IB物理 波动 简谐运动 行波 驻波 多普勒 | IB Physics Waves Simple Harmonic Motion Travelling Waves Standing Waves Doppler

IB物理 波动 简谐运动 行波 驻波 多普勒

波动 (Waves) 是 IB 物理 Core 部分的重要章节，横跨 Topic 4 和 Topic 9。从简谐运动的数学描述到行波与驻波的物理本质，再到多普勒效应和波的现象学（干涉、衍射、折射），波动学建立了一个贯穿物理学各个分支的概念框架。本文带你从零开始，系统梳理 IB 物理波动章节的核心知识点。

Waves is a core topic in IB Physics, spanning Topic 4 and Topic 9. From the mathematical description of simple harmonic motion to the physical nature of traveling and standing waves, and onto the Doppler effect and wave phenomena (interference, diffraction, refraction), wave theory establishes a conceptual framework that runs through every branch of physics. This article takes you from scratch through the core knowledge points of the IB Physics waves chapter.

1. 简谐运动 Simple Harmonic Motion (SHM)

简谐运动是波动学的数学基础。当一个物体的加速度与其位移成正比且方向相反时，它就做简谐运动：a = -ω²x。其中 ω 是角频率（angular frequency），与周期 T 和频率 f 的关系为 ω = 2π/T = 2πf。

Simple harmonic motion is the mathematical foundation of wave theory. An object undergoes SHM when its acceleration is proportional to its displacement and opposite in direction: a = -ω²x. Here ω is the angular frequency, related to period T and frequency f by ω = 2π/T = 2πf.

位移随时间的变化可以用正弦或余弦函数描述：x = A cos(ωt) 或 x = A sin(ωt)，其中 A 为振幅（amplitude）。速度 v = -Aω sin(ωt)，加速度 a = -Aω² cos(ωt) = -ω²x。

The displacement as a function of time can be described by sine or cosine: x = A cos(ωt) or x = A sin(ωt), where A is the amplitude. Velocity is v = -Aω sin(ωt), and acceleration is a = -Aω² cos(ωt) = -ω²x.

SHM 的能量在动能与势能之间交替转化。总能量守恒：E_total = 1/2 mω²A²。在最大位移处（x = ±A），动能 = 0，势能最大；在平衡位置（x = 0），动能最大，势能 = 0。

Energy in SHM alternates between kinetic and potential forms. The total energy is conserved: E_total = 1/2 mω²A². At maximum displacement (x = ±A), KE = 0 and PE is maximum; at equilibrium (x = 0), KE is maximum and PE = 0.

常见 SHM 实例 Common Examples

• 弹簧振子 Mass-spring system: T = 2π√(m/k)
• 单摆 Simple pendulum: T = 2π√(L/g)
• U型管振荡 U-tube oscillation
• LC 电路 LC circuit (Topic 11)

SHM 记忆要点 Key Points

• a ∝ -x (加速度与位移成正比反向)
• T 由系统参数决定，与 A 无关 (T independent of A)
• 相位差 π/2 在 x, v, a 之间
• E_total ∝ A² (能量与振幅平方成正比)

2. 波的基本性质 Wave Properties

波是能量在空间中的传递而不伴随物质的净传输。波的基本参数包括：振幅 A（最大位移）、波长 λ（相邻同相位点之间的距离）、频率 f（每秒振动次数）、周期 T（= 1/f）、以及波速 v。它们之间的关系为 v = fλ。

A wave is the transfer of energy through space without the net transport of matter. Key wave parameters include: amplitude A (maximum displacement), wavelength λ (distance between adjacent points in phase), frequency f (oscillations per second), period T (= 1/f), and wave speed v. These are related by v = fλ.

波分为两类：横波 (transverse waves) 和 纵波 (longitudinal waves)。横波的振动方向垂直于传播方向（如电磁波、绳波）；纵波的振动方向平行于传播方向（如声波、地震 P 波）。IB 考试经常考察对两种波的区别和实例。

Waves are classified into two types: transverse waves and longitudinal waves. In transverse waves, the oscillation is perpendicular to the direction of propagation (e.g., EM waves, waves on a string); in longitudinal waves, the oscillation is parallel to the propagation direction (e.g., sound waves, P-waves). IB exams frequently test the distinction and examples of both types.

强度 Intensity

波的强度与振幅的平方成正比：I ∝ A²。对于球面波，强度与距离平方成反比：I ∝ 1/r²（能量守恒）。这是解释声音随距离衰减和光强随距离变化的物理基础。

Intensity is proportional to amplitude squared: I ∝ A². For spherical waves, intensity follows the inverse-square law: I ∝ 1/r² (energy conservation). This is the physical basis for explaining sound attenuation and light intensity variation with distance.

3. 行波 Traveling Waves

行波是能量从一处传播到另一处的波。一维行波的数学描述为：y(x,t) = A sin(kx - ωt + φ)，其中 k = 2π/λ 是波数（wave number），ω = 2πf 是角频率，φ 是初相位（initial phase）。

A traveling wave is one that transfers energy from one location to another. The mathematical description of a 1D traveling wave is: y(x,t) = A sin(kx - ωt + φ), where k = 2π/λ is the wave number, ω = 2πf is the angular frequency, and φ is the initial phase.

