Oscillations and Mechanical Waves

Big Picture

Oscillation is motion that repeats itself — back and forth around an equilibrium position. When oscillations propagate through a medium, they become waves. The math of simple harmonic motion (SHM) — sines and cosines — is the universal language of periodic phenomena. Every oscillation and wave shares the same concepts: amplitude, frequency, period, and phase.

ELI-10: Explain It Like I'm 10

A swing is an oscillator. Push it once and it goes back and forth at a steady rhythm. That rhythm is its natural frequency. If you push at just the right time each swing, the swing goes higher and higher — that is resonance. A wave is what happens when an oscillation travels: drop a pebble in a pond and the up-and-down motion of the water spreads outward in rings. The water does not travel outward — only the disturbance does.

8.1 Simple Harmonic Motion

Core Idea

Simple harmonic motion (SHM) occurs when the restoring force is proportional to displacement and always points toward equilibrium: F = -kx.

Physics and Mathematics

Position: x(t) = Acos(ωt + φ)

Velocity: v(t) = -Aωsin(ωt + φ)

Acceleration: a(t) = -Aω²cos(ωt + φ) = -ω² x(t)

Where:

• A = amplitude (m) — maximum displacement
• ω= 2πf = 2πT = angular frequency (rad/s)
• f = frequency (Hz = s⁻¹)
• T = 1/f = period (s)
• φ = phase constant (rad) — sets initial conditions

Mass on a spring: ω= √k/m, T = 2π√m/k

Simple pendulum (small angle): T = 2π√L/g

ELI-10: Explain It Like I'm 10

Tie a weight to a spring and pull it down. Let go. It bounces up and down at a steady pace. Pull it farther down and it bounces with bigger swings (larger amplitude) but the same timing (same period). The period depends only on the mass and the spring stiffness — not on how hard you pull. A pendulum clock works the same way: the swing time depends only on the length of the pendulum, not on how far back you pull it (as long as the swing is small).

Understanding Phase

The phase constant φ determines where the oscillator starts in its cycle at t = 0:

• φ= 0: starts at maximum positive displacement (x = +A).
• φ= π/2: starts at equilibrium moving negative (x = 0, moving left).
• φ= π: starts at maximum negative displacement (x = -A).

Phase is crucial when comparing two oscillators. If two identical pendulums are released from the same height but at slightly different times, they have a phase difference — one reaches the bottom before the other. In wave interference (Section 8.4), phase differences determine whether waves reinforce or cancel.

Worked Example: Mass-Spring Period

Problem: A 0.50 kg mass on a spring oscillates with a period of 0.80 s. Find the spring constant. What is the frequency?

Given: m = 0.50 kg, T = 0.80 s.

Solution: T = 2π√m/k → k = 4π² m/T² = 4π²(0.50)/(0.80)² ≈ 31 N/m. f = 1/T = 1/0.80 = 1.25 Hz.

Does it Make Sense? A period under one second with a half-kilogram mass requires a moderately stiff spring (~31 N/m). This is typical for a lab spring. If you doubled the mass, the period would increase by √2 ≈ 1.41 to about 1.13 s — heavier = slower oscillation.

Common Mistakes

• Thinking amplitude affects the period in SHM. For a mass-spring system or a small-angle pendulum, period is independent of amplitude.
• Forgetting the simple pendulum formula only works for SMALL angles (θ < ~15°).

Key Takeaway

SHM and uniform circular motion (Topic 1) are intimately connected: the projection of uniform circular motion onto a diameter is simple harmonic motion. If you shine a light on a peg rotating at constant angular speed on a wheel, its shadow on the wall moves back and forth in SHM. This geometric connection explains why ω appears in both topics and why sinusoidal functions describe both phenomena.

8.2 Energy in SHM

Core Idea

In SHM, energy continuously converts between kinetic and potential form, but the total is constant (no damping).

E_total = 12kA² = 12mv_max²

At maximum displacement: all energy is potential (U = 12kA², K = 0). At equilibrium: all energy is kinetic (K = 12mv_max², U = 0).

