The Wave Ocean

NODE 01

What is a Wave?

↓

• Propagating disturbance.
• Transfers energy/info without bulk transport of medium.
• Particles oscillate locally; disturbance travels.

NODE 02

Wave Source

• Requires a disturbance.
• Vibrating string, tuning fork, speaker.
• Source → Disturbance → Propagation.

NODE 03

Mechanical Waves

• Require a material medium.
• Sound, water waves, seismic waves.
• Medium supplies interactions for propagation.

NODE 04

EM Waves (Contrast)

MECHvsEM

• EM waves can propagate through vacuum.
• Major classification: Mechanical vs EM.
• Detailed EM belongs in Realm 21.

NODE 05

Transverse Waves

• Particles oscillate perpendicular to propagation.
• Waves on a string, EM waves.
• Crest, trough, amplitude.

NODE 06

Longitudinal Waves

• Particles oscillate parallel to propagation.
• Compression, rarefaction.
• Sound in air is standard example.

NODE 07

Mixed Wave Motion

• Real waves aren't just rigid binaries.
• Surface waves, seismic waves.
• Combined transverse/longitudinal behavior.

NODE 08

Wave Pulse

• A single disturbance.
• Shape, amplitude, speed.
• Reflection and transmission.

NODE 09

Periodic Wave

v = fλ

• A repeating disturbance.
• Period (T), Freq (f), Wavelength (λ), Speed (v).
• Core relation: v = fλ.

NODE 10

Wavelength (λ)

• Distance between corresponding points.
• Crest to crest, compression to compression.
• Spatial period. Units: metre.

NODE 11

Frequency (f)

• Complete oscillations per second (Hz).
• Connected to source vibration, pitch, energy.

NODE 12

Wave Speed (v)

• Speed disturbance propagates (v = fλ).
• CRITICAL: Particle velocity ≠ wave speed.

NODE 13

Angular Frequency

ω = 2πfk = 2π/λ

• Connects Realm 13 math.
• Wave number (k).
• v = ω/k.

NODE 14

Wave Function

y(x,t) = A sin(kx - ωt + φ)

• 1D harmonic wave.
• Sign determines propagation direction.

NODE 15

Phase of a Wave

• Same phase, opposite phase.
• Spatial/temporal phase difference.
• Δφ = 2πΔx/λ.

NODE 16

Amplitude (A)

• Max displacement from equilibrium.
• Energy transported is proportional to A².

NODE 17

Wavefronts

• Joins points in the same phase.
• Plane, Spherical, Cylindrical.
• Geometric view of propagation.

NODE 18

Rays & Wavefronts

• Ray → direction of propagation.
• Ray is perpendicular to wavefront.
• Foundation for optics (Realm 22).

NODE 19

Prop in a Medium

f ↔ SOURCEv ↔ MEDIUM

• Speed depends on physical properties (density, elasticity).
• Freq depends on source, Speed depends on medium.

NODE 20

Waves on a String

v = √(T/μ)

• v = √(T/μ). T = tension, μ = mass/length.
• Essential for standing waves on strings.

NODE 21

Reflection

• Wave encounters a boundary.
• Fixed (phase reversal) vs Free boundary.

NODE 22

Transmission

• At boundary: reflected, transmitted, absorbed.
• Transmission coefficient.

NODE 23

Refraction (Wave)

• Change direction when speed changes between media.
• Freq stays constant, wavelength changes.

NODE 24

Superposition

y = y₁ + y₂

• Multiple waves overlap, displacements add algebraically.
• Gateway to interference, standing waves, beats.

NODE 25

Constructive Inter

+

=

• Displacements reinforce → larger amplitude.
• Δφ = 2nπ, Δ = nλ.

NODE 26

Destructive Inter

+

=

• Waves oppose → cancellation.
• Δφ = (2n+1)π, Δ = (n + ½)λ.

NODE 27

Coherence

• Stable phase relationship required for interference.
• Same frequency, constant phase diff.

NODE 28

Standing Waves

• Oppositely traveling waves combine.
• Nodes, Antinodes. No net propagation.

NODE 29

Nodes

• Locations with zero displacement.
• Destructive interference. Spacing = λ/2.

NODE 30

Antinodes

• Locations of max amplitude.
• Spacing = λ/2. Node to Antinode = λ/4.

NODE 31

Waves on a String

• L = nλ/2 → fn = nv/2L.
• Fundamental, 1st harmonic, 2nd harmonic.

