Edexcel物理 波方程 驻波 光栅 光电效应
在 Pearson Edexcel A Level Physics（9PH0）的 13 个主题里，Topic 5 "Waves and Particle Nature of Light" 独占一个主题，主要落在 Paper 2，Paper 3 也会拿它做综合情境题。这一主题从 v = fλ 一路走到光电效应和德布罗意波长，中间的驻波、折射、光栅几个知识点在历年 Paper 2 里出现频率很高。下面把 9PH0 specification 中这一主题的公式、常考图像和判分点拆开讲。
In the Pearson Edexcel 9PH0 specification, Topic 5 runs from the wave equation all the way to wave-particle duality. It is examined mainly in Paper 2 and reappears synoptically in Paper 3. The formulas below are the ones that carry marks in every exam series.
波方程与相位差的换算
波的三个量之间只有一个关系要记牢：v = fλ，其中频率与周期互为倒数，T = 1/f。9PH0 的题目很少直接问 v = fλ，更常见的是给一张示波器照片或 displacement-time graph，让你先读出周期再换算频率。相位差用弧度算，两点相距 Δx 时相位差为 2πΔx/λ；同一波形上相隔整整一个波长，相位差就是 2π。很多学生会把 180 度当成 π/2 而不是 π，选择题的分数常常丢在这个换算上。
Only one equation links the three wave quantities. Speed equals frequency times wavelength, so a period of 0.02 s means 50 Hz before any substitution. Exam questions usually hide the frequency inside a graph reading. Phase difference is measured in radians. Two points separated by a full wavelength differ by 2 pi rather than pi.
横波纵波与两类图像的区别
横波的振动方向与传播方向垂直，纵波两者平行，声波是典型的纵波。考题常给两条曲线：displacement-distance graph 的横轴是位置，读出的间隔是波长；displacement-time graph 的横轴是时间，读出的间隔是周期。两张图形状几乎一样，物理含义完全不同，混淆之后所有代入都会错。另外只有横波能够偏振，这一点在偏振那一节还会再用到。
Transverse waves oscillate perpendicular to their direction of travel, while longitudinal waves oscillate along it. Plotted against distance, the graph yields wavelength; plotted against time, the identical shape yields period. Mixing the two axes up is the classic slip in multiple choice, and it also decides whether the wave can be polarised.
叠加相干与杨氏双缝条纹间距
两列波相遇时位移直接相加，这就是 superposition。要看到稳定的干涉图样，两个光源必须相干，也就是同频率且相位差恒定，所以实验里用激光而不是普通灯泡。杨氏双缝的条纹间距 w = λD/s，D 是缝到屏的距离，s 是缝间距，w 是相邻亮纹的中心距离。光程差为 nλ 处是亮纹，(n + 1/2)λ 处是暗纹。常见的错误是把 D 和 s 弄反，或者拿两个缝的间距去算条纹宽度。
Superposition simply adds the displacements of the two waves. A stable fringe pattern needs coherent sources, which is why a laser replaces a lamp in the standard demonstration. Fringe spacing follows w = lambda D / s, with D the slit-to-screen distance and s the separation of the two slits. Bright fringes sit where the path difference is a whole number of wavelengths; a half-wavelength offset gives darkness.
弦与管中的驻波与谐波
两列同频反向的波叠加形成驻波，节点始终不动，相邻节点间距是 λ/2。两端固定的弦上，波长满足 λ = 2L/n；基频对应 n = 1，此时整条弦只有两个节点和中间一个波腹。弦的基频还能写成 f = (1/2L)√(T/μ)，张力越大音调越高，μ 是单位长度质量。一端封闭的管只出现奇次谐波，λ = 4L/n（n 取 1, 3, 5）。画管中驻波时，学生最容易把开口端画成节点，实际开口端是波腹。
Standing waves form when two identical waves travel in opposite directions; nodes stay fixed and adjacent nodes sit half a wavelength apart. A string fixed at both ends obeys lambda = 2L/n, and its fundamental rises with tension and falls with mass per unit length. A pipe closed at one end supports odd harmonics only, and the open end is always an antinode.
