Oscillations and Waves: SHM, Damped and Forced Oscillations

An oscillation is any motion that repeats itself regularly in time, like a swinging pendulum or a vibrating guitar string. The simplest kind is simple harmonic motion (SHM), where the restoring force is proportional to the displacement from equilibrium. A wave is a travelling oscillation: a disturbance that carries energy from place to place without carrying matter along.

They matter because vibrations and waves are everywhere in engineering. Buildings must survive earthquakes, bridges must avoid resonant shaking, and all wireless communication rides on electromagnetic waves. Understanding one mass on a spring unlocks clocks, musical instruments, and radio alike.

Key Concepts

• Displacement (x): distance from the equilibrium position, in metres.
• Amplitude (A): the maximum displacement. Larger amplitude means more energy.
• Time period (T): time for one complete oscillation, in seconds.
• Frequency (f): oscillations per second, in hertz. f = 1/T.
• Angular frequency (omega): omega = 2 pi f = 2 pi / T, in radians per second.
• Damped oscillation: amplitude decays because of friction or resistance; can be underdamped, critically damped, or overdamped.
• Forced oscillation and resonance: pushing a system at its natural frequency makes the amplitude grow large.
• Wave speed: v = f lambda, where lambda is the wavelength in metres.

Formulas

• SHM displacement: x(t) = A sin(omega t plus phi), where phi is the phase constant.
• SHM condition: acceleration a = minus omega squared x. The minus sign means acceleration always points back toward equilibrium.
• Spring-mass period: T = 2 pi times square root of (m/k), with mass m in kg and spring constant k in N/m.
• Simple pendulum period: T = 2 pi times square root of (L/g), valid for small angles, with length L in metres and g = 9.8 m/s squared.
• SHM energy: total energy E = (1/2) k A squared, constant, sloshing between kinetic and potential.
• Wave relation: v = f lambda. For a string, v = square root of (tension divided by mass per unit length).

Worked Example

A 1 kg mass hangs from a spring of stiffness 100 N/m and is set oscillating. Find the period and frequency.

1. Use T = 2 pi times square root of (m/k) = 2 pi times square root of (1/100).
2. square root of 0.01 = 0.1, so T = 2 pi times 0.1 = 0.2 pi seconds.
3. Numerically, T is about 0.628 s.
4. Frequency f = 1/T = 1/0.628, about 1.59 Hz.
5. Check energy: if amplitude A = 0.05 m, total energy = 0.5 times 100 times 0.0025 = 0.125 J, constant throughout the motion.

Common Misconceptions

• Any back-and-forth motion is SHM. SHM specifically needs restoring force proportional to displacement. A bouncing ball is periodic but not simple harmonic.
• Resonance means the frequencies must match exactly to cause damage. Large response occurs near the natural frequency; damping controls how sharp and how large the peak is. Engineers add damping to tame it.
• Waves carry matter along with them. In a water wave, each water molecule mostly moves in a small circle; it is the energy and the pattern that travel forward.

Real-World Applications

• Timekeeping: pendulum clocks and quartz crystals rely on stable natural frequencies.
• Civil engineering: tuned mass dampers in skyscrapers counteract wind and earthquake resonance.
• Music: strings, air columns, and membranes are tuned oscillators and standing-wave systems.
• Communications: radio, microwaves, and light are electromagnetic waves obeying v = f lambda.

Simple harmonic motion is the prototype of all vibration: a linear restoring force gives sinusoidal motion with period set by mass and stiffness. Add damping and you get decay; add a periodic push near the natural frequency and you get resonance.