Superposition on a linear string: pulses that cross, phase, beats, reflection at a junction, and standing waves. In every tab the black curve is just the sum of the coloured ones.
Two pulses travel in opposite directions at the same speed. Where they overlap the string shape is the sum; afterwards each continues unchanged.
With opposite pulses of equal size, press Jump to overlap. The string is perfectly flat, yet the energy has not gone anywhere: the stretch energy (½τ(∂y/∂x)²) is zero, and all of it is kinetic (½ρ(∂y/∂t)²), concentrated where the string moves fastest. With equal pulses of the same sign the opposite happens: the string is momentarily at rest and all the energy is stored as stretch.
y₁ = A₁ sin(kx − ωt)
y₂ = A₂ sin(kx − ωt − φ)
A₁ = A₂: y = 2A₀ cos(φ/2) sin(kx − ωt − φ/2)
The right panel adds the two waves as arrows (phasors), tip to tail. All three arrows rotate together at ω, and their vertical projections are the values of y₁, y₂ and y at x = 0 (the dots on the left). The resultant length is the amplitude, which is why it does not depend on time: superposing two harmonic waves of the same ω and k gives another harmonic wave, with amplitude 2A₀cos(φ/2) and phase φ/2 when the amplitudes are equal. With unequal amplitudes the cancellation at φ = π is never complete.
y = 2A₀ cos[2πt (f₁ − f₂)/2] cos[2πt (f₁ + f₂)/2]
The top panel is drawn at the true frequencies, so at 440 Hz the individual oscillations merge into a filled band and only the envelope ±2A₀|cos[π(f₁−f₂)t]| stands out. The shaded strip is the 25 ms window enlarged below, where y₁ and y₂ drift in and out of step. The envelope goes through two maxima per period of the cosine term, hence the beat frequency is |f₁ − f₂|, not half of it. Above roughly 15 Hz the ear stops hearing distinct beats and hears roughness; much further apart, two separate tones.
R₁₂ = (Z₁ − Z₂)/(Z₁ + Z₂)
T₁₂ = 2Z₁/(Z₁ + Z₂), Z = ρv = √(τρ)
Same tension τ on both sides, so Z₂/Z₁ = v₁/v₂.
Polarity. Going onto a heavier string (Z₂ > Z₁) the reflection is inverted; onto a lighter one it is upright; the transmitted pulse is always upright. The two limits are the fixed end (Z₂ → ∞, R = −1) and the free end (Z₂ → 0, R = +1). The transmitted pulse can be taller than the incident one (T up to 2) and still carry less energy, because energy flux scales with Z·amplitude². The thickness of each string is drawn proportional to √ρ. The same formula holds for SH waves at normal incidence with Z = ρβ, and for sound with Z = ρc.
Dashpot termination. Shaking the end of a semi-infinite string at velocity u takes a force F = Z₁u, exactly like pushing a damper. So a dashpot with b = Z₁ is indistinguishable from an endless string: press Matched with the dashpot selected and the pulse disappears into it without any reflection, R = (Z₁ − b)/(Z₁ + b) = 0. This is the idea behind the viscous absorbing boundaries (dashpots ρVP, ρVS) used to truncate numerical models of ground motion.
A harmonic wave arrives from the left and reflects at x = 0 with coefficient R. For R = −1 the sum is y = 2A₀ sin(kx) cos(ωt).
Before the reflection comes back the string just carries a travelling wave. Behind the returning front every point oscillates in phase, with an amplitude fixed in space: nodes every λ/2, antinodes halfway between, λ/4 from the nearest node. With partial reflection the minima no longer reach zero; their depth is measured by the standing-wave ratio (1 + |R|)/(1 − |R|). A fixed end is always a node and a free end an antinode, which is what selects the allowed wavelengths once a second end is added: the modes of vibration.
Z = ρv (Pa·s/m, "rayl"). Plane waves at normal incidence; v = VP (sound speed in fluids) or VS for SH waves.
Each cell is the transmission loss in dB for energy crossing between the row and column media; click one to select the pair. The matrix is symmetric: the same fraction of energy crosses in both directions, even though the amplitudes do not (T₁₂ ≠ T₂₁). Velocity and pressure reflect with opposite signs: a sound wave in water hitting the sea surface reflects pressure with R ≈ −1 (pressure-release boundary) while the particle velocity doubles. Values are typical, not site-specific.
Three readings of the same table. Air → water loses about 30 dB, which is why a shout from the shore is barely heard underwater and why the middle ear needs the ossicles as an impedance transformer. Tissue → air is just as bad, which is why ultrasound needs coupling gel. Granite → soft sediment for SH waves transmits only about 14% of the energy, yet the displacement amplitude nearly doubles (T ≈ 1.9), and the free surface doubles it again: the first step of site amplification.