📚 Longitudinal Wave Graphical Representation in IB Physics | IB物理：纵波的图像表示
In IB Physics, waves are often introduced through transverse waves, where displacement is perpendicular to the direction of energy transfer. However, longitudinal waves — such as sound waves — require a different representational approach. This article explains how to draw, interpret, and convert between displacement–distance and displacement–time graphs for longitudinal waves, a key skill for both SL and HL students.
在IB物理中，波动通常先以横波引入，即位移方向与能量传播方向垂直。然而，纵波（如声波）需要不同的图像表示方法。本文旨在讲解如何绘制、解读纵波的位移-距离图与位移-时间图，并掌握二者之间的转换，这是SL和HL学生的核心技能之一。
1. What Makes a Wave Longitudinal | 什么是纵波
A longitudinal wave is one in which the particles of the medium oscillate parallel to the direction of wave propagation. The classic example is sound travelling through air: air molecules vibrate back and forth along the same line as the sound travels, creating regions of higher pressure (compressions) and lower pressure (rarefactions).
纵波是指介质中质点的振动方向与波的传播方向平行的波。最典型的例子是声音在空气中的传播：空气分子沿着声音传播的方向来回振动，形成气压较高的疏密相间区域——压缩区（密部）和稀疏区（疏部）。
Unlike a transverse wave, a longitudinal wave cannot be represented by a simple sinusoidal curve of particle displacement versus position in the same intuitive way. Instead, we use a graph of displacement against distance, where positive and negative displacements indicate particles shifted forward or backward relative to their equilibrium positions along the direction of propagation.
与横波不同，纵波不能直观地用质点位移随位置变化的正弦曲线表示。我们改用“位移-距离”图：正位移表示质点沿传播方向向前偏移，负位移表示质点向后偏移（相对于平衡位置）。
2. Displacement–Distance Graph (Snapshot at Fixed Time) | 位移-距离图（固定时刻的“快照”）
For a longitudinal wave, the displacement–distance graph plots the displacement of each particle from its equilibrium position against the distance along the wave’s direction of travel. The horizontal axis represents the equilibrium position of particles, and the vertical axis shows displacement (positive = forward, negative = backward).
对于纵波，位移-距离图以波的传播方向为横轴（表示各质点的平衡位置），以质点的位移为纵轴（正为向前，负为向后），描述某一时刻各质点的位移情况。
Consider a sinusoidal longitudinal wave at a fixed time. The graph looks like a sine or cosine curve. Where the curve crosses zero with a steep positive slope, particles are at their equilibrium positions but densely packed — this corresponds to a compression. Where the curve crosses zero with a steep negative slope, particles are moving apart — this is a rarefaction. The maximum positive displacement corresponds to particles pushed farthest forward; the maximum negative displacement corresponds to particles pushed farthest backward. Note that compressions and rarefactions are not simply the crests and troughs of the displacement graph; they occur where the gradient of the displacement graph is steepest.
以某一固定时刻的正弦纵波为例，位移-距离图呈正弦或余弦曲线。当曲线以陡峭的正斜率穿过零值时，质点位于平衡位置但间距较小，对应压缩区；当曲线以陡峭的负斜率穿过零值时，质点间距拉大，对应稀疏区。最大正位移表示质点向前偏移最远，最大负位移表示质点向后偏移最远。注意：压缩区和稀疏区并不是位移图的波峰和波谷，而是位移图斜率最陡的地方。
compression: gradient of displacement–distance graph is maximum positive
压缩区：位移-距离图的斜率为最大正值
rarefaction: gradient of displacement–distance graph is maximum negative
稀疏区：位移-距离图的斜率为最大负值
3. From Displacement Graph to Density/Pressure Graph | 从位移图到密度/压强图
Because compressions and rarefactions are regions of increased and decreased particle density, we can also represent a longitudinal wave using a pressure–distance graph. The pressure variation is proportional to the negative of the spatial derivative (gradient) of the displacement–distance graph.
由于压缩区和稀疏区分别对应粒子密度增大和减小，我们也可以用“压强-距离”图表示纵波。压强变化与位移-距离图的空间导数（斜率）的负值成正比。
For a displacement graph y = A sin(2πx/λ), the pressure variation is proportional to −d y/d x = −(2πA/λ) cos(2πx/λ). Thus, where displacement has maximum positive gradient, pressure is minimum (rarefaction); where displacement has maximum negative gradient, pressure is maximum (compression).
