A-Level Physics: Key Quantities for Describing Oscillations | A-Level 物理：振动描述的关键物理量

📚 A-Level Physics: Key Quantities for Describing Oscillations | A-Level 物理：振动描述的关键物理量

Oscillations are everywhere in physics: a swinging pendulum, a vibrating guitar string, the alternating current in a circuit, and even the atoms in a solid. To describe any oscillation precisely, we need a standard set of physical quantities. In this guide, we will define displacement, amplitude, period, frequency, angular frequency, phase, and the relationships between velocity, acceleration and energy in simple harmonic motion (SHM).

振动在物理学中无处不在：摆动的单摆、振动的琴弦、电路中的交变电流，乃至固体中的原子。要精确描述任何一种振动，我们需要一组标准的物理量。在本篇文章中，我们将定义位移、振幅、周期、频率、角频率、相位，以及简谐运动（SHM）中速度、加速度和能量之间的关系。

1. Displacement and Amplitude | 位移与振幅

Displacement, usually denoted x, is the instantaneous position of an oscillating object measured from its equilibrium position. It is a vector quantity, so it can be positive or negative depending on which side of the equilibrium position the object is on.

位移通常用 x 表示，是振动物体相对于平衡位置的瞬时位置。它是一个矢量，因此根据物体位于平衡位置的哪一侧，位移可以为正值或负值。

Amplitude, denoted A, is the maximum magnitude of displacement from the equilibrium position. Because it is a maximum magnitude, amplitude is always positive and has units of metres. The amplitude tells us how “large” the oscillation is, and for an undamped oscillator it remains constant.

振幅用 A 表示，是从平衡位置到最大位移处的大小。由于它是最大距离，振幅始终为正值，单位是米。振幅告诉我们振动有多大；对于无阻尼振子，振幅保持不变。

For example, a pendulum of length 1.0 m is pulled 0.05 m to the right before release. Its amplitude is 0.05 m, while its displacement starts at +0.05 m and changes continuously as it swings.

例如，一根 1.0 m 长的单摆被拉到平衡位置右侧 0.05 m 后释放。它的振幅是 0.05 m，而位移从 +0.05 m 开始，并随着摆动不断变化。

2. Period and Frequency | 周期与频率

The period T is the time taken for one complete cycle of oscillation. In SI units, period is measured in seconds (s). For a mass on a spring, one complete cycle means moving from the starting point, through equilibrium, to the opposite extreme, and then returning to the starting point.

周期 T 是完成一次全振动所需的时间。在国际单位制中，周期以秒（s）为单位。对于弹簧振子，一次全振动是指从起点出发，经过平衡位置到达另一个极端，再回到起点。

The frequency f is the number of complete oscillations per second, measured in hertz (Hz), where 1 Hz = 1 s⁻¹. Period and frequency are reciprocals:

频率 f 是每秒完成全振动的次数，单位为赫兹（Hz），其中 1 Hz = 1 s⁻¹。周期与频率互为倒数：

T = 1 / f and f = 1 / T

For example, if a pendulum completes 20 oscillations in 40 s, then T = 40/20 = 2.0 s and f = 0.50 Hz. In SHM, the period is independent of amplitude for small oscillations of a pendulum, which is the principle behind pendulum clocks.

例如，如果一个单摆在 40 s 内完成 20 次全振动，那么 T = 40/20 = 2.0 s，f = 0.50 Hz。在简谐运动中，对于小角度摆动的单摆，周期与振幅无关，这正是摆钟的原理。

3. Angular Frequency | 角频率

In SHM, the displacement function involves sine or cosine of an angle that increases linearly with time. The angular frequency ω (Greek letter omega) measures how rapidly the phase angle changes in radians per second. It is related to period and frequency by:

在简谐运动中，位移函数涉及随时间线性增大的角度。角频率 ω（希腊字母 omega）表示相位角变化的快慢，单位为弧度每秒。它与周期和频率的关系为：

ω = 2π / T = 2π f

Angular frequency appears in the standard SHM equations, for example x = A sin(ωt + φ). It is especially useful because it connects the time-based description of oscillation with the circular-motion analogy used to derive SHM equations.

角频率出现在标准简谐运动方程中，例如 x = A sin(ωt + φ)。它特别有用，因为它把基于时间的振动描述与用于推导简谐运动方程的圆周运动类比联系在一起。

Notice that ω has units of rad s⁻¹, not Hz. Although both frequency and angular frequency describe “how fast” something oscillates, frequency counts cycles per second, while angular frequency measures phase change per second.

注意，ω 的单位是 rad s⁻¹，而不是 Hz。虽然频率和角频率都描述振荡“多快”，但频率计算每秒多少个循环，而角频率测量每秒相位变化多少弧度。

4. Phase and Phase Difference | 相位与相位差

The phase of an oscillation describes the position within the cycle at a particular time. For the equation x = A sin(ωt + φ), the quantity (ωt + φ) is the phase, measured in radians (or degrees). The constant φ is the phase constant, which depends on where in the cycle the motion starts at t = 0.