ELI-10: Explain It Like I'm 10

A swing at the highest point is momentarily stopped — all energy is stored (potential). At the lowest point, it is moving fastest — all energy is motion (kinetic). Throughout the swing, energy trades back and forth, but the total never changes (without friction). In a real swing, friction and air resistance gradually drain energy, making the swings smaller over time — this is called damping. A heavily damped oscillator (like a swing in water) barely oscillates at all; it just slowly returns to equilibrium.

8.3 Wave Properties

Core Idea

A wave is a traveling disturbance that transfers energy through a medium. The medium itself does not travel with the wave.

Physics and Mathematics

• Transverse wave: particles vibrate perpendicular to wave direction (light waves, guitar strings).
• Longitudinal wave: particles vibrate parallel to wave direction (sound waves).

Key parameters:

• Wavelength λ (m): distance between successive crests.
• Frequency f (Hz): number of cycles per second.
• Period T = 1/f (s): time for one cycle.
• Amplitude A (m): maximum displacement from equilibrium.
• Wave speed: v = fλ= λT

Wave speed depends on the medium's properties, NOT on frequency or amplitude. For a string: v = √F_T/μ where F_T is tension and μ is linear density.

ELI-10: Explain It Like I'm 10

Shake one end of a rope and a bump travels down the rope. The rope itself does not move forward — each piece just wiggles up and down. The bump (the wave) carries energy from your hand to the far end. Faster shaking makes more bumps per second (higher frequency). The wave speed depends on how tight and how heavy the rope is — not on how fast you shake it.

8.4 Superposition and Interference

Core Idea

When two waves meet, their displacements add algebraically (superposition principle).

Constructive interference: waves in phase → amplitudes add → larger wave. Destructive interference: waves 180° out of phase → amplitudes subtract → smaller wave (possibly zero).

ELI-10: Explain It Like I'm 10

Two water ripples meet: where two peaks overlap, you get a taller peak. Where a peak meets a valley, they cancel and the water is flat. The waves pass through each other unchanged — they do not bounce or break. This is superposition.

8.5 Standing Waves and Resonance

Core Idea

A standing wave forms when two identical waves travel in opposite directions, creating fixed nodes (zero displacement) and antinodes (maximum displacement).

String fixed at both ends: λ_n = 2Ln,   f_n = nv2L = nf₁  (n = 1,2,3,…)

The fundamental frequency f₁ = v/(2L). Harmonics are integer multiples.

Resonance: when a driving frequency matches a natural frequency, amplitude grows dramatically. This can be destructive (Tacoma Narrows Bridge) or useful (musical instruments, MRI).

ELI-10: Explain It Like I'm 10

Pluck a guitar string. It vibrates in a pattern that does not travel — it stands still in place. Certain points (nodes) do not move at all. The whole string vibrates at its natural "note." Press a finger at the 12th fret and you get a note one octave higher (double the frequency, half the wavelength). That is playing harmonics. If you sing the right note near a wine glass, it can shatter — that is resonance: matching the glass's natural frequency exactly.

Standing Waves in Air Columns

Sound waves in pipes also form standing waves. The boundary conditions differ:

• Open at both ends: antinodes at both ends. λ_n = 2L/n, f_n = n v/(2L) (same pattern as a string).
• Closed at one end: node at closed end, antinode at open end. Only odd harmonics exist: λ_n = 4L/n for odd n, f_n = n v/(4L) for n = 1, 3, 5, …

This is why a clarinet (approximately closed at one end) has a different timbre from a flute (open at both ends) — the clarinet produces only odd harmonics, giving it a "hollow" sound.

8.6 Sound

Core Idea

Sound is a longitudinal mechanical wave that travels through a medium. In air at 20°C, the speed of sound is approximately 343 m/s.

Key Concepts

• Pitch is determined by frequency. Higher frequency → higher pitch.
• Intensity I = P/A (W/m²). Related to amplitude squared.
• Sound level: β= 10log₁₀(I/I₀) decibels (dB), where I₀ = 10^-12 W/m².
• Doppler effect: observed frequency shifts when source and observer move relative to each other.