当 kx - ωt = 常数时，我们看到某一特定相位在空间中移动。相位速度（波速）为 v = ω/k = fλ。波沿 +x 方向传播时取负号，沿 -x 方向传播时取正号。

When kx - ωt = constant, we see a particular phase moving through space. The phase velocity (wave speed) is v = ω/k = fλ. Waves traveling in the +x direction use the negative sign, while those in the -x direction use the positive sign.

波前的概念对于理解波传播非常重要。波前 (wavefront) 是波上同相位的点所组成的面。惠更斯原理 (Huygens' principle) 指出，波前上的每一点都可以看作次级子波的波源，这解释了波的衍射和折射现象。

The concept of wavefronts is crucial for understanding wave propagation. A wavefront is the surface connecting all points on a wave that are in phase. Huygens' principle states that every point on a wavefront acts as a source of secondary wavelets, which explains diffraction and refraction.

4. 驻波 Standing Waves

驻波是由两列频率相同、振幅相同但方向相反的波叠加形成的。驻波的特点是存在波节 (nodes)：振幅恒为零的点，以及波腹 (antinodes)：振幅最大的点。驻波不传输能量，能量在波腹之间来回振荡。

A standing wave is formed by the superposition of two waves with the same frequency and amplitude traveling in opposite directions. Standing waves are characterized by nodes: points of zero amplitude, and antinodes: points of maximum amplitude. Standing waves do not transfer energy; energy oscillates back and forth between antinodes.

驻波的数学描述为：y(x,t) = 2A sin(kx) cos(ωt)。注意这个形式将空间部分 sin(kx) 与时间部分 cos(ωt) 分离了：波节出现在 sin(kx) = 0 的位置，即 x = nλ/2；波腹出现在 |sin(kx)| = 1 的位置，即 x = (2n+1)λ/4。

The mathematical form of a standing wave is: y(x,t) = 2A sin(kx) cos(ωt). Note how the spatial part sin(kx) and the temporal part cos(ωt) are separated: nodes occur where sin(kx) = 0, i.e., x = nλ/2; antinodes occur where |sin(kx)| = 1, i.e., x = (2n+1)λ/4.

弦上的驻波 Standing Waves on a String

两端固定的弦：λ_n = 2L/n, f_n = nf_1 = nv/(2L)

n = 1 为基频 (fundamental), n > 1 为泛音 (harmonics)

两端固定，两端必为波节。相邻波节间距 = λ/2。频率 f = v/(2L) × n，其中 v = √(T/μ)，T 为张力，μ 为线密度。

Fixed at both ends: both ends are nodes. Node spacing = λ/2. Frequency f = v/(2L) × n, where v = √(T/μ), T is tension, μ is linear density.

管中驻波 Standing Waves in Pipes

闭管 (closed at one end): λ_n = 4L/n (n 为奇数), f_n = nv/(4L)

开管 (open at both ends): λ_n = 2L/n, f_n = nv/(2L)

闭管一端为波节（闭端），一端为波腹（开口端）。开管两端均为波腹。n 的取值不同。

Closed pipe: node at closed end, antinode at open end. Open pipe: antinodes at both ends. The allowed values of n differ.

5. 多普勒效应 The Doppler Effect

多普勒效应描述的是当波源和观察者之间存在相对运动时，观察者接收到的频率与波源发出的频率不同的现象。当两者相互靠近时频率升高，远离时频率降低。

The Doppler effect describes the change in observed frequency when there is relative motion between the wave source and the observer. The observed frequency is higher when they approach each other and lower when they move apart.

对于声波，当声源以速度 u_s 向静止观察者移动时，观察者接收到的频率为：f' = f × v / (v - u_s)。当观察者以速度 u_o 向静止声源移动时：f' = f × (v + u_o) / v。一般公式为 f' = f × (v ± u_o) / (v ∓ u_s)，其中上方符号用于靠近的情况。

For sound waves, when the source moves toward a stationary observer at speed u_s: f' = f × v / (v - u_s). When the observer moves toward a stationary source at speed u_o: f' = f × (v + u_o) / v. The general formula is f' = f × (v ± u_o) / (v ∓ u_s), where the upper signs apply to approaching situations.

对于光波（电磁波），由于光速恒定，多普勒效应的公式变为：f' = f × √[(c + v)/(c - v)]（相对论修正）。红移 (redshift) 发生在光源远离时（频率降低），蓝移 (blueshift) 发生在光源靠近时（频率升高）。

For light (EM waves), because the speed of light is constant, the Doppler formula becomes: f' = f × √[(c + v)/(c - v)] (relativistic correction). Redshift occurs when the source recedes (frequency decreases), and blueshift occurs when the source approaches (frequency increases).

