Resonance | 共振

📚 Resonance | 共振

In A-Level Physics, resonance is one of the most important ideas in oscillations and waves. It explains why a small periodic force can produce large-amplitude vibrations when its frequency matches the natural frequency of a system. Resonance appears in everyday technology, music, medicine, civil engineering and many exam questions on forced oscillations and damping.

在 A-Level 物理中，共振是振动与波中最重要的概念之一。它解释了为什么当周期性驱动力的频率与系统的固有频率匹配时，很小的驱动力也能产生大振幅的振动。共振出现在日常科技、音乐、医学、土木工程以及许多关于受迫振动和阻尼的考试题目中。

1. Free Oscillations and Natural Frequency | 自由振动与固有频率

A free oscillation occurs when a system is displaced from its equilibrium position and then released. No external periodic force acts after release. The system vibrates at its natural frequency f₀, which is determined by the physical properties of the system.

自由振动是指系统偏离平衡位置后被释放，释放后没有外部周期性力作用。系统以其固有频率 f₀ 振动，该频率由系统的物理特性决定。

For a mass-spring system, the natural frequency depends on the mass m and the spring constant k. For a simple pendulum, it depends on the length L and the gravitational field strength g. These are standard examples of free oscillators.

对于弹簧-质量系统，固有频率取决于质量 m 和弹簧劲度系数 k。对于单摆，固有频率取决于摆长 L 和重力场强度 g。这些都是标准的自由振子示例。

f₀ = 1/(2π) √(k/m)

f₀ = 1/(2π) √(g/L)

In real systems, free oscillations do not continue forever. Resistive forces such as air resistance remove energy, so the amplitude decreases. This energy loss is called damping, and it strongly affects how resonance behaves.

在实际系统中，自由振动不会永远持续。空气阻力等阻力会消耗能量，因此振幅会减小。这种能量损失称为阻尼，它会强烈影响共振的表现。

2. Forced Oscillations and Driving Frequency | 受迫振动与驱动频率

A forced oscillation happens when an external driver applies a periodic force to an oscillating system. The driver could be a child pushing a swing, an engine vibration, or an electromagnetic shaker in the laboratory.

当外部驱动源对振动系统施加周期性力时，就会产生受迫振动。驱动源可以是推秋千的小孩、发动机的振动，或实验室中的电磁振荡器。

After a short transient period, the system oscillates at the driving frequency f_d, not at its natural frequency. The amplitude of the steady-state oscillation depends on how close f_d is to f₀.

经过短暂的过渡过程后，系统以驱动频率 f_d 振动，而不是以其固有频率振动。稳态振动的振幅取决于 f_d 与 f₀ 的接近程度。

If the driving frequency is much lower or much higher than f₀, the amplitude is usually small. When f_d is close to f₀, the amplitude becomes large. This observation leads directly to the idea of resonance.

如果驱动频率远

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