📚 4 Wave behaviour | 4 波的行为
Wave behaviour is a central theme in IB Physics, exploring how waves interact with boundaries, other waves, and the media they travel through. This topic covers reflection, refraction, diffraction, superposition, interference, standing waves, polarisation, and the Doppler effect – all fundamental to understanding sound, light, and many modern technologies.
波的行为是IB物理的核心主题，研究波如何与边界、其他波以及传播介质相互作用。本主题涵盖反射、折射、衍射、叠加、干涉、驻波、偏振和多普勒效应，这些内容对于理解声、光以及许多现代技术都至关重要。
1. Introduction to Wave Behaviour | 波的行为概述
When a wave encounters a change in medium or an obstacle, its behaviour can be described by a set of universal principles. These principles apply to both mechanical waves, such as sound and water waves, and electromagnetic waves, including light, radio waves, and X-rays.
当波遇到介质变化或障碍物时，其行为可以用一组普适原理来描述。这些原理不仅适用于声波和水波等机械波，也适用于光、无线电波和X射线等电磁波。
Key wave phenomena include reflection (bouncing off a surface), refraction (bending due to speed change), diffraction (spreading around edges), interference (superposition of two or more waves), polarisation (restriction of oscillation direction), and the Doppler effect (frequency shift due to relative motion).
关键的波现象包括反射（从表面弹回）、折射（因速度改变而弯曲）、衍射（绕过边缘扩展）、干涉（两个或多个波的叠加）、偏振（限制振动方向）以及多普勒效应（相对运动引起的频率变化）。
Understanding wave behaviour not only explains everyday experiences like echoes and rainbows but also underpins advanced technologies such as fibre optics, medical ultrasound, and astronomical spectroscopy.
理解波的行为不仅能解释回声和彩虹等日常经验，还为光纤通信、医学超声和天文光谱学等先进技术奠定了基础。
2. Reflection of Waves | 波的反射
Reflection occurs when a wave strikes a boundary and returns into the original medium. For a straight wavefront incident on a flat surface, the law of reflection states that the angle of incidence θᵢ equals the angle of reflection θᵣ, both measured with respect to the normal to the surface.
当波撞击边界并返回原介质时，就发生反射。对于入射到平坦表面的直波前，反射定律表明入射角θᵢ等于反射角θᵣ，两者都是相对于表面法线测量的。
θᵢ = θᵣ
At a fixed boundary (e.g. a rope tied to a wall), the reflected pulse undergoes a phase change of π radians (180°), meaning it is inverted. At a free boundary, the pulse reflects without a phase change, remaining upright.
在固定边界（例如绳子系在墙上），反射脉冲会发生π弧度（180°）的相位变化，即脉冲倒置。在自由边界，脉冲反射时没有相位变化，保持原样。
In two-dimensional reflection, plane wavefronts obey the same angle rule, and ray diagrams are often used to represent the direction of energy travel. Applications range from parabolic mirrors in telescopes to sonar and radar systems.
在二维反射中，平面波前遵循相同的角度规则，通常用光线图表示能量传播方向。应用范围从望远镜中的抛物面镜到声纳和雷达系统。
3. Refraction of Waves | 波的折射
Refraction is the change in direction of a wave as it passes from one medium into another in which its speed differs. When a wave slows down, it bends towards the normal; when it speeds up, it bends away from the normal. This is described by Snell's law.
折射是指波从一种介质进入速度不同的另一种介质时方向的改变。当波减速时，它向法线弯曲；当波加速时，它远离法线弯曲。这由斯涅尔定律描述。
n₁ sinθ₁ = n₂ sinθ₂
c / v, where c is the speed of light in vacuum and v is the speed of light in the medium.
其中n₁和n₂是两种介质的折射率，θ₁和θ₂是与法线的夹角。介质的折射率定义为n = c / v，c是真空中的光速，v是光在介质中的速度。
The change in speed also causes a change in wavelength, while the frequency remains constant. For light, this explains why a straw appears bent in a glass of water, and it is the basis for lenses and optical instruments.
速度的变化也会引起波长的变化，而频率保持不变。对于光来说，这解释了吸管在水中看起来弯曲的原因，也是透镜和光学仪器的基础。
c, given by sinθ_c = n₂ / n₁ (with n₁ > n₂). This principle drives fibre optic technology.
当光从光密介质传播到光疏介质时，如果入射角超过临界角θ_c，就会发生全内反射，临界角由sinθ_c = n₂ / n₁ （其中n₁ > n₂）给出。这一原理驱动了光纤技术。
4. Diffraction of Waves | 波的衍射
Diffraction is the spreading out of a wave as it passes through a gap or around an obstacle. The amount of diffraction depends on the ratio of the wavelength λ to the size of the opening or obstacle a. Significant diffraction occurs when the wavelength is comparable to or larger than the aperture.
