拉尼板的厚度及均匀性对克拉尼图形是否有影响, 数据表格: 1(根据实验步骤1、2，写出该驻波的表达式以及叠加这一驻波的两列行波的表达式，并计算出波的传播速度。 弦线长度 cm ; 受力砝码 3 个 g measations, to ensure the project quality. 13th chapter, and the owner, general contractor, supervision and design of supportinracts in advance, and study visits to the providers, ensure the quality of supplied materials, bid can quickly supply organizs contation. Purchase advanced equipment, to provide work. Ensuring construction quality and schedule. Materials: supplier purchaseachines and equipment: using a combination of to buy or rent, rental advanced equipment, to ensure normal progress of the durrs. Merned about labor, imposed on the project earmarked to protect the interests of migrant workers, not workers ' wages in arreamaterials, adequate supply of labor resources: give full play to the company's resources, early labor organizations, and conc ve efficiency and ensure quality. Third, labour resources, machinery and equipment,made damage, improve the utilization of machines, to shorten the duration. 8, actively carry out QC group activities to impro-onstruction equipment management, eliminate mancosts. 7, strengthen c construction budget and feedback to departments, for inclusion in the settlement, to avoid losses, in order to better controlng measures in construction procedureures, and owner of fit 1, meet the requirements of construction on contract duration and quality management, take correspondi70 半波数 λ/2 振幅A 频率ν(Hz) u(m/s)波 2 5 y= 驻 y= 1 y= 2 改变频率后，根据测出的半波长及算出的波速计算电压信号频率，并与信号发生器发出的信号频率相比较。 ,,,220 ，100,,,20 2(弦线上的传播速度与弦线松紧程度的关系: 改变受力砝码个数，调节信号频率，使弦线上出现2个完整波腹，将相应的频率记录下来，并求波速，最后再就不同的弦线松紧程度对波速的影响作一讨论。 砝码数(个) 弦线状态 频率ν(Hz) u(m/s) 观察讨论 波波 较松 2 较紧 4 很紧 6 3(弦线上的传播速度与弦线长度的关系: 改变弦线长度，调节信号频率，使弦线上出现2个完整波腹，将相应的频率记录下来，并求波速，最后再就不同的弦线长度对波速的影响作一讨论。 弦线长度 频率ν(HZ) u(m/s) 观察讨论 波波 50cm 70cm 90cm 4(克拉尼图形观察 出现克拉尼图形的频率范围 频率ν(Hz) 克拉尼图形 tion on contract duration and quality management, take corresponding measures in construction procedurener, general contractor, supervision and design of supporting measures, and owner of fit 1, meet the requirements of construche owthe quality of supplied materials, bid can quickly supply organizations, to ensure the project quality. 13th chapter, and t struction quality and schedule. Materials: supplier purchases contracts in advance, and study visits to the providers, ensureng cont, rental advanced equipment, to ensure normal progress of the duration. Purchase advanced equipment, to provide work. Ensurihe interests of migrant workers, not workers ' wages in arrears. Machines and equipment: using a combination of to buy or renect tay to the company's resources, early labor organizations, and concerned about labor, imposed on the project earmarked to protensure quality. Third, labour resources, machinery and equipment, materials, adequate supply of labor resources: give full pl ve efficiency andmade damage, improve the utilization of machines, to shorten the duration. 8, actively carry out QC group activities to impro-to avoid losses, in order to better control costs. 7, strengthen construction equipment management, eliminate manconstruction budget and feedback to departments, for inclusion in the settlement, 71