📚 Wave Properties | 波的性质
Waves are fundamental in physics, transferring energy and information without the permanent displacement of matter. Understanding wave properties — from simple harmonic origins to complex interference patterns — is essential for IB Physics. This article systematically explains transverse and longitudinal waves, key parameters, the wave equation, and the behaviours of reflection, refraction, diffraction, superposition, standing waves, polarisation, the Doppler effect, and the relationship between intensity and amplitude.
波是物理学的基础，能够在不引起物质永久位移的情况下传递能量和信息。理解波的性质——从简谐运动的起源到复杂的干涉图样——是IB物理的重要内容。本文系统地阐释了横波与纵波、关键参数、波动方程，以及反射、折射、衍射、叠加、驻波、偏振、多普勒效应以及强度与振幅的关系。
1. Wave Types: Transverse and Longitudinal | 波的类型：横波与纵波
A transverse wave is one in which the particles of the medium oscillate perpendicularly to the direction of energy transfer. Examples include waves on a string, water surface waves, and all electromagnetic waves. In a longitudinal wave, particles oscillate parallel to the direction of energy propagation, producing compressions and rarefactions. Sound waves in air and seismic P-waves are longitudinal.
横波是指介质粒子的振动方向与能量传递方向垂直的波。例如绳波、水表面波以及所有电磁波。在纵波中，粒子沿能量传播方向振动，产生疏密区域。空气中的声波和地震纵波都是纵波。
Both wave types can be described by their displacement–distance or displacement–time graphs. For transverse waves the displacement is easy to visualise; for longitudinal waves, the displacement is often plotted as a variation of pressure or particle position along the propagation axis.
两种波都可以用位移–距离图或位移–时间图来描述。横波的位移很容易可视化；对于纵波，位移通常表示为沿传播轴的压力变化或粒子位置变化。
2. Basic Wave Parameters | 波的基本参数
ion, and the frequency (f) is the number of complete cycles per second, measured in hertz (Hz). They are related by f = 1 / T.
振幅（A）是粒子偏离平衡位置的最大位移，单位为米。波长（λ）是相邻同相点之间的最短距离，例如波峰到波峰或密部到密部。周期（T）是一次全振动所需的时间，频率（f）是每秒完整循环的次数，单位为赫兹（Hz）。两者满足 f = 1 / T。
The wave speed (v) is the distance travelled per unit time by the wave disturbance. In a given medium, the speed depends only on the medium's properties, not on frequency or amplitude.
波速（v）是波扰在单位时间内传播的距离。在给定介质中，波速只取决于介质的性质，与频率或振幅无关。
3. The Wave Equation | 波动方程
The fundamental relationship connecting wave speed, frequency and wavelength is the wave equation:
波速、频率和波长之间的基本关系是波动方程：
v = f λ
Here v is the speed of the wave (m s⁻¹), f is the frequency (Hz) and λ is the wavelength (m). This equation applies to all types of waves, provided the medium is uniform and the wave is periodic. Since v is usually constant for a specific medium, an increase in frequency leads to a proportional decrease in wavelength.
其中 v 是波速（米/秒），f 是频率（赫兹），λ 是波长（米）。该方程适用于所有类型的波，只要介质均匀且波是周期性的。由于在特定介质中 v 通常是常数，频率增加会导致波长成比例地减小。
4. Wavefronts and Huygens' Principle | 波阵面与惠更斯原理
A wavefront is a surface that connects all adjacent points that are in phase. For a point source, wavefronts are spherical; for a plane wave, they are parallel lines. Huygens' principle states that every point on a wavefront acts as a source of secondary spherical wavelets. The envelope of these wavelets determines the new position of the wavefront at a later time. This construction elegantly explains reflection, refraction and diffraction.
