Seeing a wave in space and time
从一个位置，看到一列波
A sinusoidal wave traveling in the positive x direction can be written as:
一列沿 x 轴正方向传播的简谐波可以写成：
A is the amplitude, k the wave number, ω the angular frequency, and φ the initial phase. Wavelength λ = 2π/k, period T = 2π/ω, and phase speed v = ω/k. Freeze time to see a sinusoidal shape across space; fix a location to see an oscillation over time.
A 是振幅，k 是波数，ω 是角频率，φ 是初相位。波长 λ = 2π/k，周期 T = 2π/ω，相速度 v = ω/k。固定时间看空间，是一条正弦曲线；固定位置看时间，则是一个振动。
The vertical displacement does not mean that particles move along the propagation direction. In a transverse wave on a string, particles oscillate vertically as the wave travels horizontally.
图中的纵向位移不是质点沿传播方向的运动。对于绳上的横波，绳上质点上下振动，而波形沿水平方向传播。
Phase: one rhythm, different starting points
相位：同一个节奏，不同的起点
A phase difference tells us how far two equal-frequency waves are offset within a cycle. At zero, crests meet crests. At π, crests meet troughs.
相位差描述两列同频波在一个周期里错开了多少。相差 0 时，波峰对波峰；相差 π 时，波峰对波谷。
Blue and pale green: two waves of equal amplitude and frequency. Bright green: their combined displacement. Coordinates are dimensionless.蓝色与浅绿色：两列等振幅、同频波；亮绿色：叠加后的位移。采用无量纲坐标。
Superposition: adding two displacements
叠加：把两个位移相加
In a linear model, the two displacements add. For waves of equal amplitude and frequency traveling in the same direction:
在线性模型中，两列波的位移可以相加。对等振幅、同频、同向传播的波：
The resulting amplitude is 2A |cos(Δφ/2)|. A phase difference of zero gives 2A; a difference of π gives complete cancellation. Move the slider to approximately 3.14 and watch the bright green curve flatten.
合成振幅为 2A |cos(Δφ/2)|。相位差为 0 时，合成振幅是 2A；相位差为 π 时，两者完全相消。把滑块拉到约 3.14，观察亮绿色曲线靠近零线。
Where the model stops
模型的边界
This experiment illustrates ideal linear superposition. It does not model damping, dispersion, reflection, or nonlinear media. The wave-number control changes the spatial period while temporal angular frequency remains fixed, so it also changes phase speed. This toy model is not the dispersion relation of a particular fixed medium.
本实验演示理想线性叠加，未模拟阻尼、色散、反射或介质的非线性。调节波数改变空间周期；演示中的时间角频率固定，因此同时改变了相速度。这个玩具模型不能直接当作某一种固定介质的色散关系。
Further reading
继续阅读
OpenStax · University Physics, 16.2 Mathematics of Waves introduces traveling waves, wavelength, period, and wave speed.
OpenStax · University Physics, 16.2 Mathematics of Waves 介绍行波表达式、波长、周期和波速之间的关系。