振动的相位描述某一时刻在振动周期中所处的位置。对于方程 x = A sin(ωt + φ)，量 (ωt + φ) 就是相位，单位为弧度（或度）。常数 φ 是初相位，取决于 t = 0 时运动从周期中的什么位置开始。

The phase difference between two oscillations tells us how much one oscillation “lags” or “leads” another. If two oscillators have the same frequency and a phase difference of 0 rad, they are in phase. If the phase difference is π rad (or 180°), they are in antiphase.

两个振动之间的相位差告诉我们一个振动比另一个“滞后”或“超前”多少。如果两个振动频率相同且相位差为 0 rad，则它们同相。如果相位差为 π rad（或 180°），则它们反相。

For example, displacement and velocity in SHM are not in phase: velocity leads displacement by π/2. Meanwhile, acceleration is in antiphase with displacement, because acceleration is always directed back toward equilibrium while displacement is measured away from equilibrium.

例如，简谐运动中的位移和速度并不同相：速度超前位移 π/2。同时，加速度与位移反相，因为加速度总是指向平衡位置，而位移是从平衡位置向外测量的。

5. Velocity in SHM | 简谐运动中的速度

For an object oscillating with SHM, the velocity is not constant. It is zero at the extreme positions and reaches its maximum speed as the object passes through the equilibrium position. The velocity at any displacement x is given by:

对于做简谐运动的物体，速度并非恒定。物体在极值处速度为零，经过平衡位置时速度达到最大值。任意位移 x 处的速度由下式给出：

v = ± ω √(A² – x²)

The plus and minus signs show that the object can be moving in either direction. The maximum speed occurs when x = 0, so:

正负号表示物体可能向两个方向中的任一方向运动。最大速度出现在 x = 0 时，因此：

v_max = ω A

If x = A sin(ωt + φ), then differentiating with respect to time gives v = Aω cos(ωt + φ). This confirms that velocity is π/2 ahead of displacement in phase.

如果 x = A sin(ωt + φ)，那么对时间求导得到 v = Aω cos(ωt + φ)。这证实了速度在相位上超前位移 π/2。

6. Acceleration in SHM | 简谐运动中的加速度

Acceleration is the rate of change of velocity. For SHM, the defining property is that acceleration is proportional to displacement but opposite in direction. Mathematically:

加速度是速度的变化率。对于简谐运动，其定义性特征是加速度与位移成正比，但方向相反。数学上可写作：

a = -ω² x

The negative sign indicates that acceleration always points toward the equilibrium position. At the extremes, x = ±A, so the acceleration has maximum magnitude a_max = ω² A. At equilibrium, x = 0, so acceleration is zero.

负号表示加速度总是指向平衡位置。在极值处，x = ±A，因此加速度的大小最大，a_max = ω² A。在平衡位置，x = 0，所以加速度为零。

This result also leads to Newton’s second law for SHM: if a mass m experiences SHM, the restoring force is F = ma = -mω² x. For a spring with force constant k, we have F = -kx, so mω² = k and therefore ω = √(k/m).

这一结果也引出了简谐运动的牛顿第二定律：如果质量为 m 的物体做简谐运动，则回复力为 F = ma = -mω² x。对于劲度系数为 k 的弹簧，F = -kx，因此 mω² = k，于是 ω = √(k/m)。

7. Energy in Oscillations | 振动中的能量

During SHM, energy constantly changes form between kinetic energy and potential energy. The total mechanical energy remains constant if there is no damping. The potential energy is stored in the spring or in the gravitational field, and the kinetic energy is carried by the moving mass.

在简谐运动过程中，能量不断在动能和势能之间转化。若没有阻尼，总机械能保持不变。势能储存在弹簧或重力场中，动能由运动的质量携带。

At displacement x, the kinetic energy and potential energy are:

在位移 x 处，动能和势能分别为：

KE = ½ m v² = ½ m ω² (A² – x²)

PE = ½ k x²

The total energy is the maximum potential energy, which occurs at x = ±A, or equivalently the maximum kinetic energy at x = 0:

总能量等于最大势能，出现在 x = ±A 处，也等于 x = 0 处的最大动能：

E_total = ½ k A² = ½ m ω² A²

For a pendulum, the same ideas apply: at the highest point of the swing, gravitational potential energy is maximum and kinetic energy is zero; at the lowest point, kinetic energy is maximum and potential energy is at its local minimum.

对于单摆，同样的思想也适用：在摆动最高点，重力势能最大而动能为零；在最低点，动能最大而势能处于局部最小值。

8. Damping and Resonance | 阻尼与共振

In real oscillations, energy is lost to friction, air resistance, or other resistive forces. This gradual loss of energy causes the amplitude to decrease over time, a process called damping. The period may remain nearly unchanged under light damping, but the amplitude decays exponentially in many practical cases.

