📚 Wave Energy: Amplitude, Intensity and Power Transfer | 波的能量：振幅、强度与能量传输
Wave energy is a central idea in A-Level Physics: waves transfer energy from a source to a detector without any bulk transfer of matter. Understanding how amplitude, frequency, power and intensity are related helps you analyse sound, light, water waves and waves on strings.
波的能量是 A-Level 物理的核心概念：波将能量从波源传递到探测器，而不发生物质的整体迁移。理解振幅、频率、功率与强度之间的关系，有助于分析声波、光波、水波和弦上的波。
1. Progressive Waves Transfer Energy Without Transferring Matter | 行波传递能量而不传递物质
A progressive wave is a disturbance that carries energy away from a source. The particles of the medium oscillate about fixed equilibrium positions, but they do not travel with the wave.
行波是一种将能量从波源向外传递的扰动。介质中的粒子围绕固定的平衡位置振动，但它们不会随波移动。
For example, when a stone is dropped into still water, the water surface moves up and down, yet the energy travels outward in circular ripples. This shows that wave motion transfers energy, not material.
例如，当石头掉入静水中时，水面上下运动，但能量以圆形波纹向外传播。这说明波动传递的是能量，而不是物质。
This distinction is frequently examined: if particles moved with the wave, the medium itself would be transported, which is not observed for mechanical waves.
这个区别是常考内容：如果粒子随波移动，介质本身就会被输运，但机械波中观察不到这种现象。
2. Energy of an Oscillating Particle in a Wave | 波中振动粒子的能量
In a sinusoidal progressive wave, each particle performs simple harmonic motion. Its total energy is constant if no energy is lost and is given by:
在正弦行波中，每个粒子都做简谐运动。如果没有能量损失，粒子的总能量保持不变，并由下式给出：
E_total = ½ mω²A²
Here m is the particle mass, ω = 2πf is angular frequency, and A is amplitude. Because energy depends on A², a small increase in amplitude causes a large increase in energy.
其中 m 是粒子质量，ω = 2πf 是角频率，A 是振幅。由于能量取决于 A²，振幅的少量增大会引起能量的大幅增加。
The kinetic energy and potential energy of the particle are interchanged during each cycle, but their sum remains ½ mω²A².
在每个周期中，粒子的动能和势能相互转化，但它们的总和保持为 ½ mω²A²。
3. Energy Carried by a Wave on a Stretched String | 弦上波传递的能量
For a stretched string, energy has two forms: kinetic energy of moving string elements and elastic potential energy due to tension. A sinusoidal wave of amplitude A, angular frequency ω and wave speed v carries average power:
对于张紧的弦，能量有两种形式：弦元的动能和由张力引起的弹性势能。振幅为 A、角频率为 ω、波速为 v 的正弦波携带的平均功率为：
P = ½ μ v ω² A²
Here μ is the mass per unit length of the string. This formula shows that the power transmitted depends on the properties of the medium, the frequency and the amplitude.
其中 μ 是弦的线密度（单位长度的质量）。这个公式表明，传递的功率取决于介质的性质、频率和振幅。
Since ω = 2πf, this can also be written as P = 2π² μ v f² A². The key point is that power is proportional to both frequency squared and amplitude squared.
由于 ω = 2πf，该式也可以写成 P = 2π² μ v f² A²。关键是功率与频率的平方和振幅的平方都成正比。
4. Distinguishing Energy, Power and Intensity | 区分能量、功率与强度
Energy is measured in joules (J). Power is the rate of energy transfer, measured in watts (W), where 1 W = 1 J s⁻¹. For a continuous wave, it is often more useful to quote power than total energy because a wave can travel indefinitely.
能量以焦耳（J）为单位。功率是能量传递的速率，以瓦特（W）为单位，其中 1 W = 1 J s⁻¹。对于连续波，通常给出功率比给出总能量更有用，因为波可以无限传播。
Intensity is the power passing through unit area perpendicular to the direction of energy flow. It is measured in watts per square metre (W m⁻²).
强度是单位面积上、垂直于能量流动方向通过的功率。其单位是瓦特每平方米（W m⁻²）。
Intensity is the quantity used to compare how concentrated a wave is at a particular point. It links energy transport to wavefront geometry.
强度用于比较波在某一特定点的集中程度。它把能量传递与波阵面的几何形状联系起来。
5. Intensity and Plane Waves | 强度与平面波
For a plane wave, the wave fronts are parallel planes. If there is no absorption or scattering, the same power passes through each plane, so intensity remains constant along the path.
对于平面波，波阵面是平行的平面。如果没有吸收或散射，每个平面通过的功率相同，因此强度沿传播路径保持不变。
In practice, plane waves are only approximations. However, a laser beam or a wave in a uniform pipe can be modelled as a plane wave over short distances.
