📚 8 Stationary and Travelling Waves | 驻波与行波
In IB Mathematics, trigonometric functions model periodic phenomena such as waves. A travelling wave propagates through space, while a stationary wave oscillates in place. Understanding their mathematical descriptions deepens your insight into function transformations, superposition, and identities.
在 IB 数学中，三角函数用于模拟波动等周期现象。行波在空间中传播，而驻波在原地振荡。理解它们的数学描述有助于加深对函数变换、叠加原理和恒等式的理解。
1. Modelling Waves with Trigonometric Functions | 用三角函数模拟波
A wave can be represented as a function of both position x and time t. The simplest form is y(x,t) = A sin(θ), where the argument θ depends on x and t. For a travelling wave, θ = kx − ωt + φ, where k is the wavenumber and ω the angular frequency. This function shifts horizontally as t increases, showing propagation. In IB problems, you often start with such a function and extract properties like amplitude, period, and speed.
波可以表示为位置 x 和时间 t 的函数。最简单的形式是 y(x,t) = A sin(θ)，其中幅角 θ 取决于 x 和 t。对于行波，θ = kx − ωt + φ，这里 k 是波数，ω 是角频率。随着 t 增加，该函数水平移动，显示传播特性。在 IB 问题中，你常常从这样的函数出发，提取振幅、周期和波速等属性。
2. Key Parameters in Wave Functions | 波函数的关键参数
The amplitude A determines the maximum displacement. The wavenumber k is related to the wavelength λ by k = 2π/λ. The angular frequency ω is connected to the period T and frequency f: ω = 2π/T = 2πf. The phase constant φ shifts the wave horizontally. These parameters allow precise mathematical description of wave properties. Note that the argument (kx − ωt) must be in radians; this is essential when using trigonometric identities.
振幅 A 决定最大位移。波数 k 与波长 λ 的关系为 k = 2π/λ。角频率 ω 与周期 T 和频率 f 有关：ω = 2π/T = 2πf。相位常数 φ 使波水平平移。这些参数能够精确地数学描述波的特性。注意幅角 (kx − ωt) 必须以弧度为单位；这在应用三角恒等式时至关重要。
3. Travelling Wave: The General Form | 行波：一般形式
A travelling wave moving in the positive x-direction is y = A sin(kx − ωt + φ). If the sign is plus, y = A sin(kx + ωt + φ), it moves in the negative x-direction. The argument (kx ∓ ωt) is constant along lines of constant phase, leading to the concept of phase velocity. The waveform maintains its shape and advances without changing amplitude, provided there is no damping.
一个沿 x 正方向传播的行波为 y = A sin(kx − ωt + φ)。若符号为加号，即 y = A sin(kx + ωt + φ)，则沿 x 负方向传播。幅角 (kx ∓ ωt) 在相位常数的线上保持不变，这引出了相速度的概念。波形保持形状并前进，振幅不变，前提是无阻尼。
4. Phase Velocity and Wave Speed | 相速度与波速
The phase velocity v is the speed at which a point of constant phase travels. By setting kx − ωt = constant and differentiating with respect to t, we obtain k (dx/dt) − ω = 0, so v = dx/dt = ω/k. This expresses the wave speed purely in terms of angular frequency and wavenumber. For a sinusoidal wave, v = fλ, since ω = 2πf and k = 2π/λ.
相速度 v 是一个固定相位点的传播速度。令 kx − ωt = 常数并对 t 求导，得到 k (dx/dt) − ω = 0，因此 v = dx/dt = ω/k。这用角频率和波数纯粹地表示了波速。对于正弦波，因为 ω = 2πf 且 k = 2π/λ，有 v = fλ。
5. Superposition Principle and Interference | 叠加原理与干涉
When two or more waves meet, the resultant displacement is the sum of individual displacements. Mathematically, y_total = y₁ + y₂. This principle applies to both travelling and stationary waves. Superposition leads to interference patterns, which are used to create stationary waves from two identical travelling waves moving in opposite directions. The algebraic combination relies on trigonometric identities.
当两个或更多的波相遇时，合位移是个别位移之和。数学上，y_total = y₁ + y₂。这一原理既适用于行波也适用于驻波。叠加导致干涉图样，用于从两列相同但反向传播的行波构建驻波。代数合并依赖三角恒等式。
6. Deriving a Stationary Wave Equation | 驻波方程的推导
Consider two travelling waves of same amplitude and frequency, moving opposite: y₁ = A sin(kx − ωt) and y₂ = A sin(kx + ωt). Using the sum-to-product identity sin P + sin Q = 2 sin[(P+Q)/2] cos[(P−Q)/2], we get y = 2A sin(kx) cos(ωt). This is the equation of a stationary wave. The spatial part sin(kx) defines a fixed profile, while cos(ωt) causes it to oscillate in time without moving. Notice the separation of variables x and t into different factors.
