Standing Waves on a String

Standing waves are produced on a string when equal waves travel in opposite directions. When the proper conditions are met, the interference between the traveling waves causes the string to move up and down in segments, as illustrated below. This segment vibration gives no appearance of motion along the length of the string. The phenomenon is called a standing wave or stationary wave and corresponds to a resonant vibration of the string.

驻波是在弦上由相向传播的相等波形成。当满足适当条件时，传播波之间的干涉使弦在若干段内上下振动，如图所示。这种段振动在弦的长度方向上没有明显的运动。这种现象称为驻波或 stationary wave，并对应于弦的共振振动。

If a light, flexible string is attached to a vibrator and the other end passed overr a fixed pulley to a weight hanger, the necessary conditions for standing waves can be met. Waves from the vibrator travel down the string and are reflected at the pulley. When the tension and length of the string are properly adjusted, these two oppositely directed wave trains superimpose to give alternate regions of no vibration, N (see figure) and regions of maximum vibration, A. These regions N and A are called nodes and antinodes, respectively, and the segment between two nodes is called a loop. The velocity, v, of a wave on a stretched string is given by:

若将一根轻而柔韧的弦系在振动器上，另一端经过一个固定滑轮连接到一个重物托盘，就可以满足形成驻波的条件。来自振动器的波沿弦传播，并在滑轮处反射。当弦的张力和长度适当调节时，这两个方向相反的波列叠加，形成交替的无振动区域N（见图）和最大振动区域A。这些区域N和A分别称为节点和抗节点，两个节点之间的部分称为一个波节。在拉紧的弦上，波的速率v由下式给出：

where T is the stretching force (tension) and m is the mass per unit length of the string. If the general wave equation:

其中 T 是拉伸力（张力），m 是弦的单位长度质量。如果一般的波动方程：

is combined with the above equation, the frequency of the vibrator is given by the relation:

结合上述方程，振动器的频率由下式给出：

where λ is the wavelength of the wave. Examination of the figure, in which the full line represents the original wave train and the dashed line the reflected wave train, will show that the distance between two nodes is one half wavelength. In this experiment, you will be given the frequency f = 60Hz and can then determine the quantity T/f²λ². The mass per unit length m of the string is to be calculated from:

其中λ是波的波长。通过观察图中，实线表示原波列，虚线表示反射波列，可以发现两个节点之间的距离是半波长。在本实验中，你将被给出频率f = 60Hz，并可以由此确定量T/f 2 λ 2。弦的单位长度质量m应从以下公式计算：

1. Attach the string to the vibrator and add weights to the weight holder until a distinct standing wave with about six loops is produced (or as many loops as possible). If the weights available are not small enough for the fine adjustment, a small movement of the pulley toward or away from the vibrator may accomplish the desired result. For best accuracy, make small adjustments in the tension to get the maximum amplitude of vibration.

1. 将弦固定在振动器上，并在重物托盘上添加重物，直到产生明显的驻波（约六个波节）（或尽可能多的波节）。如果可用的重物不足以进行精细调节，将滑轮向振动器方向移动或远离振动器可能达到 desired 结果。为了获得最佳精度，应通过微调张力来获得最大振动振幅。

2. Omitting the loop nearest the vibrator, measure and record the total length of the remaining loops. Record the number of loops and the stretching force

省略靠近振动器的那一个环，测量并记录剩余环的总长度。记录环的数量和拉伸力。

3. Keep the distance between the pulley and the vibrator the same, except for small adjustments, and increase the stretching force until a distinct standing wave is produced with one less loop. Again measure the loops and record the data as in step 2.

保持滑轮与振动器之间的距离相同（仅允许进行小的调整），直到产生一个波形少一个波腹的明显驻波。再次测量波腹数并记录数据，如步骤2所述。

4. Continue the process of reducing the number of loops by one until you have added enough to produce only two loops. Record all data as before.

继续减少循环数，直到只剩下两个循环为止。记录所有数据，如前所述。

5. 计算每个试验的波长。

6. Convert the stretching force to newtons by multiplying the mass in kilograms by the acceleration of gravity, g = 9.8 m/sec².

将拉伸力转换为牛顿，方法是将质量（以千克为单位）乘以重力加速度g = 9.8 m/s²。

7. Calculate the values of mass per unit length m = T/f²λ² for each of the different loop patterns.

Standing Waves on a String

Equipment: String Resonance