对于位移图 y = A sin(2πx/λ)，压强变化与 −d y/d x = −(2πA/λ) cos(2πx/λ) 成正比。因此，位移图斜率最大正值处对应压强最小（稀疏区），斜率最大负值处对应压强最大（压缩区）。
This conversion is often tested in IB exam questions. Students are expected to sketch the pressure graph given the displacement graph, or vice versa, and to identify the phase relationship: pressure and displacement are 90° out of phase.
这一转换是IB考试中常见的考点。学生需要能够根据位移图画出压强图，或根据压强图画出位移图，并识别相位关系：压强与位移的相位差为90°。
4. Wavelength and Amplitude on a Longitudinal Wave Graph | 纵波图中的波长与振幅
On a displacement–distance graph of a longitudinal wave, the wavelength λ is the distance between two consecutive points that are in phase, for example, the distance between two successive maximum positive displacements, or two successive zero crossings with the same slope direction.
在纵波的位移-距离图上，波长λ是指两个相邻同相点之间的距离，例如相邻两个最大正位移之间的距离，或两个相隔一个周期且斜率方向相同的零值点之间的距离。
The amplitude A is the maximum magnitude of displacement from equilibrium. This is the peak value of the displacement graph. It represents the maximum displacement of the particles from their rest positions along the direction of propagation.
振幅A是质点偏离平衡位置的最大位移量，即位移图的峰值。它表示质点沿传播方向离开静止位置的最大距离。
It is important to remember that the amplitude of a sound wave is related to its loudness, while the frequency (or wavelength) is related to its pitch. Doubling the amplitude quadruples the intensity, since intensity is proportional to the square of amplitude.
务必记住：声波的振幅与响度相关，频率（或波长）与音调相关。振幅加倍时，强度变为原来的4倍，因为强度与振幅的平方成正比。
5. Displacement–Time Graph for a Longitudinal Wave | 纵波的位移-时间图
A displacement–time graph for a longitudinal wave shows how the displacement of a single particle varies with time at a fixed position. This graph is identical in form to the displacement–time graph for a transverse wave, because it records the oscillation of one particle regardless of wave type.
纵波的位移-时间图表示某一固定位置处单个质点的位移随时间的变化。这种图形与横波的位移-时间图在形式上完全一致，因为它记录的是单个质点的振动，与波的类型无关。
From this graph, you can directly read the period T (the time for one complete oscillation) and the amplitude A (maximum displacement). The frequency f is the reciprocal of the period: f = 1/T. The phase of the particle at any instant can also be determined from the graph.
从位移-时间图中可以直接读出周期T（完成一次全振动所需的时间）和振幅A（最大位移）。频率f是周期的倒数：f = 1/T。还可以确定任意时刻质点的相位。
f = 1/T
To find the wave speed v, combine information from both types of graphs: use the wavelength λ from the displacement–distance graph and the period T (or frequency f) from the displacement–time graph, then apply v = fλ.
要计算波速v，需要结合两种图像的信息：从位移-距离图中读取波长λ，从位移-时间图中读取周期T（或频率f），然后应用 v = fλ。
v = fλ
6. Particle Motion vs Wave Motion | 质点运动与波动的区别
In a longitudinal wave, the particles oscillate back and forth about fixed equilibrium positions. They do not travel with the wave. The wave itself transfers energy and momentum through the medium, but the average displacement of any particle over a full cycle is zero.
在纵波中，质点围绕各自的平衡位置来回振动，并不随波迁移。波通过介质传递能量和动量，但任意质点在一个完整周期内的平均位移为零。
On the displacement–distance graph, each point on the horizontal axis represents a different particle at the same instant. On the displacement–time graph, the curve represents one particle at different times. Confusing these two is a common mistake in IB exams.
在位移-距离图中，横轴上的每个点代表同一时刻的不同质点；在位移-时间图中，曲线代表同一质点在不同时刻的位移。混淆这两种图像是IB考试中常见的错误。
For a longitudinal wave, when a particle is at its maximum forward displacement, the particle just ahead of it may be at equilibrium, leading to a compression. When a particle is at its maximum backward displacement, the particle behind it may be at equilibrium, creating a rarefaction. Practising these spatial relationships helps solidify understanding.