折射率与斯涅尔定律
折射率定义 n = c/v，c 是真空中光速 3.00 x 10^8 m/s，所以 n 永远大于 1。界面处用 n1 sinθ1 = n2 sinθ2，角度一律从法线量起；水约 1.33，玻璃约 1.52，这些数值在 9PH0 数据表里可以直接引用。光从光密介质进光疏介质时折向法线外侧，频率保持不变，波长与速度同时改变。审题时注意题目给的是入射角还是与界面的夹角，基准量错整题归零。
Refractive index is defined as n = c / v, so it can never fall below one. Snell's law pairs the two indices with the sines of the angles measured from the normal, and the data sheet supplies values such as 1.33 for water. Frequency is unchanged across a boundary while speed and wavelength change, which is why colour does not shift on refraction.
全内反射与光导纤维
光从光密射向光疏介质时，入射角超过临界角后不再有折射光，只剩反射。临界角由 sinC = n2/n1 给出，界面另一侧是空气时可近似写成 sinC = 1/n。玻璃到空气的临界角约 41 度，这个数字在光纤题里反复出现。光纤靠全内反射把光锁在纤芯中，包层折射率必须低于纤芯。模态色散和材料色散会拉宽脉冲，这是长距离链路必须加中继的原因。
Past the critical angle no light leaves the denser medium, and total internal reflection takes over. With air on the far side the condition reduces to sin C = 1/n. An optical fibre traps light because the cladding has a lower index than the core, and pulse spreading from dispersion is why long links need repeaters.
波的强度与振幅平方关系
波的强度定义为单位时间内垂直通过单位面积的能流，点光源均匀辐射时 I = P/(4πr²)，距离加倍强度只剩四分之一。强度与振幅的平方成正比，I 随 A² 同步变化，所以振幅减半意味着强度降到原来的 25%。题目里强度与振幅经常混着问，先确认问的是哪一个再代公式。双缝图样中亮纹中心强度最大，向两侧降到接近零，这也是能量重新分布的结果。
Intensity is the power crossing unit area, so a point source spreading evenly obeys an inverse square law: twice the distance leaves a quarter of the intensity. It also scales with amplitude squared. In an interference pattern energy is redistributed rather than created, which is why dark fringes are dark.
薄透镜成像与放大率
薄透镜公式 1/f = 1/u + 1/v，u 是物距，v 是像距，实像取正、虚像取负，放大率 m = v/u。凸透镜在 u > f 时成倒立实像，u < f 时成正立放大的虚像，放大镜就是这么工作的。9PH0 的题目常配光路图，要求标出主光轴、焦点与二倍焦距的位置。画图时三条特殊光线任选两条就能定像，漏画光线箭头是最常见的失分点。
The thin lens equation links focal length, object distance and image distance, while magnification is the ratio of the two distances. A converging lens forms a real inverted image when the object lies outside the focal point, and a magnified virtual image when it lies inside. Two of the three standard rays are enough to locate the image on a ray diagram.
脉冲回波测距与超声成像
脉冲回波法先发出一个短促的波脉冲，再记录反射回来的时刻，距离用 s = vt/2 计算，除以 2 是因为脉冲走了来回两趟。超声成像用 MHz 级频率，波长越短分辨率越高，衰减也越快。9PH0 常给一张发射与回波的时间轴图，让你读出时间差再换算深度。扫查时探头与皮肤之间要涂耦合剂，否则空气界面几乎全反射，信号根本进不去。
Pulse-echo timing sends a short burst out and waits for the reflection, so the depth is half the product of speed and time. Higher frequencies buy finer resolution at the cost of faster attenuation, which is why medical scanners work in the megahertz range. Coupling gel matters because an air gap reflects almost the whole pulse.
偏振与马吕斯定律
偏振只发生在横波上，纵波没有可偏振的横向振动分量，所以声波无法偏振。偏振片只让某一方向的振动通过，两片偏振片夹角为 θ 时透射强度按 I = I0 cos²θ 变化，这就是 Malus 定律。9PH0 里偏振还用来检测应力：透明塑料在受力区会旋转偏振面，两片正交偏振片之间会看到彩色条带。判断偏振题时先看光是否来自反射或散射，再决定第二片偏振片是否完全消光。
Polarisation belongs to transverse waves only, which is why sound cannot be polarised. Crossing two filters cuts the transmitted intensity according to Malus's law. Rotating the analyser through a right angle extinguishes the beam. Stressed plastics show colour bands between crossed filters. That is because the stress rotates the plane of vibration.
