📚 Graphs of Longitudinal Waves | 纵波的图像
Understanding displacement–position graphs for longitudinal waves is a core skill in IB Physics. These graphs look nearly identical to those of transverse waves, yet they encode entirely different physical information about particle motion parallel to wave propagation. This article explains how to interpret such graphs, locate compressions and rarefactions, and relate them to pressure and density variations.
理解纵波的位移–位置图像是IB物理的核心技能。这些图像看起来与横波的图像几乎相同，但编码了关于粒子沿波传播方向运动的完全不同的物理信息。本文将解释如何解读这类图像、定位压缩与稀疏区域，并将它们与压力和密度变化联系起来。
1. What Is a Longitudinal Wave? | 什么是纵波？
A longitudinal wave is a wave in which the particles of the medium oscillate parallel to the direction of energy transfer. Sound waves in air, seismic P-waves, and ultrasound are typical examples. The oscillation creates alternating regions of high pressure (compressions) and low pressure (rarefactions). Unlike transverse waves, longitudinal waves cannot be drawn as simple wiggles in space; their graphical representation requires a careful choice of axes.
纵波是介质粒子沿能量传递方向振动的波。空气中的声波、地震P波和超声波是典型例子。振动产生交替的高压（压缩）和低压（稀疏）区域。与横波不同，纵波不能简单地画成空间中的摆动曲线；其图形表示需要谨慎选择坐标轴。
2. Displacement–Position Graph: Basic Setup | 位移–位置图：基本设置
A displacement–position graph for a longitudinal wave plots the displacement y of particles from their equilibrium positions against their equilibrium positions x along the direction of propagation. The positive direction of displacement is conventionally chosen to be the same as the direction of wave travel. Thus, a positive y means the particle has moved forward, and a negative y means it has moved backward.
纵波的位移–位置图将粒子偏离平衡位置的位移y与沿传播方向的平衡位置x作图。位移的正方向通常选定为与波行进方向相同。因此，正的y表示粒子向前移动，负的y表示粒子向后移动。
y = A sin(kx − ωt)
y = A sin(kx − ωt)
At a snapshot in time (e.g. t = 0), this reduces to y = A sin(kx), which looks exactly like a transverse wave snapshot. The key difference lies in what the graph represents physically: motion along the x-axis, not perpendicular to it.
在某一时刻的快照（例如t = 0），方程简化为 y = A sin(kx)，这看起来与横波的快照完全一样。关键差别在于图像所代表的物理量：沿x轴的运动，而非垂直于x轴。
3. Interpreting Compressions and Rarefactions | 解读压缩与稀疏
In a displacement–position graph, points where y spond to the centres of compression. Here, particles on the left have moved further right than those on the right, squeezing the medium together. Conversely, points where y d the gradient is positive mark the centres of rarefaction, where the medium is stretched.
在位移–位置图中，y = 0 且梯度为负的点对应于压缩中心。在那里，左侧粒子比右侧粒子向右移动得更多，挤压介质。相反，y = 0 且梯度为正的点标志着稀疏中心，此处介质被拉伸。
More formally, the pressure variation ΔP is proportional to the negative of the displacement gradient: ΔP ∝ −∂y/∂x. Therefore, wherever the displacement curve has a steep negative slope, pressure peaks; where the slope is steep and positive, pressure dips to a minimum.
更正式地，压力变化ΔP 正比于位移梯度的负值：ΔP ∝ −∂y/∂x。因此，位移曲线有陡峭负斜率的地方，压力达到峰值；在有陡峭正斜率的地方，压力降至最低。
4. Pressure–Position and Density–Position Graphs | 压力–位置图与密度–位置图
Pressure–position and density–position graphs for a sinusoidal longitudinal wave are also sinusoidal, but they are shifted relative to the displacement graph. If the displacement is y = A sin(kx), then ΔP ∝ −cos(kx). This means pressure maxima occur at displacement zero-crossings with negative gradient, while pressure minima occur at displacement zero-crossings with positive gradient.
