IB Physics: Intensity of Waves — Definition and Calculation | IB物理：波的强度定义与计算

📚 IB Physics: Intensity of Waves — Definition and Calculation | IB物理：波的强度定义与计算

In the IB Physics syllabus, wave intensity is a fundamental concept that links energy transfer to observable wave properties. This article defines intensity, derives its key relationships, and shows you how to calculate it in both plane and spherical wave scenarios. You will also find worked examples and common pitfalls to avoid in exams.

在IB物理课程中，波的强度是一个将能量传递与可观测波性质联系起来的核心概念。本文将定义强度，推导其关键关系，并展示如何在平面波和球面波情境中计算强度。此外，你还会看到典型例题和考试中需要避免的常见误区。

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1. What Is Wave Intensity? | 什么是波的强度？

Intensity describes how much energy a wave carries through a given area per unit time. In everyday terms, it measures how “loud” a sound is or how “bright” a light appears, assuming all other factors remain fixed.

波强度描述的是单位时间内通过某一面积的波所携带的能量。通俗地说，它衡量声音有多“响”或光有多“亮”，当然前提是其他条件保持不变。

Mathematically, intensity I is defined as the rate of energy transfer (power) per unit area perpendicular to the direction of wave propagation. The base unit for intensity is the watt per square metre (W m⁻²).

从数学上讲，强度 I 定义为单位面积上的能量传递速率（即功率），该面积与波的传播方向垂直。强度的基本单位是瓦特每平方米（W m⁻²）。

Since a wave spreads energy continuously, intensity is a scalar quantity that tells us the local “concentration” of wave energy at a point in space.

由于波不断向外传播能量，强度是一个标量，它告诉我们空间某一点波能量的“集中程度”。

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2. The Basic Formula: I = P / A | 基本公式：I = P / A

The defining equation for intensity is simple but powerful. If total power P passes uniformly through an area A perpendicular to the wave direction, then the intensity is:

强度的定义式简单而有力。如果总功率 P 均匀通过垂直于波传播方向的面积 A，则强度为：

I = P / A

Here, P is measured in watts (W), A in square metres (m²), and therefore I in watts per square metre (W m⁻²). A larger power passing through the same area gives a larger intensity; spreading the same power over a larger area reduces the intensity.

其中，P 的单位是瓦特（W），A 的单位是平方米（m²），因此 I 的单位是瓦特每平方米（W m⁻²）。相同面积上通过的功率越大，强度越大；相同功率分布在更大的面积上，强度就会减小。

In many IB problems, you are given the power of a source and the area of a detector or a spherical surface. In that case, simply divide the power by the area to obtain the intensity.

在许多IB题目中，你会已知波源的功率以及探测器或球面的面积。此时，直接用功率除以面积即可得到强度。

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3. Intensity and Amplitude: I ∝ A² | 强度与振幅：I ∝ A²

For any harmonic wave, the energy carried per unit volume is proportional to the square of the amplitude. Because intensity is the flux of energy through an area, it follows that intensity is proportional to the square of the amplitude:

对于任何简谐波，单位体积内携带的能量都与振幅的平方成正比。由于强度是通过单位面积的能量通量，因此强度与振幅的平方成正比：

I ∝ A²

This relationship is one of the most frequently tested ideas in IB Physics. If you double the amplitude of a wave, the intensity becomes four times larger; if you reduce the amplitude to one-third, intensity drops to one-ninth.

这个关系是IB物理中最常考的知识点之一。如果把波的振幅加倍，强度会变为原来的四倍；如果把振幅减为三分之一，强度则降为原来的九分之一。

For a plane progressive wave, the intensity can be written explicitly as:

对于平面行波，强度可以明确写成：

I = ½ ρ A² ω² v

In this expression, ρ is the density of the medium, A is the displacement amplitude, ω is the angular frequency, and v is the wave speed. You will meet this formula when studying sound and mechanical waves.

在该表达式中，ρ 是介质的密度，A 是位移振幅，ω 是角频率，v 是波速。你在学习声音和机械波时会遇到这个公式。

Since ω = 2πf, an alternative form for sound waves is I = 2π²ρ f² A² v. This shows that intensity grows with the square of frequency as well as with amplitude, which is why high-frequency, large-amplitude sounds are especially energetic.

由于 ω = 2πf，声波的强度还可写成 I = 2π²ρ f² A² v。这表明强度不仅随振幅的平方增长，也随频率的平方增长，这也是为什么高频、大振幅的声音尤其有能量。

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4. Intensity and Distance: The Inverse Square Law | 强度与距离：平方反比定律

Many IB questions involve a point source that emits waves uniformly in all directions. At a distance r from the source, the wave energy is spread over the surface of a sphere of radius r, whose area is 4πr².

许多IB题目涉及向四面八方均匀发射波的点波源。在距离源 r 处，波的能量分布在半径为 r 的球面 4πr² 上。

Therefore, the intensity at distance r is:

因此，在距离 r 处的强度为：

I = P / (4πr²)

This is the inverse square law. It shows that I is inversely proportional to the square of the distance from the source. If you triple the distance, intensity falls to one-ninth of its original value.

这就是平方反比定律。它表明 I 与到源距离的平方成反比。如果把距离扩大到原来的三倍，强度会降到原来的九分之一。

It is crucial to understand that the inverse square law applies only to point sources in a uniform, non-absorbing medium. For plane waves, intensity remains constant as the wave travels, assuming no energy is absorbed.

需要特别强调的是，平方反比定律只适用于均匀、无吸收介质中的点波源。对于平面波，在假设没有能量吸收的情况下，传播过程中强度保持不变。

When the distance changes from r₁ to r₂, the ratio of intensities is given by I₂/I₁ = (r₁/r₂)². This ratio form often appears in exam questions because it does not require you to know the source power.

当距离从 r₁ 变为 r₂ 时，强度之比为 I₂/I₁ = (r₁/r₂)²。这种比例形式常出现在考试中，因为你无需知道波源的功率。

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5. Intensity, Frequency and the Wave Equation | 强度、频率与波动方程

A wave transfers energy by making particles of the medium oscillate. The rate of energy transfer depends not only on amplitude, but also on how fast the particles vibrate (frequency)

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