Freeze time: a spatial snapshot
Freeze position: one point's time history
- Wavelength λ
- Period T
- Current disturbance u(x0, t)
One field, two cuts, one moving phase
u(x, t), in centimetersHorizontal cuts are spatial snapshots. Vertical cuts are time histories. The gold line follows one crest's constant phase, not a material particle. Both axes retain the same physical 0–2 m and 0–2 s windows as parameters change.
The event (x0, t)
- Spatial contribution kx0
- Temporal contribution −ωt
- Phase constant φ
- Total phase θ
- Equivalent phase modulo 2π
- k and ω
Reading a traveling wave
A pattern travels; each marked point stays at its x
For this transverse-displacement illustration, material points oscillate up and down at fixed horizontal coordinates. The waveform moves through those points. A crest is a recognizable phase feature, not a particle being carried along the curve. Horizontal distance is in meters; vertical displacement is in centimeters, so the drawing is not a same-scale picture of a particular string.
With k > 0 and ω > 0, kx − ωt + φ moves toward +x, while kx + ωt + φ moves toward −x. Changing φ shifts the initial cycle but does not reverse propagation.
Space and time answer different questions
A snapshot u(x, t0) varies position while holding time fixed. Its spatial repeat length is λ = 2π/k. A record u(x0, t) holds position fixed and varies time. Its temporal repeat interval is T = 2π/ω, and f = 1/T.
A quarter-period advance shifts any given crest by a quarter-wavelength in the chosen direction. The gold crest keeps the same identity even after it leaves the displayed window; it is not wrapped back to the other side. One local time history alone need not reveal direction: at suitable phases, opposite-moving waves can produce the same record.
Phase is an angle, not a distance or a time
The complete phase is θ(x,t) = kx − ωt + φ for rightward propagation. Each contribution is an angle: (rad/m)×m, (rad/s)×s, and φ in radians. Adding them is dimensionally consistent. The amplitude lies outside the trigonometric function and does not set the spatial or temporal repeat interval.
Sine and cosine are equivalent conventions because cos θ = sin(θ + π/2). Switching the displayed form adjusts φ by 90° to preserve the physical field. The numerical phase then changes, but the disturbance, crests, direction and measured periods do not.
The article's worked events
Example 1 has A = 0.030 m, k = 4π rad/m, ω = 10π rad/s and φ = π/6. It therefore has λ = 0.50 m, T = 0.20 s and f = 5 Hz, and the minus temporal sign means rightward motion.
Example 2 uses u = 2 cos(πx − 2πt + π/3) cm. At x = 1 m and t = 0.25 s, θ = 5π/6 and u = −√3 cm, approximately −1.732 cm. At zero amplitude, the field is flat: λ, T, direction and phase remain nominal generator settings, with no observable crest or smallest spatial/temporal repeat.
WM07 is kinematic. These are exact prescribed sinusoids, not simulated masses, a force-derived string model, an energy-transport calculation or a numerical wave-equation solver. Constant-phase tracking connects to WM08; wave-speed derivations and medium dynamics belong to the later articles.