正弦纵波的压力–位置图和密度–位置图也是正弦曲线，但与位移图有相位差。如果位移为 y = A sin(kx)，则 ΔP ∝ −cos(kx)。这意味着压力最大值出现在位移过零点且负梯度处，而压力最小值出现在位移过零点且正梯度处。
In terms of phase, the pressure graph lags the displacement graph by 90° (π/2 rad) if we consider the cosine–sine relationship. Density follows the same pattern as pressure because density rises where particles are compressed. An examiner may ask you to sketch a pressure graph given a displacement–position graph; remember to align the compression centres with pressure peaks.
就相位而言，考虑到余弦与正弦关系，压力图落后于位移图90°（π/2弧度）。密度图遵循与压力相同的模式，因为在被压缩的地方密度升高。考官可能要求你根据位移–位置图绘制压力图；记住将压缩中心与压力峰值对齐。
5. Wavelength and Amplitude from the Graph | 从图像中读取波长和振幅
Wavelength λ is the distance along the x-axis between two successive points that are in phase, such as two adjacent compressions or two adjacent rarefactions. On a displacement–position graph, this is simply the distance between two consecutive points where the pattern repeats, e.g. from one zero-crossing with the same slope direction to the next.
波长λ是沿x轴两个相继同相点之间的距离，例如两个相邻压缩中心或两个相邻稀疏中心。在位移–位置图上，这就是两个连续模式重复点之间的距离，例如从一个具有相同斜率方向的过零点到下一个过零点。
Amplitude A is the maximum magnitude of displacement from equilibrium. It is read directly off the vertical axis as the maximum positive or negative y-value. Always include units, and if the graph is uncalibrated, express the amplitude in terms of A as shown.
振幅A是偏离平衡位置的最大位移的量值。可直接从纵轴读取，即最大正或负的y值。务必注明单位；如果图像未标刻度，则用所示A表示振幅。
6. Phase Difference Along the Wave | 沿波的相位差
Phase difference φ between two particles separated by distance Δx can be found using φ = (2π/λ) · Δx. On the displacement–position graph, if you pick two particles at different x to their separation. For example, particles separated by λ/2 are in antiphase (φ = π rad).
相距Δx的两个粒子之间的相位差φ可通过 φ = (2π/λ) · Δx 求得。在位移–位置图上，如果选取不同x位置的两个粒子，它们相位角的差正比于它们的间距。例如，相距λ/2的粒子处于反相（φ = π弧度）。
This is identical to the treatment for transverse waves, but the physical consequence is different: in a longitudinal wave, two particles in antiphase move in opposite directions along the line of travel, creating alternating compression and rarefaction.
这与横波的处理方法相同，但物理结果不同：在纵波中，处于反相的两个粒子沿传播线反向运动，从而交替产生压缩和稀疏。
7. Comparing with Transverse Wave Graphs | 与横波图的比较
A transverse wave displacement–position graph plots particle displacement perpendicular to propagation against position. Superficially, it looks the same as a longitudinal wave graph. However, in a transverse wave, crests and troughs correspond to maximum displacement perpendicular to the medium, whereas in a longitudinal wave, what appears as a 'crest' in the graph actually indicates a region where particles have moved forward maximally, contributing to a rarefaction on one side and a compression on the other.
横波的位移–位置图将垂直于传播方向的粒子位移对位置作图。表面上看，它与纵波图相同。然而，在横波中，波峰和波谷对应于介质中垂直方向的最大位移；而在纵波中，图中的“波峰”实际上指示粒子向前移动最大的区域，从而导致一侧稀疏、另一侧压缩。
This distinction is a frequent topic in IB multiple-choice questions. Always check the axes labels: if the y-axis is labelled 'displacement' and the wave is longitudinal, the graph does not show the physical shape of the wave but rather a mathematical plot of particle displacement along the direction of propagation.
这种区别是IB选择题中的常见考点。务必检查坐标轴标签：如果纵轴标有“位移”且波是纵波，则图像并未显示波的物理形状，而是沿传播方向粒子位移的数学绘图。
8. Graphical Representation of Standing Longitudinal Waves | 纵驻波的图像表示
Standing longitudinal waves can form in air columns, such as in organ pipes. The displacement–position graph for a standing longitudinal wave reveals nodes (points of zero displacement) and antinodes (points of maximum displacement). For a sound wave in an open pipe, the displacement antinodes are at the open ends, while pressure nodes are found there. In a closed pipe, the closed end is a displacement node and a pressure antinode.