NODE 32

Open & Closed Pipes

• Open: both ends open. Closed: one end closed.
• Closed pipe resonances: L = (2n-1)λ/4 (odd harmonics).

NODE 33

Resonance in Air

• Driving freq excites natural mode.
• Flutes, pipes, bottles, Helmholtz resonators.

NODE 34

Sound

• Mechanical wave from vibrating source.
• Primarily longitudinal in gases (air).

NODE 35

Speed of Sound

v = √(γP/ρ)v = √(γRT/M)

• Depends on medium, temp, density, elasticity.
• Faster in solids than gases.

NODE 36

Sound Characteristics

PITCH (f)LOUDNESS (A)TIMBRE (WAVE)

• Pitch ~ Freq. Loudness ~ Amplitude/Intensity.
• Timbre ~ Waveform/Harmonics.

NODE 37

Intensity of Sound

• Power per unit area (I = P/A).
• Spherical spreading: I ∝ 1/r².

NODE 38

Sound Level (β)

10 log₁₀(I/I₀) dB

• Logarithmic scale (Decibels).
• Doubling intensity ≠ doubling decibel level.

NODE 39

Human Hearing

• Audible range: ~20Hz to 20kHz.
• Distinguish physical freq from perceived capability.

NODE 40

Ultrasound

• Above human hearing (>20kHz).
• Medical imaging, NDT, distance measurement.

NODE 41

Infrasound

• Below human hearing (<20Hz).
• Atmospheric phenomena, volcanoes, seismic.

NODE 42

Beats

• Two close frequencies interfere.
• Alternating loud/soft: f_beat = |f₁ - f₂|.
• Tuning instruments.

NODE 43

Doppler Effect

• Source/observer relative motion.
• Observed freq differs from emitted freq.

NODE 44

Doppler Formula

f' = f(v ± v_o)/(v ∓ v_s)

• Depends on relative motion direction.
• Establish sign convention before substitution.

NODE 45

Doppler Apps

🚑📡🩺

• Ambulances, Radar, Medical ultrasound.
• Astronomy, Weather sensing.

NODE 46

Shock Waves

• Source travels faster than wave speed (Mach > 1).
• Cone-shaped wavefront forms.

NODE 47

Sonic Boom

• Macroscopic consequence of supersonic motion.
• Pressure disturbance, shock wave arrival.

NODE 48

Seismic Waves

• P waves (longitudinal), S waves (transverse).
• Surface waves. Bridge to Earth science.

NODE 49

Water Waves

• Surface waves, gravity/surface tension effects.
• Particle motion can be complex (not purely transverse).

NODE 50

Dispersion (Look)

• Diff frequencies propagate at diff speeds in media.
• Phase vs Group velocity. Wave packets.

NODE 51

Phase & Group Vel

v_pω/k

v_gdω/dk

• Phase vel: speed of a single frequency.
• Group vel: speed of wave packet/energy transport.

NODE 52

Wave Energy

• Wave transports energy, NOT bulk matter.
• Depends on amplitude, intensity, geometrical spreading.

NODE 53

Wave Power

P ∝ A²

• Power transfer depends on Amp, Freq, Speed, Medium.
• Stronger waves transport more energy.

NODE 54

Wave Equation

• Standard 1D wave equation.
• Bridge between mathematical physics and wave picture.

NODE 55

Wave → Optics Bridge

WAVE→OPTICS

• Reflection, refraction, interference, diffraction.
• Light itself behaves as a wave (Realms 21-24).

NODE 56

Experiment Lab

λ

f

v

Δ

• Wave speed on string (T, μ). Standing waves.
• Resonance tube, Beats, Doppler simulation.

NODE 57

Graph Chamber

• v vs √(T/μ). Amp vs freq.
• Intensity vs dist. Beat freq vs diff.

NODE 58

Misconceptions

X

• "Particles travel with wave" → No, local oscillation.
• "Wave speed = particle speed" → No.
• "Sound travels in vacuum" → No.

NODE 59

Solving Engine

• ID Source/Medium → Periodic? v=fλ.
• Superposition? Inter/Beats. Boundaries? Refl/Stand.
• Relative motion? Doppler.

NODE 60

Master Map

SOURCE → DISTURBANCEPROPAGATION → v = fλSUPERPOS → SOUND / DOPPLER

• Oscillation → Disturbance → Propagation.
• Wavelength, Frequency, Speed. Superposition.
• Interference, Standing Waves, Sound, Doppler.