6. 波的干涉、衍射和折射 Interference, Diffraction, and Refraction

干涉 (Interference)：当两列同频率的波在空间中相遇时，它们按叠加原理组合。当相位差 = 2π 的整数倍时发生相长干涉 (constructive interference)（振幅增强）；当相位差 = π 的奇数倍时发生相消干涉 (destructive interference)（振幅减弱）。

Interference: When two waves of the same frequency meet in space, they combine according to the superposition principle. Constructive interference (amplitude enhancement) occurs when the phase difference is an integer multiple of 2π; destructive interference (amplitude reduction) occurs when the phase difference is an odd multiple of π.

杨氏双缝实验 (Young's double-slit experiment) 是干涉的经典示范。亮条纹的条件：d sin θ = nλ；暗条纹的条件：d sin θ = (n + 1/2)λ。条纹间距为 Δy = λD/d，其中 d 是缝间距，D 是屏距。

Young's double-slit experiment is the classic demonstration of interference. Bright fringes: d sin θ = nλ; dark fringes: d sin θ = (n + 1/2)λ. Fringe spacing is Δy = λD/d, where d is slit separation and D is screen distance.

衍射 (Diffraction)：波绕过障碍物或通过狭缝时扩展的现象。单缝衍射的暗纹位置：a sin θ = nλ，其中 a 是缝宽。中央亮纹的角宽度约 2λ/a。衍射光栅 (diffraction grating) 的亮纹：d sin θ = nλ（与双缝相同但更锐利）。

Diffraction: The spreading of waves as they pass around obstacles or through apertures. Single-slit dark fringes: a sin θ = nλ, where a is the slit width. The angular width of the central maximum is approximately 2λ/a. Diffraction grating maxima: d sin θ = nλ (same form as double-slit but sharper).

折射 (Refraction)：波从一种介质进入另一种介质时速度改变导致方向改变的现象。斯涅尔定律 (Snell's law) 描述为：n_1 sin θ_1 = n_2 sin θ_2。折射率 n = c/v，即光在真空中的速度除以在介质中的速度。

Refraction: The change in direction of a wave when it passes from one medium to another due to a change in speed. Snell's law: n_1 sin θ_1 = n_2 sin θ_2. The refractive index is n = c/v, the ratio of the speed of light in vacuum to its speed in the medium.

7. IB 考试技巧 IB Exam Tips

Paper 1 选择题策略

记住：SHM 中 T 与 A 无关（常考点）。波的频率由波源决定，波速由介质决定（f source-dependent, v medium-dependent）。判断是否发生干涉的关键是频率相同。

Remember: T is independent of A in SHM (frequently tested). A wave's frequency is determined by the source, while its speed is determined by the medium. The key condition for interference is same frequency.

Paper 2 计算题要点

v = fλ 是最重要的等式：已知任意两个，求第三个。多普勒公式注意 ± 符号：靠近取上方（± 同向），远离取下方（∓ 反向）。驻波题务必先画图标注波节位置。

v = fλ is the most important equation: given any two, find the third. Doppler formula: be careful with ± signs: approaching uses upper signs, receding uses lower. For standing wave problems, always sketch the nodes first.

IB 考察波动学的方式比 A-Level 更深入，特别是 SHM 的严格数学推导（微分方程）、相位差的理解、以及定量应用多普勒公式。HL 学生还可能遇到波的偏振（polarization）、马吕斯定律（Malus's law: I = I₀ cos²θ），以及声强级（sound intensity level: dB = 10 log₁₀(I/I₀)）等内容。

IB tests wave theory more deeply than A-Level, especially the rigorous mathematical derivation of SHM (differential equation), understanding of phase difference, and quantitative application of the Doppler formula. HL students may also encounter wave polarization, Malus's law (I = I₀ cos²θ), and sound intensity level (dB = 10 log₁₀(I/I₀)).

关键公式速查 Key Formula Reference

简谐运动 SHM

a = -ω²x

T = 2π√(m/k)

T = 2π√(L/g)

E = ½mω²A²

波的基本方程 Waves

v = fλ

y = A sin(kx - ωt)

k = 2π/λ, ω = 2πf

I ∝ A², I ∝ 1/r²

驻波 Standing Waves

弦: λ_n = 2L/n

弦: f_n = nv/(2L)

闭管: λ_n = 4L/n (n 奇)

开管: λ_n = 2L/n

多普勒 Doppler

f' = f(v±u_o)/(v∓u_s)

干涉: d sinθ = nλ

折射: n₁sinθ₁=n₂sinθ₂

光栅: d sinθ = nλ

🛍️ 微信小店好物推荐

fx-991CN X 中文版计算器-老王推荐

¥168.00 🛒 微信小店购买

直击高考专项突破 高中物理+高中数学二级公式快解

¥59.80 🛒 微信小店购买

🎓 Need one-on-one tutoring? 需要一对一辅导?

📱 16621398022 同微信

Follow tutorhao on WeChat for more learning resources 关注公众号获取更多学习资源

更多咨询请联系16621398022（同微信）