# 波的性质

> 爱德思国际普通中等教育证书（IGCSE）物理 · 4PH1 2017
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-physics-s3-properties-of-waves/

本指南覆盖爱德思IGCSE物理（4PH1）考纲3.1至3.9要求的所有波的核心性质，包括波的类型、核心定义、相关计算、多普勒效应以及波的基础行为。

**先修:** [基础国际单位制（SI）单位换算能力](https://www.owlsprep.com/zh/study/edexcel-igcse-physics-s1-si-units/)

## 学习目标

- 区分横波与纵波，并掌握二者的现实实例
- 定义波的核心术语，包括振幅、波长、频率、周期与波前
- 记忆并应用波速公式与频率-周期公式解答计算类题目
- 定性解释多普勒效应与波的核心行为（反射、折射）

## 波的类型与核心性质

所有波都可以传递能量与信息，且不会发生净物质的转移。例如，当水波经过时，漂浮的软木塞只会上下浮动，不会被波带到岸边。

- **Transverse waves**: Oscillations are *perpendicular* to the direction of energy transfer. Examples include all electromagnetic waves, water ripples, and waves on a stretched string.
- **Longitudinal waves**: Oscillations are *parallel* to the direction of energy transfer, made of alternating compressions (high particle density) and rarefactions (low particle density). Examples include sound waves and seismic P-waves.

**例题:** 对光波和声波进行分类，并分别说明分类理由。

1. 1. 光是一种电磁波：其电场与磁场的振荡方向与传播方向垂直，因此属于横波。
2. 2. 声波通过空气粒子的振动传播，粒子振动方向与传播方向平行，存在压缩区和稀疏区，因此属于纵波。

> **应试技巧**
>
> 考试中遇到相关问题时，一定要为每种波的类型给出具体实例，这是典型的2分题。牢记电磁波为横波、声波为纵波即可快速得分。

*计算器:* forbidden

## 波的核心术语、单位与公式

**波的核心物理量** — 振幅（A）：相对于静止位置的最大位移，单位为米（m）。波长（$\lambda$）：两个相邻同相位点之间的距离，单位为米（m）。频率（f）：每秒通过的波的数量，单位为赫兹（Hz）。周期（T）：一个完整波的传播时间，单位为秒（s）。波前：连接所有同相位振荡点的线。

*记法:* A, \lambda, f, T

*例:* A 10 Hz sound wave has 10 complete waves passing a point every second, so its time period is 0.1 s.

本章节的两个公式不会出现在考试提供的公式表中，你必须熟练记忆：

$$v = f \times \lambda$$

$$f = \frac{1}{T}$$

**例题:** 某波的周期为0.02 s，计算其频率；若其波长为1.5 m，求该波的波速。

1. 1. 使用公式$f = 1/T$计算频率：
2. $$f = \frac{1}{0.02} = 50 \text{ Hz}$$
3. 2. 使用公式$v = f\lambda$计算波速：
4. $$v = 50 \times 1.5 = 75 \text{ m/s}$$

> **warning**
>
> 计算前务必将所有单位转换为国际单位制：将千赫兹（kHz）乘以1000得到赫兹（Hz），将厘米（cm）乘以0.01得到米（m），否则会得到错误结果。

*计算器:* allowed

## 不同场景下的波的计算

你需要在不同场景下应用波的相关公式，包括声波与电磁波。所有电磁波在真空中的传播速度均为$3 \times 10^8$ m/s，所有电磁波相关计算题目都可以使用该数值。

**例题:** 某调频无线电波的频率为95 MHz，计算其波长。

1. 1. 将频率转换为国际单位制单位（Hz）：
2. $$95 \text{ MHz} = 95 \times 10^6 = 9.5 \times 10^7 \text{ Hz}$$
3. 2. 对公式$v = f\lambda$变形，求解波长：
4. $$\lambda = \frac{v}{f}$$
5. 3. 代入数值（电磁波的波速取$v = 3 \times 10^8$ m/s）：
6. $$\lambda = \frac{3 \times 10^8}{9.5 \times 10^7} \approx 3.16 \text{ m}$$

**概念自测**

1. What is the period of a 200 Hz sound wave?

   - A) 0.005 s
   - B) 0.05 s
   - C) 200 s

   *解析:* Use $T = 1/f = 1/200 = 0.005$ s. Remember period is the inverse of frequency.