纵驻波可以在空气柱中形成，例如管风琴。纵驻波的位移–位置图显示出波节（位移为零的点）和波腹（位移最大的点）。对于开管中的声波，位移波腹位于开口端，而压力波节也在此处。在闭管中，封闭端是位移波节和压力波腹。
When constructing these graphs, remember the 90° phase shift between displacement and pressure: a displacement antinode corresponds to a pressure node. IB questions may ask you to sketch both displacement and pressure standing wave patterns for a given harmonic.
绘制这些图像时，请记住位移与压力之间相位差为90°：位移波腹对应压力波节。IB考题可能要求你为某一谐频绘制位移和压力驻波模式。
9. Worked Example: Analysing a Displacement–Position Graph | 例题：分析位移–位置图
Scenario: A displacement– entify the positions of two compressions and two rarefactions between x = 0 and x = 1.6 m.
情景： 一个纵波在t = 0时的位移–位置图显示一条振幅为2.0 mm、波长为0.80 m的正弦曲线。正x方向向右（波传播方向）。请标出x = 0到x = 1.6 m之间两个压缩中心和两个稀疏中心的位置。
Solution: g: x = 0.80 m (positive gradient) → rarefaction; x = 1.20 m (negative gradient) → compression; x = 1.60 m (positive gradient) → rarefaction. Thus compressions at x ≈ 0.40 m and 1.20 m; rarefactions at x = 0, 0.80 m, and 1.60 m.
解答： 图像为y = 2.0 sin(kx) mm，其中k = 2π/0.80 = 7.85 rad/m。位移零点位于x = 0, 0.40 m, 0.80 m, 1.20 m, 1.60 m。在x = 0处，梯度为正（cos 0 = 1），所以这是稀疏中心。在x = 0.40 m处，sin(π) = 0且梯度为负（cos π = −1），标志着压缩中心。继续：x = 0.80 m（正梯度）→ 稀疏；x = 1.20 m（负梯度）→ 压缩；x = 1.60 m（正梯度）→ 稀疏。因此压缩中心在约0.40 m和1.20 m处；稀疏中心在x = 0, 0.80 m和1.60 m处。
10. Common Misconceptions | 常见误区
- Misconception: The displacement–position graph shows the actual shape of the longitudinal wave. Truth: It is a mathematical construct; a longitudinal wave has no 'up and down' shape.
- Misconception: Higher displacement always means higher pressure. Truth: Pressure depends on the displacement gradient, not displacement itself.
- Misconception: The graph for a longitudinal wave is completely different from that of a transverse wave. Truth: The mathematical form is identical; only the physical interpretation differs.
- 误区： 位移–位置图显示了纵波的实际形状。真相： 它是一种数学构造；纵波没有“上下”的形状。
- 误区： 位移越大，压力越高。真相： 压力取决于位移梯度，而非位移本身。
- 误区： 纵波的图像与横波的图像完全不同。真相： 数学形式是相同的；仅物理诠释不同。
11. Summary and Exam Tips | 总结与应试技巧
When faced with a longitudinal wave graph in IB exams, first confirm the wave type. Identify the direction of propagation and the sign convention for displacement. Use the zero-crossings and gradients to locate compressions and rarefactions. Remember the phase relationship between displacement and pressure. Practice sketching the corresponding pressure–position graph: align pressure maxima with compressions (zero displacement, negative gradient). Finally, always read the axes and units carefully to avoid losing marks through simple oversight.
在IB考试中遇到纵波图像时，首先确认波的类型。明确传播方向以及位移的正负号约定。利用过零点和梯度来定位压缩与稀疏区域。牢记位移与压力之间的相位关系。练习绘制相应的压力–位置图：将压力峰值与压缩中心（零位移、负梯度）对齐。最后，务必仔细阅读坐标轴和单位，避免因简单疏忽而失分。
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