16.4
绘制波函数图
假设沿正 x 方向移动的正弦波的波动方程。 波动方程是位置和时间的函数。 根据波动方程，可以绘制出两个不同的图形。
如果选择了特定时间，例如t= 0，则表示波浪的“快照”，得而到的图就是t=0时波的形状。 这图称为位移与位置图，表示粒子相对于其平衡位置的位移作为位置的函数。 可以从该图中推断出波长。…
考虑一个沿 x 方向传播的正弦波。
由于其波动方程是位移和时间的函数，介质中粒子的运动可以通过位移-位置图和位移-时间图进行图形化表示。
在固定时间，粒子的位移随位置变化。
这表示粒子相对于其平衡位置的位移。随后可从该图中推导出波长。
以弦上的横波为例，该图像表示在某一特定时刻弦的实际形状。
当选择特定坐标时，绘制波动方程会得到位移-时间图像。
利用该图像可以推导出周期——即波传播一个波长所需的时间。
在波动方程中，余弦函数的自变量称为波的相位。
相速度是指波在保持相位恒定的情况下传播的速度。
对时间求导，得到相速度的表达式。
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Q1: What does a displacement versus position graph show for a wave?
A displacement versus position graph represents a snapshot of the wave at a fixed time, showing how particle displacement varies with position. For a transverse wave on a string, this graph depicts the actual shape of the string at that instant. The wavelength can be determined by measuring the distance between two consecutive crests or troughs with the same height and slope.
Q2: How do you find the period of a wave from a displacement-time graph?
A displacement-time graph shows particle displacement as a function of time at a specific position. The period is the time required for one complete oscillation of the particle. By observing the graph, you can identify the time interval between successive identical points in the wave cycle to determine the period.
Q3: What is the phase of a wave and what does it represent?
The phase is the argument of the cosine function in the wave equation, measured in radians. It determines which part of the sinusoidal cycle is occurring at a particular point and time. For a crest, the phase equals 0, 2π, 4π, etc.; for a trough, the phase equals π, 3π, 5π, etc.
Q4: What is phase velocity and how does it relate to wave motion?
Phase velocity is the speed at which a wave moves while keeping the phase constant. It describes how fast the wave pattern propagates through the medium. The phase velocity can be derived by taking the derivative of the wave equation with respect to time, providing a mathematical expression for wave propagation speed.
Q5: How are crests and troughs identified on a wave graph?
The crest is the highest point of the wave above equilibrium, while the trough is the lowest point below equilibrium. On a displacement versus position graph, crests appear as maximum positive displacements and troughs as maximum negative displacements. The distance between consecutive crests or troughs with identical height and slope equals the wavelength.
Q6: Why are two different graphs needed to fully describe a sinusoidal wave?
A sinusoidal wave depends on both position and time. The displacement versus position graph shows the wave's spatial shape at one instant, revealing wavelength. The displacement versus time graph shows how a single particle oscillates, revealing the period. Together, these graphs provide complete information about wave behavior in space and time.
Q7: How does the wave equation relate to graphical representations of waves?
The wave equation is a mathematical function of both position and time. By fixing time, you obtain a displacement versus position graph showing spatial wave structure. By fixing position, you obtain a displacement versus time graph showing temporal oscillation. These graphical representations make the wave equation's behavior visually interpretable for analyzing wave parameters.