波阵面是连接所有相邻同相点的曲面。对于点源，波阵面呈球形；对于平面波，波阵面是平行线。惠更斯原理指出，波阵面上的每一个点都可以看作发射球面次波的子波源。这些子波的包络面决定了下一时刻波阵面的新位置。这一构造方法可以优雅地解释反射、折射和衍射现象。
5. Reflection of Waves | 波的反射
When a wave encounters a boundary, it can be reflected. The law of reflection states that the angle of incidence θᵢ equals the angle of reflection θᵣ, both measured with respect to the normal. The wavelength and frequency remain unchanged, but the direction of propagation changes. Huygens' principle shows that the circular wavelets originating from the boundary construct a new wavefront obeying this law.
当波遇到界面时会发生反射。反射定律指出，入射角 θᵢ 等于反射角 θᵣ，两者都是相对于法线测量的。波长和频率保持不变，但传播方向改变。惠更斯原理表明，从边界产生的圆形子波会构建出遵从该定律的新波阵面。
Fixed-end and free-end reflections introduce phase changes: a pulse reflected from a fixed end undergoes a phase inversion of π rad (180°), while a free-end reflection produces no phase change.
固定端和自由端的反射会引入相位变化：从固定端反射的脉冲发生π弧度（180°）的相位反转，而自由端反射没有相位变化。
6. Refraction of Waves | 波的折射
Refraction is the change in direction of a wave as it passes from one medium to another where its speed differs. The frequency remains constant, so the wavelength changes. Snell's law quantifies the relationship:
折射是波从一种介质进入波速不同的另一种介质时传播方向发生改变的现象。频率保持不变，因此波长会改变。斯涅尔定律定量地描述了这个关系：
n₁ sin θ₁ = n₂ sin θ₂
ve entering a denser medium slows down and bends towards the normal. Refraction explains phenomena such as the apparent bending of a stick in water.
其中 n₁ 和 n₂ 是介质的折射率，θ₁ 和 θ₂ 是相对于法线的角度。因为 n = c / v，波进入光密介质时速度减慢并向法线偏折。折射解释了诸如水中筷子看起来弯曲等现象。
7. Diffraction of Waves | 波的衍射
to the size of the aperture or obstacle, a. Significant spreading occurs when λ ≥ a. If λ ≪ a, the wave casts a sharp shadow with little diffraction. Huygens' in principle again provides a clear explanation: edge wavelets interfere to produce the curved wavefronts.
衍射是波通过缝隙或绕过障碍物时发生的展宽现象。衍射的程度取决于波长 λ 与缝隙或障碍物尺寸 a 之比。当 λ ≥ a 时会发生明显的展宽。如果 λ ≪ a，波会形成清晰的阴影，衍射很小。惠更斯原理再次给出了清晰的解释：边缘的子波相互干涉形成弯曲的波阵面。
um and alternating dark and bright fringes, with the first minimum satisfying a sin θ = λ.
单缝衍射图样呈现明亮的中央极大以及交替的暗纹和亮纹，第一级极小满足 a sin θ = λ。
8. Superposition and Interference | 波的叠加与干涉
The principle of superposition states that when two or more waves overlap, the resultant displacement is the vector sum of the individual displacements. Constructive interference occurs when waves meet in phase, resulting in a larger amplitude; destructive interference occurs when they meet exactly out of phase, leading to cancellation.
叠加原理指出，当两个或多个波重叠时，合位移是各个位移的矢量和。当波同相相遇时发生相长干涉，振幅增大；当波完全反相相遇时发生相消干涉，导致抵消。
an integer multiple of λ: d sin θ = n λ, where d is the slit separation and n = 0, ±1, ±2 …
双缝实验展示了光的干涉。当光程差为 λ 的整数倍时出现亮纹：d sin θ = n λ，其中 d 是缝间距，n = 0, ±1, ±2……
d sin θ = n λ
For white light, coloured fringes are produced because different wavelengths interfere constructively at different angles.
对于白光，因为不同波长在不同角度产生相长干涉，会形成彩色条纹。
9. Standing Waves | 驻波
A standing wave is formed when two identical waves travelling in opposite directions interfere. It is characterised by nodes (points of zero displacement) and antinodes (points of maximum displacement). The distance between adjacent nodes or antinodes is half a wavelength, λ/2. Standing waves store energy, unlike travelling waves that transmit it.