2. A sound wave travels at 340 m/s with a wavelength of 0.17 m. What is its frequency?

   - A) 20 Hz
   - B) 200 Hz
   - C) 2000 Hz

   *解析:* Rearrange $v = f\lambda$ to $f = v/\lambda = 340 / 0.17 = 2000$ Hz.

*计算器:* allowed

## 多普勒效应与波的核心行为

**多普勒效应** — 当波源相对于静止观察者运动时，观察者观测到的波的频率与波长发生变化的现象。波源自身发射的频率并未发生改变。

*例:* An ambulance siren sounds higher pitched when approaching you, and lower pitched when driving away from you.

- Source moving *towards* observer: Wavefronts bunch up in front of the source, observed wavelength is shorter, observed frequency is higher.
- Source moving *away from* observer: Wavefronts stretch out behind the source, observed wavelength is longer, observed frequency is lower.

所有波（包括横波与纵波）都可以发生反射（在表面反弹）与折射（在两种不同介质间传播时改变速度与传播方向）。

**例题:** 请解释为什么静止的观察者听到救护车驶近时的警笛频率更高，而救护车驶离时的警笛频率更低。

1. 1. 驶近阶段：救护车向观察者运动，因此声波的波前会在波源前方聚集。
2. 2. 更短的观测波长会导致更高的观测频率，因此警笛的音调听起来更高。
3. 3. 驶离阶段：波前在波源后方被拉伸，导致观测波长更长、观测频率更低，因此警笛的音调听起来更低。

> **考试提示**
>
> 本主题不会要求你计算多普勒频移的具体数值，仅要求解释其定性效应。作答时务必将频率变化与波长变化关联起来，才能拿到满分。

*计算器:* forbidden

## 常见错误

- **错误做法:** Mixing up oscillation directions for transverse and longitudinal waves
  - 原因: This is a common 1-2 mark exam question, mixing the two loses easy marks
  - 正确做法: Remember transverse = perpendicular, longitudinal = parallel; use examples (EM = transverse, sound = longitudinal) to check your answer
- **错误做法:** Using non-SI units directly in wave calculations (e.g. kHz, cm)
  - 原因: Wave formulae only give correct answers when using SI units (Hz, m, m/s)
  - 正确做法: Convert all units to SI first: multiply kHz by 1000 to get Hz, multiply cm by 0.01 to get m
- **错误做法:** Stating that waves transfer matter as they travel
  - 原因: This is a common misconception tested in multiple choice questions
  - 正确做法: Recall that waves only transfer energy and information, with no net movement of matter particles
- **错误做法:** Using $f = T$ instead of $f = 1/T$ for frequency calculations
  - 原因: The formula is not provided on the formula sheet, so misremembering it leads to wrong answers
  - 正确做法: Recall frequency is the number of waves per second, so divide 1 by the time for one wave (period) to get frequency
- **错误做法:** Claiming the Doppler effect changes the frequency emitted by the source
  - 原因: Examiners regularly test that only observed frequency changes, not source frequency
  - 正确做法: Clarify the source emits a constant frequency, only the observed frequency and wavelength change with relative movement

## 速查表

| Quantity | Symbol | Unit | Formula/Definition |
| --- | --- | --- | --- |
| Amplitude | A | m | Max displacement from rest position |
| Wavelength | $\lambda$ | m | Distance between two adjacent in-phase points |
| Frequency | f | Hz | Number of waves per second; $f = 1/T$ |
| Time Period | T | s | Time for one complete wave; $T = 1/f$ |
| Wave Speed | v | m/s | $v = f \times \lambda$ |
| Transverse Wave | - | - | Oscillations perpendicular to energy transfer |
| Longitudinal Wave | - | - | Oscillations parallel to energy transfer |
| Doppler (Approaching Source) | - | - | Observed f higher, $\lambda$ shorter |
| Doppler (Receding Source) | - | - | Observed f lower, $\lambda$ longer |

## 下一步

Now that you have mastered core wave properties, you are ready to progress to the remaining sub-topics in the Edexcel IGCSE Physics waves unit. Next, you will learn about the full electromagnetic spectrum, including its ordering, common uses, and associated safety risks, as well as how EM wave properties apply to real-world technologies. Following that, you will explore detailed wave behaviours including the laws of reflection and refraction, ray diagrams, total internal reflection, and specific properties of sound waves. This foundational knowledge will also be critical when you later study astrophysics topics, including the Doppler red-shift evidence for the expansion of the universe. Make sure you memorize the two wave formulae and unit conversion rules, as they will appear frequently across all these subsequent topics.

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