A-Level Physics: Resonance and Its Conditions | A-Level 物理：共振现象及其产生条件

📚 A-Level Physics: Resonance and Its Conditions | A-Level 物理：共振现象及其产生条件

Resonance is one of the most elegant and practical ideas in A-Level Physics. It explains why a small repeated push can make a heavy swing rise higher and higher, why some bridges vibrate dangerously in the wind, and why a note from a singer can shatter a glass. In this revision article, we will define resonance precisely, examine the conditions required for it to occur, explore the resonance curve and damping, and look at real-life applications and dangers.

共振是 A-Level 物理中既优美又实用的概念之一。它可以解释为什么持续的小推力能让沉重的秋千越荡越高、为什么有些桥梁会在风中危险地振动、为什么歌手的一个音符可以震碎玻璃。在这篇复习文章中，我们将精确定义共振、分析其产生条件、研究共振曲线和阻尼，并考察现实中的应用与危害。

1. What Is Resonance? | 什么是共振？

Resonance is the phenomenon in which a system oscillates with a much larger amplitude when the frequency of an applied periodic force is equal to the natural frequency of the system. At resonance, the driving force transfers energy to the system in the most efficient way, so the amplitude reaches a maximum value.

共振是当外加周期力的频率等于系统固有频率时，系统的振幅显著增大的现象。在共振时，驱动力以最高效率向系统传递能量，因此振幅达到最大值。

For example, a child on a swing has a fixed natural frequency. If a parent pushes the swing at exactly this frequency, each push adds energy at the right moment and the swing goes higher and higher. If the pushes come at the wrong time, the swing barely moves.

例如，荡秋千的孩子有一个固定的固有频率。如果家长恰好以这个频率推秋千，每一次推力都在正确的时刻输入能量，秋千就会越荡越高。如果推的时机不对，秋千几乎不会升高。

2. Key Terms: Free Vibration and Natural Frequency | 关键术语：自由振动与固有频率

A free vibration occurs when a system oscillates on its own after being displaced from equilibrium, with no external periodic force acting on it. The frequency of this vibration is called the natural frequency, written as f₀. It depends on the physical properties of the system, such as mass, stiffness, or length.

自由振动是指系统偏离平衡位置后，在没有任何外部周期力作用的情况下自行振动。这种振动的频率称为固有频率，记作 f₀。它取决于系统本身的物理性质，例如质量、劲度系数或长度。

For a simple pendulum, the period is related to its length:

对单摆而言，其周期与摆长有关：

T = 2π√(l/g)

For a mass on a spring, the period is related to the mass m and the spring constant k:

对于弹簧振子，其周期与质量 m 和劲度系数 k 有关：

T = 2π√(m/k)

In both cases, the natural frequency is f₀ = 1/T. You should be able to state these formulas and explain what happens when l, m or k changes.

在这两种情况下，固有频率都是 f₀ = 1/T。你应该能够写出这些公式，并解释当 l、m 或 k 改变时会发生什么。

3. Forced Vibration and Driving Frequency | 受迫振动与驱动力频率

A forced vibration happens when an external periodic force continuously drives an oscillator. The system is then made to vibrate at the driving frequency, also called the forcing frequency or applied frequency, not necessarily at its own natural frequency.

受迫振动是指外部周期力持续驱动振子时所发生的振动。此时系统被迫以驱动频率振动，而非一定以其固有频率振动。驱动频率也叫受迫频率或施加频率。

After a short initial period, the system settles into a steady state in which it oscillates at the driving frequency. The size of the steady amplitude depends strongly on how close the driving frequency is to the natural frequency.

经过短暂的起始阶段后，系统会进入稳态，并以驱动频率振动。稳态振幅的大小强烈依赖于驱动频率与固有频率的接近程度。

If the driving frequency is very different from f₀, the response is small. If the driving frequency is close to f₀, even a small force can produce a very large amplitude. This is the key idea behind resonance.

如果驱动频率与 f₀ 相差很远，系统的响应很小；如果驱动频率接近 f₀，即使很小的力也能产生很大的振幅。这就是共振背后的核心思想。

4. Conditions for Resonance | 共振的产生条件

Four conditions are usually required for resonance to be observed clearly:

要清楚地观察到共振，通常需要满足以下四个条件：

1. The system must have a natural frequency f₀.

系统必须具有固有频率 f₀。
2. An external periodic driving force must act on the system.

