W · Oscillations and Waves — Reference Library

Simple Harmonic Motion

Any restoring influence proportional to displacement produces the same sinusoidal motion, whatever is oscillating.

W1.1The Defining Equation

Simple harmonic motion arises whenever the restoring acceleration is proportional to displacement and directed back toward equilibrium.

a = - ω^{2} x, x (t) = A cos (ω t + ϕ)

Definition(W1.1.1)Restoring acceleration proportional to displacement

W1.2Energy in an Oscillation

Total mechanical energy in an oscillator is constant, continuously exchanged between kinetic and potential forms.

E = \frac{1}{2} k A^{2} = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2}

Result(W1.2.1)Total energy of an oscillator from its amplitude

W1.3Pendulums

For small swing angles, a pendulum's restoring torque is proportional to angular displacement, satisfying the SHM condition.

T = 2 π \frac{L}{g}

Result(W1.3.1)Period of a simple pendulum

W1.4Damping and Resonance

Friction or drag removes energy from a real oscillator, gradually decaying its amplitude — underdamped (oscillates while decaying), critically damped (returns to equilibrium fastest without oscillating), or overdamped (returns slowly, no oscillation). Driving an oscillator periodically produces resonance: the amplitude peaks sharply when the driving frequency matches the system's natural frequency.

Wave Motion

A wave carries energy and pattern through a medium without transporting the medium itself.

W2.1The Wave Function

A travelling wave is a disturbance that repeats in both space (wavelength λ) and time (period T), moving at fixed shape and speed.

y (x, t) = A sin (k x - ω t)

Definition(W2.1.1)Displacement of a travelling sinusoidal wave

W2.2Speed on a String

Wave speed on a string is set by the balance between the restoring effect of tension and the inertia of the string's mass.

v = \frac{F _{T}}{μ}

Result(W2.2.1)Wave speed from tension and mass density

W2.3Energy Transport

A wave carries energy at the rate its source supplies it, proportional to the square of both amplitude and frequency.

P = \frac{1}{2} μ ω^{2} A^{2} v

Result(W2.3.1)Average power carried by a wave on a string

W2.4Reflection and Transmission

A wave meeting a boundary between two media partially reflects and partially transmits. Reflecting from a denser medium (a fixed end) inverts the wave; reflecting from a less dense medium (a free end) does not. The fraction reflected versus transmitted depends on how sharply the wave impedance changes at the boundary.

Sound

Sound is a longitudinal pressure wave; its speed, loudness scale, and frequency shifts all follow from that.

W3.1Pressure Waves

Sound is a longitudinal wave: alternating compressions and rarefactions of the medium, oscillating along the same direction the wave travels rather than perpendicular to it.

v = \frac{B}{ρ}

Result(W3.1.1)Speed of sound from a medium's stiffness and density

W3.2Intensity and Decibels

The audible range of intensities spans many orders of magnitude, compressed onto a manageable logarithmic scale.

β = 10 lo g_{10} (\frac{I}{I _{0}}) dB

Definition(W3.2.1)Sound level on the decibel scale

W3.3The Doppler Effect

Relative motion between a sound source and an observer shifts the observed frequency.

f^{'} = f (\frac{v \pm v _{o}}{v \mp v _{s}})

Result(W3.3.1)Frequency shift from relative motion of source and observer

W3.4Shock Waves

A source moving faster than the wave speed leaves its wavefronts unable to outrun it, piling them into a cone.

sin θ = \frac{v}{v _{s}}

Result(W3.4.1)Half-angle of the shock cone behind a supersonic source

Superposition

Waves add linearly; the pattern that results — reinforcement, cancellation, or a beat — depends only on their relative phase.

W4.1Interference

Two coherent waves reaching the same point combine constructively or destructively depending on the difference in the distance each has travelled.

Δ r = mλ (constructive), Δ r = (m + \frac{1}{2}) λ (destructive)

Result(W4.1.1)Constructive or destructive condition from path difference

W4.2Standing Waves

Two identical waves travelling in opposite directions combine into a pattern that oscillates in place rather than travelling.

y (x, t) = 2 A sin (k x) cos (ω t)

Result(W4.2.1)Displacement pattern of a standing wave

W4.3Normal Modes

A string fixed at both ends supports only wavelengths that fit an integer number of half-wavelengths between the fixed points.

f_{n} = \frac{n v}{2 L} (n = 1, 2, 3, \dots)

Result(W4.3.1)Allowed frequencies of a string fixed at both ends

W4.4Beats

Two sound sources of nearly equal frequency produce a slowly pulsing loudness, rather than two separately audible tones.

f_{beat} = ∣ f_{1} - f_{2} ∣

Result(W4.4.1)Pulsing frequency from two close source frequencies