Waves, Oscillations & Acoustics

Nodes & Antinodes

Specific fixed positions on a standing wave where displacement amplitude is zero (nodes) or maximum (antinodes).

Node Spacing = λ / 2, Antinode Spacing = λ / 2 m

Non-linear Oscillations

Oscillatory systems where restoring forces do not obey linear Hooke's law, leading to amplitude-dependent frequency and chaotic behavior.

m (d²x/dt²) + k x + β x³ = 0 (Duffing Equation) Various

Waves, Oscillations & Acoustics

Normal Modes

Characteristic independent natural patterns of motion in which all parts of an oscillating system move sinusoidally at the same natural…

f_n = n f1 = n v / (2 L) Hz

Waves, Oscillations & Acoustics

Open Organ Pipes

A cylindrical acoustic pipe open at both ends, containing displacement antinodes at both openings.

f_n = n v / (2 L) (n = 1, 2, 3, ...) Hz

Waves, Oscillations & Acoustics

Overtones

Any resonant frequency higher than the fundamental frequency in a musical sound.

First Overtone = 2nd Harmonic (for strings) Hz

Waves, Oscillations & Acoustics

Phase

The fractional position of a point within a wave cycle, expressed as an angle relative to a reference origin.

Φ(x,t) = k x - ω t + φ0 rad (radians)

Waves, Oscillations & Acoustics

Phase Difference

The angular difference between corresponding points on two waves or between different points on a single wave.

Δφ = (2π / λ) Δx = (2π / T) Δt rad (radians)

Waves, Oscillations & Acoustics

Phase Velocity

The speed at which a single monochromatic wave crest or phase front advances through a medium.

v_p = ω / k = f λ m/s

Waves, Oscillations & Acoustics

Pitch

The subjective psychoacoustic perception of a sound's frequency by human listeners.

Pitch ∝ f Hz (or Musical Notes)

Waves, Oscillations & Acoustics

Principle of Superposition

The principle stating that when two or more linear waves overlap at a point, total resultant displacement equals vector sum of individual…

y_total(x,t) = y1(x,t) + y2(x,t) + ... + yn(x,t) m

Waves, Oscillations & Acoustics

Resonance

The phenomenon where an oscillating system driven at its natural frequency absorbs maximum energy, resulting in large-amplitude vibrations.

f_driven = f_natural = (1 / 2π) √(k / m) Hz

Waves, Oscillations & Acoustics

Reverberation

The persistence of sound in an enclosed space caused by multiple continuous reflections from room boundaries after the original source…

RT60 = 0.161 V / A (Sabine's Formula) s

Waves, Oscillations & Acoustics

Shock Wave

A extremely thin propagating pressure disturbance characterized by a nearly instantaneous jump in pressure, temperature, and density.

P2 / P1 = 1 + (2γ / (γ + 1)) (M1² - 1) Pa

Waves, Oscillations & Acoustics

Sonar

Sound Navigation and Ranging—a technique utilizing acoustic wave propagation underwater to navigate, communicate, or detect objects.

d = v_water t / 2 m

Waves, Oscillations & Acoustics

Sonic Boom

An explosive impulsive acoustic thunderclap heard on the ground when an object travels faster than the speed of sound.

sin α = 1 / M (Mach Cone Angle) dB / Pa

Waves, Oscillations & Acoustics

Sound Intensity

The time-averaged rate of sound energy flow per unit area perpendicular to the direction of propagation.

I = P / A = p_rms² / (ρ v) W/m²

Waves, Oscillations & Acoustics

Speed of Sound

The speed at which a pressure wave travels through an elastic medium.

v = √(K / ρ) (Fluids), v = √(γ R T / M) (Ideal Gas) m/s

Waves, Oscillations & Acoustics

Standing Waves

A stationary wave pattern formed by the superposition of two identical waves traveling in opposite directions along the same medium.

y(x,t) = [2 A sin(k x)] cos(ω t) m

Waves, Oscillations & Acoustics

Threshold of Hearing

The minimum sound pressure level or intensity audible to a healthy human ear at 1000 Hz.

I0 = 10⁻¹² W/m² (0 dB SPL, P0 = 20 µPa) W/m² (0 dB)

Waves, Oscillations & Acoustics

Threshold of Pain

The upper sound intensity limit above which acoustic vibrations cause physical discomfort and immediate structural damage to human ears.

I_pain ≈ 1 W/m² (120 - 130 dB SPL) W/m² (120-130 dB)

Waves, Oscillations & Acoustics

Timbre

The subjective tone quality or color of a musical sound that distinguishes different instruments playing the same fundamental pitch and…

Fourier Spectrum Balance N/A

Waves, Oscillations & Acoustics

Transverse Wave

A wave in which medium particles oscillate perpendicular to the direction of wave energy propagation.

y(x,t) = A sin(k x - ω t) m

Waves, Oscillations & Acoustics

Tsunami

A series of extremely long-wavelength ocean water waves generated by sudden vertical displacement of the seafloor.

v = √(g d) m/s

Waves, Oscillations & Acoustics

Van der Pol Oscillator

A non-linear oscillator with non-linear damping that exhibits stable self-sustained limit cycle vibrations.

d²x/dt² - μ (1 - x²) dx/dt + x = 0 N/A