Wave Speed | 波速

📚 Wave Speed | 波速

Wave speed is a core concept in CIE A-Level Physics. It links the frequency and wavelength of any periodic wave and appears in topics from sound and strings to electromagnetic radiation. This article explains how wave speed is defined, derived, measured, and applied in different wave types.

波速是 CIE A-Level 物理中的核心概念。它把任何周期性波的频率和波长联系起来，并出现在从声音、弦波到电磁辐射等多个主题中。本文讲解波速的定义、推导、测量方法以及在不同波型中的应用。

1. What is Wave Speed? | 什么是波速？

Wave speed is the distance travelled by a wave disturbance or wave energy per unit time. It is not the same as the speed of individual particles in the medium. In a transverse wave, particles oscillate perpendicular to the direction of energy transfer, but the wave profile advances at the wave speed.

波速是波扰动或波的能量在单位时间内传播的距离。它不同于介质中单个粒子的速度。在横波中，粒子垂直于能量传递方向振动，但波的轮廓以波速向前推进。

For example, when a ripple spreads across a pond, the water molecules mainly move up and down while the circular wave pattern moves outward. The speed of this pattern is the wave speed.

例如，当水波在池塘中扩散时，水分子主要上下运动，而圆形波图案向外移动。这个图案的移动速度就是波速。

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2. The Wave Equation v = fλ | 波速方程 v = fλ

The fundamental equation for all periodic waves is v = f λ, where v is wave speed, f is frequency, and λ is wavelength. Frequency is the number of complete oscillations per second; wavelength is the distance between two consecutive points in phase, such as two adjacent crests.

所有周期性波的基本方程是 v = f λ，其中 v 是波速，f 是频率，λ 是波长。频率是每秒完整振动的次数；波长是两个相邻同相点之间的距离，例如两个相邻波峰之间的距离。

To derive this, recall that the period T is the time for one complete wavelength to pass a fixed point: T = 1/f. Since speed is distance divided by time, one wavelength λ passes in time T, so v = λ / T = f λ.

推导该式时，周期 T 是一个完整波长通过固定点所需的时间：T = 1/f。由于速度等于距离除以时间，在时间 T 内通过一个波长 λ，因此 v = λ / T = f λ。

v = f λ

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3. Transverse and Longitudinal Speeds | 横波与纵波的速度

The equation v = f λ applies to both transverse and longitudinal waves. Transverse waves include waves on strings, water ripples, and electromagnetic waves. Longitudinal waves include sound waves in air and pressure waves in springs.

公式 v = f λ 既适用于横波，也适用于纵波。横波包括弦上的波、水波和电磁波。纵波包括空气中的声波和弹簧中的压力波。

In most mechanical waves, the wave speed is set by the properties of the medium, not by the source frequency. If frequency is increased, the wavelength decreases so that v remains constant, provided the medium is non-dispersive. In a dispersive medium, wave speed can depend on frequency or wavelength.

在大多数机械波中，波速由介质的性质决定，而不是由波源频率决定。如果频率增大，波长会减小，使 v 保持不变，前提是介质为非色散介质。在色散介质中，波速可能依赖于频率或波长。

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4. Wave Speed on a Stretched String | 弦上的波速

For a transverse wave on a stretched string, the wave speed is given by v = √(T/μ), where T is the tension in the string and μ is the mass per unit length. This formula is derived from Newton’s second law applied to a small string element.

对于弦上的横波，波速由 v = √(T/μ) 给出，其中 T 是弦中的张力，μ 是弦的线密度（单位长度的质量）。该公式由牛顿第二定律应用于弦的小段元推导得出。

v = √(T / μ)

A greater tension pulls the string back to equilibrium more strongly, so the wave travels faster. A larger mass per unit length increases inertia, so the wave travels more slowly. Changing the amplitude has no first-order effect on wave speed.

张力越大，弦回到平衡位置的恢复力越强，因此波传播得越快。线密度越大，惯性越大，因此波传播得越慢。改变振幅对波速没有一阶影响。

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5. Speed of Sound in Air | 空气中的声速

Sound waves are longitudinal pressure waves. In air at about 20 °C, the speed of sound is approximately 343 m/s; at 0 °C it is about 331 m/s. The speed increases with temperature because warmer air molecules move faster and transmit compressions more quickly.