在实际振动中，能量会因摩擦、空气阻力或其他阻力而损失。这种能量的逐渐损失导致振幅随时间减小，这一过程称为阻尼。在轻阻尼下，周期可能几乎保持不变，但在许多实际情形中振幅按指数规律衰减。

There are three useful categories of damping. Under light damping, the system oscillates with gradually decreasing amplitude. Under heavy damping, the system returns to equilibrium without oscillating. At critical damping, the system returns to equilibrium in the shortest possible time without oscillating, which is ideal for door closers and suspension systems.

阻尼有三种常用分类。在轻阻尼下，系统振幅逐渐减小并继续振动。在重阻尼下，系统不振动地回到平衡位置。在临界阻尼下，系统以最短时间回到平衡位置且不发生振动，这非常适合门吸和悬挂系统。

Resonance occurs when the driving frequency of an external periodic force equals the natural frequency of the system. At resonance, energy transfer is maximised, causing a sharp increase in amplitude. Uncontrolled resonance can be destructive, as in the case of a bridge oscillating strongly under wind or marching soldiers.

当外部周期性驱动的频率等于系统固有频率时，就会发生共振。共振时能量传递最大，导致振幅急剧增大。不可控制的共振可能具有破坏性，例如桥梁在风力或士兵齐步走作用下剧烈振动。

9. Key Equations Summary | 关键公式总结

To succeed in exam questions, you need to know which formula to apply in each situation. The table below summarises the most important equations for describing oscillations.

要在考试中取得好成绩，你需要知道在每种情况下应用哪个公式。下表总结了描述振动最重要的公式。

Quantity \| 物理量	Equation \| 公式
Period and frequency \| 周期与频率	T = 1 / f
Angular frequency \| 角频率	ω = 2π / T = 2π f
Displacement \| 位移	x = A sin(ωt + φ)
Velocity \| 速度	v = ± ω √(A² – x²); v_max = ω A
Acceleration \| 加速度	a = -ω² x; a_max = ω² A
Mass-spring period \| 弹簧振子周期	T = 2π √(m / k)
Pendulum period \| 单摆周期	T = 2π √(l / g)
Total energy \| 总能量	E = ½ k A² = ½ m ω² A²

10. Common Exam Pitfalls | 常见考试易错点

One common mistake is confusing displacement with amplitude. Amplitude is constant for undamped SHM, while displacement varies with time. Another error is forgetting the phase difference between displacement, velocity and acceleration. In a graph question, students often misidentify which curve reaches its maximum first.

常见错误之一是把位移与振幅混淆。对于无阻尼简谐运动，振幅恒定，而位移随时间变化。另一个错误是忘记位移、速度和加速度之间的相位差。在做图题中，学生常常无法正确判断哪条曲线先达到最大值。

Students also frequently misapply the formula for period. The pendulum period does not depend on the mass or the amplitude for small angles, but the mass-spring period does depend on mass and spring constant. Always check the physical situation before substituting numbers.

学生还常常错误套用周期公式。单摆周期在小角度下与质量或振幅无关，但弹簧振子的周期确实与质量和劲度系数有关。在代入数值之前，务必先判断物理情境。

Finally, with energy calculations, remember that the total energy is constant only when damping is negligible. If damping is present, the amplitude and total energy decrease over time, so you cannot use E = ½ k A² with the initial amplitude for later times.

最后，在能量计算中，请记住只有当阻尼可忽略时总能量才守恒。如果存在阻尼，振幅和总能量都随时间减小，因此不能在之后时刻继续用初始振幅代入 E = ½ k A²。

11. Conclusion | 结论

The key quantities for describing oscillations form the foundation of SHM in A-Level Physics. Displacement and amplitude describe the geometry of motion; period, frequency and angular frequency describe its timing; phase describes its alignment with other oscillators; and velocity, acceleration and energy describe the dynamical behaviour. Mastering these definitions and their relationships will allow you to solve both calculation-style and reasoning-style exam questions with confidence.

描述振动的关键物理量构成了 A-Level 物理中简谐运动的基础。位移和振幅描述运动的几何特征；周期、频率和角频率描述运动的时间特征；相位描述与其他振子的相对位置；而速度、加速度和能量描述运动的动力学行为。掌握这些定义及其相互关系，将帮助你自信地解决计算型和推理型考试题目。

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Quantity \| 物理量	Equation \| 公式
Period and frequency \| 周期与频率	T = 1 / f
Angular frequency \| 角频率	ω = 2π / T = 2π f
Displacement \| 位移	x = A sin(ωt + φ)
Velocity \| 速度	v = ± ω √(A² – x²); v_max = ω A
Acceleration \| 加速度	a = -ω² x; a_max = ω² A
Mass-spring period \| 弹簧振子周期	T = 2π √(m / k)
Pendulum period \| 单摆周期	T = 2π √(l / g)
Total energy \| 总能量	E = ½ k A² = ½ m ω² A²

A-Level Physics: Key Quantities for Describing Oscillations | A-Level 物理：振动描述的关键物理量