For sound: f' = f(v ± v_ov ∓ v_s), where v is sound speed, v_o is observer velocity, v_s is source velocity. (Signs depend on direction — approaching raises frequency.)

ELI-10: Explain It Like I'm 10

Sound is air wiggling. When a speaker cone vibrates, it pushes air molecules together and pulls them apart, creating traveling zones of compressed and stretched air. Your eardrum detects these wiggles. Higher-pitched sounds wiggle faster. Louder sounds wiggle harder (bigger amplitude). An ambulance siren sounds higher as it approaches you and lower after it passes — the sound waves get squished together ahead of it and stretched out behind. That is the Doppler effect.

Worked Example: Doppler Effect

Problem: An ambulance siren emits a constant 800 Hz tone. The ambulance approaches a stationary observer at 30 m/s. What frequency does the observer hear? (Speed of sound = 343 m/s.)

Given: f = 800 Hz, v_s = 30 m/s (approaching), v = 343 m/s, v_o = 0.

Equation: f' = f vv - v_s (source approaching, denominator reduces → frequency increases).

Solution: f' = 800 343343 - 30 = 800 343313 ≈ 877 Hz.

After the ambulance passes (receding): f' = 800 343343 + 30 = 800 343373 ≈ 736 Hz.

Does it Make Sense? The observed frequency is higher on approach and lower on recession — the classic "EEEE-yoooow" of a passing siren. The 141 Hz shift is a clearly audible pitch change.

Assumptions and Model Limitations

Oscillations and wave models in this topic assume:

• SHM: restoring force is exactly proportional to displacement (F = -kx). Real springs deviate at large stretches.
• Small-angle pendulum: period formula T = 2π√L/g is only accurate for θ< ∼ 15^°.
• Ideal string: perfectly flexible, uniform linear density, tension constant.
• No damping: real oscillators lose energy to friction/air resistance; amplitude decays exponentially.
• Plane waves: wavefronts are treated as flat; valid far from the source.
• Constant sound speed: v_sound varies with temperature and medium.

Limiting Cases for Waves

• What if frequency goes to zero? Wavelength becomes infinite — the "wave" is essentially a static displacement. DC (Topic 11) is the zero-frequency limit of AC.
• What if tension in a string goes to zero? Wave speed goes to zero (v = √F_T/μ). A completely slack string cannot transmit waves.
• What if the driving frequency equals the natural frequency? Resonance — amplitude grows dramatically, limited only by damping. This can shatter a wine glass or destroy a bridge (Tacoma Narrows).

Topic Summary

• SHM occurs when F ∝ -x. Position, velocity, and acceleration are sinusoidal with frequency independent of amplitude.
• Energy in SHM continuously converts between kinetic and potential. Total E = 12kA².
• Waves transfer energy without bulk matter transport. v = fλ. Speed depends on medium, not frequency.
• Superposition means waves add; constructive and destructive interference result.
• Standing waves form from opposite-traveling waves; nodes and antinodes are fixed in space. Harmonics are integer multiples of the fundamental.
• Sound is a longitudinal wave. Doppler effect: approaching source → higher frequency.

Essential Equations

Concept Check

1. If you double the amplitude of a mass-spring oscillator, what happens to its period? Its total energy?
2. A wave travels from a thick rope to a thin rope. The frequency stays the same. What changes — speed, wavelength, or both?
3. Why do soldiers break step when marching across a bridge?
4. A guitar string is shortened to half its length (by pressing at the 12th fret). How does the frequency change?
5. An ambulance approaches you with its siren on. You hear a higher pitch. What do you hear after it passes? Why?
6. If you take a pendulum clock from Earth to the Moon (g_Moon ≈ g/6), does it run faster or slower? By approximately what factor?
7. An open pipe and a closed pipe have the same length. Which has the lower fundamental frequency? Why?

Open Educational References

• OpenStax, College Physics, Chapters 16–17: Oscillatory Motion, Physics of Hearing
• OpenStax, University Physics, Volume 1, Chapters 15–17: Oscillations, Waves, Sound