考虑两列振幅、频率相同且反向传播的行波：y₁ = A sin(kx − ωt) 和 y₂ = A sin(kx + ωt)。利用和差化积恒等式 sin P + sin Q = 2 sin[(P+Q)/2] cos[(P−Q)/2]，得到 y = 2A sin(kx) cos(ωt)。这就是驻波方程。空间部分 sin(kx) 定义了一个固定的轮廓，而 cos(ωt) 使其随时间振荡而不移动。注意到变量 x 和 t 被分离到不同的因子中。
y = 2A sin(kx) cos(ωt)
7. Nodes and Antinodes: Fixed Positions | 波节与波腹：固定位置
Nodes are points where the displacement is always zero. For the stationary wave y = 2A sin(kx) cos(ωt), nodes occur when sin(kx) = 0, i.e., kx = nπ, so x = nπ/k = nλ/2 (n integer). Antinodes are points of maximum oscillation amplitude, where |sin(kx)| = 1, giving kx = (2n+1)π/2, so x = (2n+1)λ/4. These positions are fixed, unlike in travelling waves where crests and troughs move. Knowing node locations helps in sketching the wave profile.
波节是位移始终为零的点。对于驻波 y = 2A sin(kx) cos(ωt)，节点出现在 sin(kx) = 0 时，即 kx = nπ，因此 x = nπ/k = nλ/2（n 为整数）。波腹是振幅最大的点，此时 |sin(kx)| = 1，即 kx = (2n+1)π/2，所以 x = (2n+1)λ/4。这些位置是固定的，与行波中波峰和波谷移动的情况不同。了解节点位置有助于绘制波的轮廓。
8. Comparison of Travelling and Stationary Waves | 行波与驻波的对比
We can summarise differences with a table. A travelling wave transfers energy along its direction of motion, while a stationary wave stores energy in local oscillations and does not transport energy. The mathematical form also reveals that the travelling wave argument is (kx − ωt), linking space and time, whereas the stationary wave separates space and time into multiplied factors. All particles between two nodes of a stationary wave move in phase, whereas in a travelling wave the phase varies continuously with position.
我们可以用一个表格总结差异。行波沿其运动方向传输能量，而驻波将能量储存在局部振荡中，不传输能量。数学形式也表明，行波幅角为 (kx − ωt)，将空间和时间联系起来，而驻波将空间和时间分离为相乘因子。驻波两节点之间的所有质点同相运动，而行波中相位随位置连续变化。
| Property | Travelling Wave | 行波 | Stationary Wave | 驻波 |
|---|---|---|
| Mathematical form | y = A sin(kx − ωt + φ) | y = 2A sin(kx) cos(ωt) |
| Phase behaviour | Phase varies with x and t | All points oscillate in phase or strictly out of phase |
| Energy transport | Yes, in direction of travel | No net transport |
| Nodes | Move with wave | Fixed positions |
| Amplitude profile | Constant across all x | Varies with x, zero at nodes |
9. IB Exam-Style Problems | IB 考试题型
IB Mathematics frequently asks students to identify amplitude, period, phase shift from wave equations, or to show that superposition leads to a standing wave. You may need to find node positions or prove that a given combination forms a stationary wave. These problems test trigonometric identities and understanding of function transformations. Always express your answer in exact form using π unless otherwise stated.
IB 数学经常要求学生从波方程中识别振幅、周期、相移，或者证明叠加形成驻波。你可能需要求节点位置或证明给定的组合构成驻波。这些问题考查三角恒等式和对函数变换的理解。除非特别说明，始终用包含 π 的精确形式给出答案。
Example: Two waves y₁ = 3 sin(2x − 5t) and y₂ = 3 sin(2x + 5t) are superposed. Show that the resultant is a stationary wave and find the positions of the first three nodes from x = 0.
示例：两列波 y₁ = 3 sin(2x − 5t) 和 y₂ = 3 sin(2x + 5t) 叠加。证明结果是驻波，并求离 x = 0 的前三个节点位置。
Solution: Using the identity, y = 2 × 3 sin(2x) cos(5t) = 6 sin(2x) cos(5t). This matches the stationary form y = 2A sin(kx) cos(ωt) with A=3, k=2, ω=5. Nodes when sin(2x) = 0 → 2x = nπ → x = nπ/2. For n = 0, 1
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