对于纵波，当某质点处于最大正向位移时，它前方的质点可能正经过平衡位置，从而形成压缩区；当某质点处于最大负向位移时，它后方的质点可能正经过平衡位置，从而形成稀疏区。多加练习这类空间关系有助于巩固理解。
7. Relating Compression/Rarefaction to Displacement Graph Slopes | 将压缩/稀疏区与位移图斜率关联
Let us examine a specific example. Consider the displacement–distance graph of a longitudinal wave shown as a sine function: y = A sin(kx), where k = 2π/λ. At x = 0, the displacement is zero and the slope is positive. Particles on either side are moving toward x = 0, so this is a compression. At x = λ/2, displacement is again zero but the slope is negative. Particles are moving away from x = λ/2, so this is a rarefaction.
我们来看一个具体例子。设纵波的位移-距离图为正弦函数：y = A sin(kx)，其中 k = 2π/λ。在 x = 0 处，位移为零且斜率为正，两侧质点向 x = 0 处靠近，因此这里是压缩区。在 x = λ/2 处，位移同样为零但斜率为负，质点背离 x = λ/2 处运动，因此这里是稀疏区。
Thus, the compressions and rarefactions are located at the zero-displacement points of the displacement graph, not at the maxima or minima. The spacing between two consecutive compressions (or two consecutive rarefactions) is one wavelength.
因此，压缩区和稀疏区位于位移图的零位移点处，而不是波峰或波谷处。相邻两个压缩区（或相邻两个稀疏区）之间的距离为一个波长。
When converted to a pressure–distance graph, the compressions appear as maximum pressure peaks and the rarefactions as minimum pressure troughs. The pressure graph is therefore a cosine function if the displacement graph is a sine function.
转换为压强-距离图时，压缩区对应压强最大值（波峰），稀疏区对应压强最小值（波谷）。因此，如果位移图为正弦函数，则压强图为余弦函数。
8. Sketching and Interpreting Graphs in Exams | 考试中绘制与解读图像
IB exam questions on longitudinal waves often ask you to sketch the displacement–distance graph from a description of compression and rarefaction positions, or to mark the positions of compressions and rarefactions on a given displacement graph. You may also be asked to convert between displacement and pressure graphs, or to determine wave speed from a pair of graphs.
IB考试中关于纵波的题目通常要求：根据压缩区和稀疏区的位置画出位移-距离图；或在给定的位移图上标出压缩区和稀疏区；也可能要求你在位移图和压强图之间转换，或从一组图像中求波速。
Useful tips for exam success:
考试实用技巧：
- Always label axes with correct quantities and units (displacement / m, distance / m, time / s). 始终正确标注坐标轴的物理量和单位（位移/m、距离/m、时间/s）。
- Mark one full wavelength clearly on the distance graph. 在距离图上清晰标出一个完整波长。
- Mark the amplitude on both types of graphs. 在两种图像上都标出振幅。
- Remember that compressions occur where the displacement–distance graph has maximum positive slope, not at maximum displacement. 记住压缩区出现在位移-距离图斜率最大正值处，而不是最大位移处。
- When converting to pressure graphs, use the negative gradient of displacement to find pressure variation. 转换为压强图时，用位移图的负斜率表示压强变化。
- Check whether the question asks about a fixed time (distance graph) or a fixed position (time graph). 判断题目问的是固定时刻（距离图）还是固定位置（时间图）。
9. Worked Example: Reading a Longitudinal Wave Graph | 例题：解读纵波图像
Suppose a longitudinal wave has the displacement–distance graph described by y = 3.0 sin(2πx/0.40), where y is in millimetres and x is in metres. The wave travels at 340 m s⁻¹. Determine the amplitude, wavelength, frequency, and the position of the first compression to the right of x = 0.
设一纵波的位移-距离图为 y = 3.0 sin(2πx/0.40)，其中 y 以毫米为单位，x 以米为单位。波速为 340 m s⁻¹。试求振幅、波长、频率以及 x = 0 右侧第一个压缩区的位置。
Solution: The amplitude is 3.0 mm = 3.0 × 10⁻³ m. The wavelength is 0.40 m. The frequency is f = v/λ = 340 / 0.40 = 850 Hz. The first compression to the right of x = 0 occurs where the displacement is zero and the slope is positive. For y = A sin(2πx/λ), this happens at x = 0, x = λ, x = 2λ, etc. Thus the first compression is at x = 0 and the next one is at x = 0.40 m. If you are asked for the first compression strictly to the right of x = 0, then it is at x = λ = 0.40 m.