声波是纵波。在约 20 °C 的空气中，声速约为 343 m/s；在 0 °C 时约为 331 m/s。声速随温度升高而增大，因为较暖的空气分子运动更快，能更快地传递压缩。

In general, sound travels faster in liquids than in gases and faster still in solids. For example, in water the speed is about 1500 m/s, and in steel it can be around 5000 m/s. This is due to stronger intermolecular forces and higher elastic modulus.

一般来说，声音在液体中比在气体中快，在固体中更快。例如，水中声速约为 1500 m/s，钢中约为 5000 m/s。这是因为固体有更强的分子间作用力和更高的弹性模量。

A simple laboratory method involves measuring the time for a sound pulse to travel to a wall and back. If the distance to the wall is d and the echo time is t, the speed of sound is v = 2d / t.

一种简单的实验室方法是测量声脉冲传到墙壁并返回的时间。如果到墙壁的距离为 d，回声时间为 t，则声速为 v = 2d / t。

v = 2d / t

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6. Electromagnetic Wave Speed | 电磁波速度

All electromagnetic waves travel at the same speed in a vacuum, denoted by c. Its value is 3.00 × 10⁸ m/s. This includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

所有电磁波在真空中的速度相同，用 c 表示，其值为 3.00 × 10⁸ m/s。这包括无线电波、微波、红外线、可见光、紫外线、X 射线和伽马射线。

In a material medium, electromagnetic waves slow down. The refractive index n of a medium is defined as n = c / v, where v is the speed of light in that medium. Since n ≥ 1, v ≤ c.

在物质介质中，电磁波会减慢。介质的折射率 n 定义为 n = c / v，其中 v 是该介质中的光速。由于 n ≥ 1，因此 v ≤ c。

n = c / v

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7. Water Waves and Depth Effects | 水波与水深影响

Water waves are surface waves involving both transverse and longitudinal particle motion. Their speed depends on water depth. In shallow water, the wave speed decreases as the depth decreases, according to the approximate relation v ≈ √(g d), where g is gravitational field strength and d is water depth.

水波是表面波，涉及粒子的横向和纵向运动。其速度取决于水深。在浅水中，波速随深度减小而减小，近似关系为 v ≈ √(g d)，其中 g 是重力场强度，d 是水深。

v ≈ √(g d)

When water waves move from deep to shallow water, they slow down. If they approach at an angle, the slowing causes refraction, and the wavelength decreases while the frequency remains the same.

当水波从深水进入浅水时，速度减慢。如果波以一定角度入射，减速会导致折射，同时波长减小而频率保持不变。

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8. Measuring Speed with a Ripple Tank | 用波纹槽测量波速

A ripple tank can be used to measure the speed of water waves. A vibrating dipper produces plane waves with a known frequency. The wavelength is measured by freezing the wave pattern with a stroboscope or by taking a photograph with a ruler in the frame.

波纹槽可用于测量水波速度。振动振子以已知频率产生平面波。通过频闪仪冻结波图，或在照片中放置尺子，可以测量波长。

Once frequency f and wavelength λ are known, the speed is calculated from v = f λ. This is a direct application of the wave equation and reinforces that frequency remains constant when water depth changes.

一旦已知频率 f 和波长 λ，就可以用 v = f λ 计算速度。这是波速方程的直接应用，也说明水深变化时频率保持不变。

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9. Measuring Wave Speed on a String | 测量弦上的波速

To measure the speed of a wave on a string, a vibration generator is connected to one end of the string and the other end passes over a pulley with a hanging mass to provide tension. When the generator frequency matches a harmonic, a clear standing wave is formed.

为了测量弦上波速，将振动发生器连接到弦的一端，另一端绕过滑轮悬挂质量以提供张力。当发生器频率与某一谐波频率匹配时，会形成清晰的驻波。

The wavelength can be measured from the distance between adjacent nodes. The frequency is read from the generator. Then v = f λ gives the wave speed

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