必须有一个外部周期力作用在系统上。
3. The driving frequency must be equal to the natural frequency.

驱动频率必须等于固有频率。
4. The damping must be small enough for the amplitude to grow large.

阻尼必须足够小，振幅才能增大到明显程度。

In examination answers, the standard definition is usually enough: resonance occurs when the driving frequency is equal to the natural frequency, causing maximum amplitude and maximum energy transfer.

在考试作答中，通常写出标准定义即可：当驱动频率等于固有频率时发生共振，此时振幅最大，能量传递效率最高。

5. The Resonance Curve | 共振曲线

The relationship between amplitude and driving frequency is shown by a resonance curve. The horizontal axis represents the driving frequency, while the vertical axis represents the steady amplitude of the forced oscillator.

振幅与驱动频率之间的关系可以用共振曲线表示。横轴代表驱动频率，纵轴代表受迫振子的稳态振幅。

As the driving frequency approaches f₀, the amplitude rises sharply. At f = f₀, the amplitude is at its maximum. With very little damping, the peak is very tall and narrow. With greater damping, the peak is shorter and broader.

当驱动频率接近 f₀ 时，振幅急剧增大。当 f = f₀ 时，振幅达到最大值。阻尼很小时，峰值又高又窄；阻尼较大时，峰值较低且较宽。

In CIE questions, you may be asked to sketch this curve for two different amounts of damping. Remember: the undamped or lightly damped curve has a much higher and sharper peak around f₀.

在 CIE 考试中，你可能会被要求画出两种不同阻尼情况下的共振曲线。请记住：无阻尼或轻阻尼曲线的峰值在 f₀ 附近更高、更尖锐。

6. Energy Transfer at Resonance | 共振时的能量传递

At resonance, the driving force does maximum positive work on the oscillator. This happens because the force is always in phase with the velocity of the oscillator, so each push acts in the direction of motion and adds energy.

在共振时，驱动力对振子做最大的正功。这是因为驱动力始终与振子的速度同相，每一次推动都沿着运动方向，从而不断输入能量。

If there were no damping, the amplitude would keep increasing without limit. In the real world, damping is always present. The amplitude stops growing when the energy supplied per cycle equals the energy lost per cycle due to damping.

如果没有阻尼，振幅将无限增大。但在现实世界中，阻尼总是存在的。当每个周期输入的能量等于每个周期因阻尼而损失的能量时，振幅就不再增大了。

Therefore, at resonance the amplitude is large and constant because of an energy balance. This idea is often examined in explanation-based questions.

因此，共振时的振幅大而稳定，是因为能量收支达到平衡。这一思路常出现在解释类考题中。

7. Phase Relationship at Resonance | 共振时的相位关系

The phase relationship between the driving force and the displacement is important in describing resonance.

驱动力与位移之间的相位关系对于描述共振非常重要。

At very low driving frequency, the displacement is nearly in phase with the driving force. At very high driving frequency, the displacement is almost exactly out of phase with the driving force, lagging by about 180°.

当驱动频率很低时，位移与驱动力几乎同相。当驱动频率很高时，位移与驱动力几乎反相，位移大约滞后 180°。

At resonance, the phase difference between the driving force and displacement is 90°. This means the driving force is in phase with velocity, which is the condition for maximum energy input.

在共振时，驱动力与位移之间的相位差为 90°。这意味着驱动力与速度同相，这正是能量输入最大的条件。

8. Effect of Damping on Resonance | 阻尼对共振的影响

Damping removes energy from an oscillating system. It can be caused by air resistance, friction, or internal forces in a material.

阻尼会使振动系统损失能量。空气阻力、摩擦或材料内部作用力都可能引起阻尼。

When damping increases, the maximum amplitude of the resonance curve decreases. The curve becomes flatter and broader, so resonance becomes less sharp. With very heavy damping, resonance may barely be noticeable.

当阻尼增大时，共振曲线的最大振幅减小。曲线变得更平坦、更宽，因此共振变得不那么尖锐。如果阻尼很重，共振可能几乎观察不到。

In addition, the frequency at which maximum amplitude occurs shifts slightly downward compared with the undamped natural frequency. In many A-Level questions, you only need to state that increased damping reduces the amplitude and makes the peak broader.