衍射是指波穿过狭缝或绕过障碍物时扩展开来的现象。衍射的程度取决于波长λ与开口或障碍物大小a的比值。当波长与孔径相当或更大时，就会发生显著的衍射。
A narrow gap comparable to the wavelength causes almost circular spreading of the wavefronts, while a wide opening produces only slight bending at the edges. This behaviour is observed for all wave types, including water waves rippling through harbour entrances and sound waves bending around a door.
与波长相当的窄缝会使波前几乎以圆形扩散，而宽开口仅会在边缘产生轻微的弯曲。所有类型的波都表现出这种行为，包括水波穿过港口入口和声波绕过门框。
Diffraction limits the resolution of optical instruments such as microscopes and telescopes because it causes light from a point source to form a small disk rather than a sharp point. The diffraction grating, on the other hand, exploits diffraction to separate light into its spectral components with high precision.
衍射限制了显微镜和望远镜等光学仪器的分辨率，因为它使点光源的光形成一个弥散斑而非锐利点。另一方面，衍射光栅利用衍射将光高精度地分离成光谱成分。
5. Superposition and Interference | 叠加与干涉
The principle of superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. This leads to interference effects where waves can reinforce or cancel each other.
叠加原理指出，当两个或多个波在某点相遇时，合位移等于各个位移的矢量和。这就会产生干涉效应，波可以相互增强或抵消。
y meet exactly out of phase, with path difference ΔL = (n + ½)λ, giving minimum amplitude.
当波同相相遇时，即它们之间的路径差为波长的整数倍：ΔL = nλ，就发生相长干涉，振幅加大。当它们完全反相相遇，路径差为ΔL = (n + ½)λ 时，发生相消干涉，振幅最小。
Interference can be observed with all wave types. For coherent light sources, Young's double-slit experiment produces a stable pattern of bright and dark fringes. In acoustics, two loudspeakers playing the same pure tone can create loud and quiet zones in a room due to interference.
所有类型的波都可以观察到干涉现象。对于相干光源，杨氏双缝实验产生稳定的明暗条纹图案。在声学中，两个播放相同纯音的扬声器会因干涉在房间内产生声音响度高低不同的区域。
6. Young's Double-Slit Experiment | 杨氏双缝实验
Young's double-slit experiment provided definitive evidence for the wave nature of light. A monochromatic light source illuminates two closely spaced slits, producing two coherent sources. The overlapping waves create an interference pattern on a distant screen.
杨氏双缝实验为光的波动性提供了确凿证据。单色光源照射两条靠得很近的狭缝，产生两个相干波源。重叠的波在远处的屏幕上形成干涉条纹。
The fringe spacing Δy (the distance between adjacent bright or dark fringes) is given by the simple formula:
条纹间距Δy（相邻亮纹或暗纹之间的距离）由以下简单公式给出：
Δy = λD / d
where λ is the wavelength, D is the distance from the slits to the screen, and d is the slit separation. This relationship allows the wavelength of light to be measured very accurately.
其中λ是波长，D是狭缝到屏幕的距离，d是狭缝间距。这个关系使得我们可以非常精确地测量光的波长。
If white light is used instead of monochromatic light, a central white fringe is observed, flanked by spectra of colours on each side, because different wavelengths produce fringes at different positions. This demonstrates dispersion through interference.
如果使用白光代替单色光，会观察到中央白色条纹，两侧分布着彩色光谱，因为不同的波长在不同的位置形成条纹。这展示了干涉引起的色散。
7. Standing Waves | 驻波
A standing wave is formed when two identical travelling waves moving in opposite directions superpose, typically as a result of reflection from a boundary. The wave pattern appears to stay in place, with points of zero displacement called nodes and points of maximum displacement called antinodes.
驻波是由两列相同但方向相反的波叠加形成的，通常是由于边界反射造成的。波形看似原地不动，存在位移为零的节点和位移最大的腹点。
In a string fixed at both ends, standing waves can form at certain frequencies, producing harmonics. The fundamental frequency f₁ corresponds to a pattern with one antinode at the centre and nodes at the ends, while higher harmonics are integer multiples of the fundamental.
在两端固定的弦上，驻波可以在特定频率下形成，产生谐波。基频f₁对应中心一个腹点、两端节点的模式，而更高次谐波则是基频的整数倍。
For a fixed string: fₙ = n × (v / 2L) = n × f₁
antinode at the open end, giving only odd harmonics: fₙ = n × (v / 4L) where n = 1,3,5...
对于两端开放的管中的声波，边界条件允许开口端为腹点，其谐波序列与固定弦类似。在一端封闭的管中，封闭端总是节点，开口端为腹点，因此只产生奇次谐波：fₙ = n × (v / 4L)，其中n = 1,3,5…
Standing waves are essential to musical instruments, from guitar strings to organ pipes, and also appear in microwave cavities and laser resonators.