解答：振幅为 3.0 mm = 3.0 × 10⁻³ m。波长为 0.40 m。频率为 f = v/λ = 340 / 0.40 = 850 Hz。x = 0 右侧的第一个压缩区出现在位移为零且斜率为正值处。对于 y = A sin(2πx/λ)，这发生在 x = 0、x = λ、x = 2λ 等处。因此第一个压缩区在 x = 0，下一个在 x = 0.40 m。如果问题要求严格位于 x = 0 右侧的第一个压缩区，则其位置为 x = λ = 0.40 m。
This example illustrates how to extract quantitative information from a longitudinal wave graph and connect it to wave properties.
此例题展示了如何从纵波图像中提取定量信息，并将其与波的物理量联系起来。
10. Common Misconceptions | 常见误区
Many students mistakenly think that a compression corresponds to the peak of the displacement–distance graph. In fact, at the peak (maximum positive displacement), the particle is furthest forward, but the particles around it are not necessarily crowded together. The compression is where the gradient is steepest, because that is where particles are closest together.
许多学生误认为压缩区对应位移-距离图的波峰。实际上，在波峰（最大正位移）处，该质点向前偏移最远，但其周围质点并不一定最密集。压缩区出现在斜率最陡处，因为那里质点间距最小。
Another common error is treating the displacement–time graph as if it shows a snapshot of the wave in space. A displacement–time graph is for one particle over time; it does not show the spatial arrangement of particles. To visualise compressions and rarefactions, you must use a displacement–distance graph.
另一个常见错误是把位移-时间图当作波在空间中的快照。位移-时间图描述的是一个质点随时间的变化，并不显示质点在空间中的排列。要直观看到压缩区和稀疏区，必须使用位移-距离图。
Finally, students often forget that in a longitudinal wave, the pressure graph is phase-shifted by 90° relative to the displacement graph. Remember this relationship when converting between the two representations.
最后，学生常常忘记纵波中压强图与位移图存在90°相位差。在两种表示之间转换时，务必记住这一关系。
11. Summary of Key Equations and Relationships | 关键公式与关系总结
The following table summarises the essential relationships and graph interpretations for longitudinal waves in IB Physics.
下表总结了IB物理中纵波的基本关系与图像解读要点。
| Quantity 物理量 | Symbol 符号 | Relationship / Graph feature 关系/图像特征 |
| Wavelength 波长 | λ | Distance between successive compressions or rarefactions 相邻压缩区或稀疏区之间的距离 |
| Amplitude 振幅 | A | Maximum displacement from equilibrium in displacement graph 位移图中离开平衡位置的最大位移 |
| Frequency 频率 | f | f = 1/T, where T is period from displacement–time graph f = 1/T，T 为位移-时间图中的周期 |
| Wave speed 波速 | v | v = fλ |
| Pressure variation 压强变化 | Δp | Proportional to −(gradient of displacement–distance graph) 与位移-距离图的斜率的负值成正比 |
Δp ∝ −(Δy/Δx) at fixed time
12. Final Advice for IB Students | 给IB学生的最终建议
Mastering longitudinal wave graphs requires practice in translating between physical situations and graphical representations. Start by sketching displacement–distance graphs for given compression/rarefaction patterns, then convert them to pressure–distance graphs. Next, draw displacement–time graphs for a specific particle and extract period and frequency.
掌握纵波图像需要在物理情境与图像表示之间反复转换练习。首先根据给定的压缩/稀疏区分布画出位移-距离图，再转换为压强-距离图。然后画出某一质点的位移-时间图，并从中读出周期和频率。
Always double-check the type of graph you are working with. Ask yourself: “Is the horizontal axis distance or time?” This simple question prevents most errors. Also, remember that for longitudinal waves, the wave direction is parallel to particle oscillation — this is the fundamental distinction from transverse waves.
始终确认你正在处理的是哪种图像。问问自己：“横轴是距离还是时间？”这个简单的问题可以避免大多数错误。同时，记住纵波的传播方向与质点振动方向平行——这是与横波的根本区别。
With systematic practice, you will be able to interpret and sketch longitudinal wave graphs quickly and accurately in the IB exam.
通过系统练习，你将能够在IB考试中快速而准确地解读和绘制纵波图像。
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