此外，最大振幅对应的频率会相对于无阻尼固有频率略微下移。在许多 A-Level 考题中，你只需说明增大阻尼会降低振幅并使峰值变宽。

9. Applications of Resonance | 共振的应用

Resonance is used deliberately in many devices. Learning a few examples helps you answer application questions quickly.

共振在许多设备中被有意利用。掌握几个典型例子有助于你快速回答应用类问题。

Radio tuning: in a radio receiver, the electrical circuit has a natural frequency. You change the capacitance until the circuit resonates with the signal from a particular station, making that station much stronger than others.

收音机调谐：收音机接收电路具有固有频率。你改变电容大小，直到电路与某个电台的信号发生共振，使该电台的信号比其他台强得多。
Microwave ovens: water molecules in food absorb microwaves strongly because the frequency of the microwaves is close to the natural rotational frequency of the water molecules. This raises the temperature of the food.

微波炉：食物中的水分子强烈吸收微波，因为微波频率接近水分子的自然转动频率。这会使食物的温度升高。
Musical instruments: the air column in a flute or the string in a violin vibrates at its natural frequency. When another part of the instrument provides a matching driving frequency, the sound is amplified by resonance.

乐器：长笛中的空气柱或小提琴中的琴弦以固有频率振动。当乐器的其他部分提供相匹配的驱动频率时，声音会通过共振而增强。

10. Dangers of Resonance and Engineering Prevention | 共振的危害与工程防护

Resonance can also damage structures. Engineers try to avoid resonance whenever possible.

共振也可能破坏结构。工程人员会尽可能避免共振。

One classic example is soldiers marching across a bridge. If their marching frequency matches the bridge’s natural frequency, the bridge can vibrate violently. To prevent this, soldiers are ordered to break step when crossing a bridge.

一个典型例子是士兵列队过桥。如果他们的步频与桥梁的固有频率相同，桥梁可能会剧烈振动。为了防止这种情况，士兵们过桥时会被命令打乱步伐。

Suspension bridges can also be set into oscillation by wind. The Tacoma Narrows Bridge collapse in 1940 is a famous case of large-amplitude oscillation caused by wind and resonance-like effects. Modern bridges are designed with a natural frequency far away from likely driving frequencies, and some use tuned mass dampers to absorb unwanted vibration.

吊桥也可能因风而产生强烈振荡。1940 年塔科马海峡大桥坍塌就是风致大幅振荡的著名案例。现代桥梁在设计时会让固有频率远离可能的驱动频率，有些还会使用调谐质量阻尼器来吸收有害振动。

In general, engineers can reduce the risk of resonance by changing the natural frequency, increasing damping, or avoiding periodic forces at dangerous frequencies.

一般来说，工程人员可以通过改变固有频率、增大阻尼或避免在危险频率下施加周期力来降低共振风险。

11. Examination Tips | 考试要点

When answering CIE resonance questions, the most common mistake is to state only that amplitudes become large, without mentioning the equality of the two frequencies. A complete answer should say:

在回答 CIE 共振类题目时，最常见的错误是只提到振幅变大，而没有说明两个频率相等。完整的回答应该这样写：

“Resonance occurs when the driving frequency equals the natural frequency, causing maximum amplitude and maximum energy transfer.”

Remember the symbol conventions used in exam papers: f₀ is the natural frequency, while the driving frequency may be written as fᵈ or simply stated in words. Always label your axes clearly when sketching a resonance curve: frequency on the x-axis and amplitude on the y-axis.

请注意考试试卷中的符号约定：f₀ 是固有频率，驱动频率可能写作 fᵈ，或直接用文字说明。在画共振曲线时，一定要标清坐标轴：横轴为频率，纵轴为振幅。

Finally, be ready to link resonance to energy. When examiners ask why the amplitude becomes large, the best answer is that the driving force transfers energy into the system at the correct frequency, and the energy gained per cycle balances the energy lost due to damping.

最后，要准备好将共振与能量联系起来。当考官问为什么振幅会变大时，最好的答案是：驱动力以正确的频率向系统输入能量，每个周期获得的能量与阻尼损失的能量达到平衡。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

A-Level Physics: Resonance and Its Conditions | A-Level 物理：共振现象及其产生条件