驻波对于乐器至关重要，从吉他弦到管风琴，也出现在微波腔和激光谐振腔中。
8. Polarisation of Waves | 波的偏振
Polarisation is a property unique to transverse waves, in which the oscillations are confined to a single plane. Longitudinal waves, such as sound, cannot be polarised. Polarisation of light provides strong evidence that light is a transverse wave.
偏振是横波独有的性质，其振动被限制在一个平面内。纵波（如声波）不能偏振。光的偏振提供了光是横波的有力证据。
A polarising filter transmits only the component of the wave oscillating in its transmission axis. When unpolarised light passes through an ideal polariser, its intensity is reduced by half. If a second polariser (analyser) is placed after the first, the transmitted intensity follows Malus's law.
偏振滤光片只允许振动方向与透射轴平行的分量通过。当非偏振光通过理想偏振片时，光强减半。如果在第一个偏振片之后放置第二个偏振片（检偏器），透射光强遵循马吕斯定律。
I = I₀ cos²θ
the transmission axes of the polariser and analyser. At θ = 90°, no light is transmitted.
其中I₀是入射到检偏器的光强，θ是起偏器和检偏器透射轴之间的夹角。当θ = 90°时，没有光透过。
Polarisation can also occur by reflection at a specific angle known as Brewster's angle, at which the reflected ray is completely plane-polarised. This is exploited in polarising sunglasses and camera filters to reduce glare.
偏振也可以通过反射产生，在特定角度（称为布儒斯特角）下，反射光完全成为平面偏振光。这一原理被用于偏光太阳镜和相机滤镜，以减少眩光。
9. The Doppler Effect | 多普勒效应
The Doppler effect is the apparent change in frequency (and wavelength) of a wave due to relative motion between the source and the observer. When the source and observer move closer together, the observed frequency is higher (blue shift for light); when moving apart, the observed frequency is lower (red shift).
多普勒效应是指由于波源和观察者之间的相对运动，导致波的频率（以及波长）发生视变化。当波源和观察者相互靠近时，观察到的频率变高（光的蓝移）；相互远离时，频率变低（红移）。
For sound waves, the observed frequency f' is given by:
对于声波，观察到的频率f'由下式给出：
f' = f × (v ± vₒ) / (v ∓ vₛ)
where f is the source frequency, v is the speed of sound, vₒ is the observer's speed, and vₛ is the source's speed. The upper signs correspond to motion towards each other (approaching), and the lower signs to motion apart.
其中f是波源频率，v是声速，vₒ是观察者的速度，vₛ是波源的速度。上面的符号对应相互靠近（接近），下面的符号对应相互远离。
The Doppler effect is widely used: in radar speed guns, medical ultrasound to measure blood flow, and in astronomy to determine the recessional velocities of galaxies, providing evidence for the expansion of the universe.
多普勒效应应用广泛：雷达测速枪、医学超声测量血流、以及天文学中测定星系的退行速度，为宇宙膨胀提供了证据。
10. Practical Applications of Wave Behaviour | 波的行为的实际应用
The principles of wave behaviour are embedded in countless everyday and scientific devices. Optical fibres rely on total internal reflection to transmit data over long distances with minimal loss, forming the backbone of the internet.
波的行为原理融入了无数日常和科学设备中。光纤依赖全内反射以极低损耗长距离传输数据，构成了互联网的主干。
Interference phenomena are harnessed in anti-reflective coatings on lenses, which use thin-film interference to cancel reflections and improve transmission. The Michelson interferometer, which splits and recombines light beams, can measure tiny distances and was used in the historic detection of gravitational waves.
干涉现象被用于透镜的抗反射涂层，利用薄膜干涉消除反射来提高透射率。迈克尔逊干涉仪将光束分裂再合并，可以测量微小距离，并曾用于历史性的引力波探测。
Diffraction gratings separate light into its constituent wavelengths with high resolution, enabling spectroscopy in chemistry and astrophysics. Standing waves allow precise measurement of acoustic frequencies and are fundamental to the design of concert halls and musical instruments.
衍射光栅能够高分辨率地将光分解为各个波长，使化学和天体物理中的光谱学成为可能。驻波可以实现对声频的精确测量，也是音乐厅和乐器设计的基础。
Polarisation is critical in liquid crystal displays (LCDs), where controlling the polarisation state of light produces images. In medicine, Doppler ultrasound imaging combines the Doppler effect with ultrasound to visualise blood flow and heart function in real time.
偏振在液晶显示器中至关重要，通过控制光的偏振状态来产生图像。在医学上，多普勒超声成像将多普勒效应与超声波结合，实时观察血流和心脏功能。
A thorough understanding of wave behaviour thus not only satisfies intellectual curiosity but also empowers innovation across physics, engineering, and beyond.
因此，透彻理解波的行为不仅能满足求知欲，还能推动物理、工程等众多领域的创新。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图