Waves & Sound
Wave propagation, interference, Doppler effect, and sound visualization
Acoustic Levitation (Schematic)
Standing wave cos(kx)cos(ωt); pressure nodes marked; bead cartoon near a node.
Acoustic Phased Array (Line)
N sources, spacing d/λ and phase ramp: steer audio/sonar-like beams (array factor sketch).
Chladni Figures
Plate mode sin(mπx)sin(nπy); nodal contrast + grains drift to nodes (model).
Cymatics: Circular Membrane
Drum eigenmodes J_m(k_{mn}r) cos(mθ); angular m, radial n, shimmer.
Echo & Echosounder
Round-trip time t = 2d/v; pulse to a wall and back with adjustable sound speed.
Wave Speed: String vs Rod
v = √(T/μ) for a string vs v ≈ √(E/ρ) for longitudinal bar waves.
LC Oscillator (Undamped)
Ideal series LC: q(t), I(t), ω₀ = 1/√(LC); U_C + U_L constant; vs RLC AC.
Larsen Effect (Feedback Loop)
Mic + speaker + delay: loop gain and saturation in a toy discrete feedback model.
Duffing Oscillator
m x¨+cx˙+kx+k₃x³=F cos ωt; soft/hard spring; A(ω) scan vs IC.
Wave Packet & Dispersion
Superpose cos(kx−ωt); ω=ck+αk²; spreading vs Wave on String PDE.
Mach Cone (Schematic)
M = v/c > 1: Huygens pulses + cone sin μ = 1/M; not CFD shock.
Monochord / Sonometer
f₁ = (1/2L)√(T/μ), harmonics, note label, Pluck; vs Standing Waves shapes.
Huygens Principle (Slit)
Secondary sources on slit; superposition & diffraction sketch; not full optics.
Forced Nonlinear Pendulum
θ¨+γθ˙+(g/L)sinθ=A cosωt; phase plot; vs Double Pendulum 2-DOF.
Seismic P and S (Schematic)
Longitudinal vs transverse particle motion; v_P, v_S sliders — not layered Earth.
Water Wave Dispersion ω(k)
Shallow k√(gh), deep √(gk), full tanh(kh) — three curves.
Tsunami & Shallow Water (1D)
Linear η,u on H(x): c = √(gH) drops over shelf; Gaussian uplift impulse.
Kelvin Ship Wake (Cusp Angle)
Deep-water wake V: arms at arcsin(1/3) ≈ 19.47° each side of the track (idealized).
KdV Solitons (Exact)
u_t + 6uu_x + u_xxx = 0; Hirota two-soliton collision or one sech² pulse.
Wave on a String
Shake one end, adjust frequency and amplitude. Standing waves and reflections.
Transverse vs Longitudinal
Side-by-side comparison of wave types with particle motion.
Wave Interference
Two sources creating constructive and destructive patterns. 2D ripple tank.
Standing Waves
Find harmonics on a string. Nodes and antinodes highlighted.
Doppler Effect
Moving source with pitch change via Web Audio. Wavefront compression visible.
Light Doppler & Redshift
EM vacuum Doppler: f/f₀ and z vs v/c; linear Δλ/λ ≈ v/c vs exact √(1+β)/√(1−β). Ties to spectral lines.
Room Reverberation (2D rays)
Specular rays in a shoebox plan; impulse response; RT60 vs Sabine. Absorption per bounce.
Active Noise Cancellation (1D)
Two tones same f: amplitudes and phase; RMS sum vs ideal π phase. Destructive interference demo.
Beam Modal Analysis (Euler–Bernoulli)
First three bending modes: λ from BC; f ∝ (λ/L)²√(EI/μ). Pinned, cantilever, clamped.
Group & Phase Velocity
Two-wave beat: ω(k)=ck+αk²; v_g=Δω/Δk vs v_p=ω̄/k̄; envelope vs carrier.
Organ Pipe (harmonic series)
Open–open vs stopped: f_n formulas; pressure shape; table of modes; Web Audio.
Sound Wave Visualizer
Real-time waveform and frequency spectrum from microphone.
Beat Frequency
Two slightly different frequencies creating audible beats.
Resonance Tube
Open vs closed pipe harmonics. Hear fₙ and see standing pressure wave.
Hearing & Loudness (sketch)
Qualitative threshold vs frequency; compare to equal-loudness ideas.
2D Wave Equation (Membrane)
Explicit FDM on a clamped rectangle: Gaussian pulse, circular fronts, edge reflections.