单缝衍射与光栅方程
单缝衍射给出又宽又亮的中央极大，缝越窄图样铺得越开；光栅则是大量等宽等距的缝，条纹细而锐利。光栅方程 d sinθ = nλ，d 是相邻缝间距，等于 1/N，N 为单位长度上的刻线数，n 是级次。零级是各色光重合的位置，级次越高角色散越明显。核心实验里用激光垂直照射光栅，先测第一级衍射角再用 d sinθ = λ 求波长，这一步最容易读错分光计的零位。
A single slit produces a broad central maximum that widens as the gap narrows. A grating, with hundreds of lines per millimetre, instead gives narrow bright orders satisfying d sin theta = n lambda, where d is one over the line density. Measuring the first-order angle is the standard core practical route to the wavelength of a laser.
光电效应与遏止电压
在光电效应里，光强只改变光电子的数目，频率才决定单个电子能否逸出。爱因斯坦方程 hf = φ + Ek(max)，φ 是逸出功，Ek(max) 由遏止电压测出，即 Ek(max) = eVs。入射频率低于截止频率时，无论光多强都没有电子出来，经典波动说解释不了这一点，这是支持光子模型的直接证据。作图题会给 Ek(max) 对 f 的直线，斜率是普朗克常量 h，纵轴截距是负的 φ。
Intensity controls how many photoelectrons appear, while frequency decides whether any appear at all. Einstein's equation sets the photon energy against the work function plus the maximum kinetic energy, the latter read from a stopping potential. Below the threshold frequency nothing is emitted however bright the light, and that outcome defeated the classical wave model.
光子能量与原子线谱
单个光子的能量 E = hf = hc/λ，h = 6.63 x 10^-34 J s。原子在两个能级之间跃迁时，吸收或放出的能量恰好等于两能级之差，所以发射光谱是一条条分立亮线而不是连续彩带。计算时把焦耳换成电子伏更省事，1 eV = 1.60 x 10^-19 J，Edexcel 的题目常让答案先用 eV 表示再换算。吸收谱是连续谱上的暗线，位置与同种原子的发射线一一对应，这个对应关系经常作为推导题出现。
A photon carries energy h f, equivalently h c over lambda. Line spectra appear because atoms exchange energy only in amounts equal to the gap between levels. Working in electronvolts keeps the arithmetic short, since one electronvolt is 1.60 times ten to the minus nineteen joules, and absorption lines sit exactly where the emission lines of the same element do.
德布罗意波长与电子衍射
德布罗意把波性给了实物粒子，λ = h/p，p 是动量。电子被电压 V 加速后，由 eV = 1/2 mv² 可推出 λ = h/√(2meV)，加速电压越高波长越短。电子穿过薄石墨膜后出现同心衍射环，这是波粒二象性最直接的实验证据。宏观物体的 λ 小到完全无法观测，所以日常看不到桌子衍射。计算时单位要统一成 kg、m、s，忘记把 eV 换成焦耳是最常见的失分点。
De Broglie assigned a wavelength to matter through lambda = h over p. Accelerating electrons through a known potential difference gives a wavelength that shrinks as the voltage rises. Concentric rings from a graphite film are the standard evidence for wave-particle duality, and any everyday object has a wavelength far too small to detect.
Paper 2 与 Paper 3 的答题切法
Topic 5 在 Paper 2 里通常配一道计算加一道解释，Paper 3 会把它放进实验情境，让你评估数据或改进装置。公式表已经给出的量不必重复推导，把时间放在单位换算和有效数字上更划算。写解释题时把关键词摆出来：相干、光程差、相位差、波腹、截止频率，判分点认的是这些词而不是句子长度。检查时回看题目问的是数值还是方向，很多丢分与物理无关。
Expect one calculation and one explanation in Paper 2, then a practical context in Paper 3 where you evaluate data or suggest improvements. Mark schemes reward technical vocabulary, so write coherence, path difference and threshold frequency instead of long sentences. Unit conversion and significant figures are where marks quietly disappear.
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