实际上，平面波只是近似。然而，激光束或均匀管道中的波在短距离内可以近似为平面波。
6. Intensity Proportional to Amplitude Squared | 强度与振幅的平方成正比
A fundamental CIE result is: for any progressive wave, intensity is directly proportional to the square of the amplitude:
CIE 的一个基本结论是：对于任何行波，强度与振幅的平方成正比：
I ∝ A²
Therefore, if the amplitude is doubled, intensity increases by a factor of 2² = 4. If the amplitude is tripled, intensity increases by a factor of 9.
因此，如果振幅加倍，强度增加为原来的 2² = 4 倍。如果振幅增加到 3 倍，强度增加为原来的 9 倍。
This relationship assumes the medium and frequency are unchanged. It follows from the fact that the energy carried by each oscillating particle is proportional to A².
这个关系假定介质和频率不变。它源于每个振动粒子携带的能量与 A² 成正比。
7. Spherical Waves and the Inverse Square Law | 球面波与平方反比定律
A point source emitting waves uniformly in all directions produces spherical wave fronts. The total power P is spread over a sphere of radius r, whose surface area is 4πr².
一个向所有方向均匀发射波的点源会产生球面波阵面。总功率 P 分布在一个半径为 r 的球面上，其表面积为 4πr²。
I = P / (4πr²)
Thus intensity obeys an inverse square law: I ∝ 1/r². Doubling the distance from the source reduces intensity to one quarter.
因此，强度遵循平方反比定律：I ∝ 1/r²。距离波源加倍会使强度减小到原来的四分之一。
Combining I ∝ A² with I ∝ 1/r² shows that for a spherical wave the amplitude is inversely proportional to distance, A ∝ 1/r.
将 I ∝ A² 与 I ∝ 1/r² 结合起来，可知对于球面波，振幅与距离成反比，即 A ∝ 1/r。
8. Energy in Different Types of Wave | 不同类型波中的能量
Wave energy can be stored in different forms depending on the type of wave. The table below summarises the main energy forms for common wave types.
波的能量根据波的类型可以储存为不同形式。下表总结了常见波类型的主要能量形式。
| Type of wave | Main energy forms |
|---|---|
| Mechanical wave on a string | Kinetic energy + elastic potential energy |
| Sound wave | Kinetic energy + pressure potential energy |
| Water wave | Kinetic energy + gravitational potential energy |
| Electromagnetic wave | Electric field energy + magnetic field energy |
In all cases, the total energy is proportional to amplitude squared for a given frequency and medium.
在所有情况下，对于给定的频率和介质，总能量都与振幅的平方成正比。
9. Damping, Absorption and Energy Loss | 阻尼、吸收与能量损失
Real waves lose energy by absorption and scattering. As energy is removed, the amplitude decreases, so damping reduces both amplitude and intensity.
真实波会因吸收和散射而损失能量。能量被带走后，振幅会减小，因此阻尼会同时减小振幅和强度。
In an absorbing medium, intensity often decreases roughly exponentially with distance travelled. This means a constant fraction of energy is lost per unit length in many materials.
在吸收介质中，强度通常随传播距离近似指数下降。这意味着在许多材料中，每单位长度损失的能量比例是恒定的。
In forced oscillation and resonance, at steady state the energy input per cycle equals the energy dissipated per cycle. This is why damping controls the maximum amplitude at resonance.
在受迫振动和共振中，达到稳态时每个周期输入的能量等于耗散的能量。这就是阻尼会控制共振最大振幅的原因。
10. Worked Examples and Exam Strategy | 例题与备考策略
Example 1: A water wave has amplitude 2.0 mm and intensity 6.0 W m⁻². Find the new intensity if the amplitude becomes 6.0 mm.
例题 1：一个水波的振幅为 2.0 mm，强度为 6.0 W m⁻²。如果振幅变为 6.0 mm，求新的强度。
Since I ∝ A², the intensity ratio is (6.0 / 2.0)² = 3² = 9. Therefore the new intensity is 6.0 × 9 = 54 W m⁻².
因为 I ∝ A²，强度比为 (6.0 / 2.0)² = 3² = 9。因此新的强度为 6.0 × 9 = 54 W m⁻²。
Example 2: A small loudspeaker emits sound uniformly with a power of 12 W. Calculate the intensity at 2.0 m from the speaker, assuming no absorption.
例题 2：一个小扬声器以 12 W 的功率均匀发射声音。假设没有吸收，计算距扬声器 2.0 m 处的强度。
I = P / (4πr²) = 12 / (4π × 2.0²) = 12 / (16π) ≈ 0.24 W m⁻².
I = P / (4πr²) = 12 / (4π × 2.0²) = 12 / (16π) ≈ 0.24 W m⁻²。
Exam tip: Always quote intensity in W m⁻², not watts. State I ∝ A² clearly before using ratio calculations. Do not claim that wave speed increases with amplitude; for a given medium, wave speed is fixed by medium properties.
备考提示：强度始终用 W m⁻² 表示，而不是瓦特。在使用比例计算前，要明确写出 I ∝ A²。不要认为波速会随振幅增大；对于给定介质，波速由介质的性质决定。
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