Energy in a String Wave
弦波中的能量
The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength.
在绷紧的弦上行波所关联的能，方便地表示为每波长的能。
The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment. The differential form of the elastic potential energy is
弦上一小段的能量可以表示为该段的动能和弹性势能之和。弹性势能的微分形式为
Using the description of a traveling wave
利用行波的描述
the potential energy expression becomes
势能表达式变为
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is
一个完整波长的能量可以通过在给定时间对这个表达式进行积分来找到，最方便的是将t=0作为积分的起点。弦上一个波长的能量是
The differential kinetic energy is
微分动能是
Using the velocity expression
使用速度表达式
the kinetic energy takes the form
动能以某种形式存在
and again setting t=0 for simplification
再次为了简化设t=0
The total energy associated with a wavelength is
与波长相关的总能
Since this amount of energy is transported a distance of one wavelength along the string in one period, this expression can be used to calculate the power transmitted along a string.
由于这种能量在一个周期内沿弦传播一个波长的距离，这个表达式可以用来计算沿弦传输的功率。
Both potential energy and kinetic energy minimum at peak of wave?
波峰处势能和动能均最小？
A much-discussed example of energy in string waves is the situation where at the peak of a traveling sinusoidal string wave both the kinetic energy and potential energy are minimum. See Halliday, Resnick & Walker or the Stack Exchange discussion.
弦波中能量的一个备受讨论的例子是，在行进的正弦弦波的波峰处，动能和势能都达到最小值。参见Halliday、Resnick与Walker或Stack Exchange的讨论。 The situation of a traveling wave moving along the string is energetically different from the statically displaced string. For a static stretched string, the straight configuration is one of minimum potential energy. It takes work to displace the string perpendicularly, and results in a longer string which increases the tension. So the displaced string has a higher potential energy.
traveling wave 沿着弦传播的情况在能量上与静止位移的弦不同。对于静止拉伸的弦，直线配置是势能最小的状态。要垂直位移弦需要做功，结果导致弦变长，张力增加。因此，位移后的弦具有更高的势能。 The unique point in the case of the traveling wave in the string is the element of the string that is at the maximum displacement as the wave passes. That element has a zero instantaneous velocity perpendicular to the straight string configuration, and as the wave goes "over the hump", it also has minimum tension. So that element of the string has the minimum energy compared to other elements along the sinusoidal contour. This does not imply that a wavelength of the sinusoidally contoured string has less potential energy than the straight string. The sinusoidally contoured string is longer than a wavelength of straight string and will have greater elastic potential energy. As that sinusoidal contour moves along the string, it transports energy.
在弦上行波的情况下，唯一的点是当波经过时弦上最大位移的元素。该元素在垂直于弦的直线配置方向上具有零瞬时速度，并且当波经过波峰时，它也具有最小的张力。因此，该弦上的元素相比其他沿正弦轮廓的元素具有最小的能量。这并不意味着正弦形轮廓的弦的波长比直弦具有更少的势能。正弦形轮廓的弦比直弦的波长更长，因此将具有更大的弹性势能。当该正弦形轮廓沿弦移动时，它会传输能量。 |