驻波由两列同频率、同振幅、相向传播的波干涉形成。其特征是波节（位移为零的点）和波腹（位移最大的点）。相邻波节或波腹之间的距离为半个波长 λ/2。与传输能量的行波不同，驻波储存能量。
For a string fixed at both ends, the resonant frequencies are given by:
对于两端固定的弦，共振频率由下式给出：
fₙ = n v / (2 L), n = 1, 2, 3 …
where L is the string length and v is the wave speed. The fundamental frequency is f₁, and higher harmonics are integer multiples.
其中 L 是弦长，v 是波速。基频为 f₁，高次谐波是基频的整数倍。
10. Polarisation | 偏振
Polarisation is a property unique to transverse waves. It describes the orientation of oscillations in the plane perpendicular to the direction of propagation. Unpolarised light has oscillations in all possible transverse directions; a polarising filter transmits only the component parallel to its transmission axis. Longitudinal waves cannot be polarised because their oscillations are already one-dimensional.
偏振是横波独有的性质。它描述了在垂直于传播方向的平面内振动的取向。非偏振光在所有可能的横向方向都有振动；偏振滤光片只允许平行于其透射轴的分量通过。纵波无法被偏振，因为其振动本来就只沿一维方向。
ity is I = I₀ / 2. For a second polariser (analyser) set at an angle θ to the first, Malus's law applies:
当强度为 I₀ 的非偏振光通过理想偏振片时，透射强度为 I = I₀ / 2。若第二个偏振片（检偏器）与第一个偏振片的透射轴夹角为 θ，则适用马吕斯定律：
I = I₀ cos² θ
Polarisation is used in sunglasses, LCD screens and stress analysis.
偏振应用于太阳镜、液晶显示屏和应力分析中。
11. The Doppler Effect | 多普勒效应
The Doppler effect is the change in observed frequency when a wave source and an observer move relative to each other. For sound, the observed frequency fₒ is given by:
多普勒效应是波源和观察者相对运动时观测频率发生变化的现象。对于声波，观测频率 fₒ 如下：
fₒ = fₛ (v ± vₒ) / (v ± vₛ)
Here fₛ is the source frequency, v is the wave speed in the medium, vₒ is the observer's speed relative to the medium, and vₛ is the source's speed. The signs are chosen to reflect approach (higher frequency) or recession (lower frequency). For light, the relativistic Doppler effect must be used, as the classical formula fails at high speeds.
其中 fₛ 是波源频率，v 是波在介质中的速度，vₒ 是观察者相对于介质的速度，vₛ 是波源速度。正负号的选取取决于接近（频率升高）还是远离（频率降低）。对于光波，必须使用相对论多普勒效应，因为经典公式在高速下失效。
The Doppler effect is applied in radar speed guns, medical ultrasound imaging and astronomy (redshift).
多普勒效应应用于雷达测速枪、医学超声成像和天文学（红移）。
12. Wave Intensity and Amplitude | 波的强度与振幅
The intensity I of a wave is the power transmitted per unit area perpendicular to the propagation direction. For a point source emitting uniformly in three dimensions, intensity decreases as the inverse square of the distance: I ∝ 1 / r². Importantly, intensity is proportional to the square of the amplitude:
波的强度 I 是沿传播方向垂直的单位面积上传递的功率。对于在三维空间中均匀发射的点源，强度随距离的平方反比衰减：I ∝ 1 / r²。重要的是，强度与振幅的平方成正比：
I ∝ A²
This relationship holds for all types of waves and explains why a doubling of amplitude results in a quadrupling of energy transfer. In combination with the inverse-square law, it links observed loudness or brightness to the energy emitted by the source.
这一关系对所有类型的波都成立，它解释了为什么振幅加倍会导致能量传递变为四倍。结合平方反比定律，它将观测到的响度或亮度与波源发射的能量联系起来。
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