Describing Progressive Waves

One function describes two different observations

Label each undisturbed string element by its equilibrium coordinate x. Its transverse displacement at time t is y(x,t), measured from equilibrium, not the distance travelled. Two independent inputs are needed because neighbouring elements need not have the same displacement at the same instant. Holding t fixed gives a spatial snapshot y versus x; holding x fixed gives the displacement history of one element, y versus t. A sinusoid can describe both, but one horizontal axis is distance and the other is time.

The sine varies between −1 and +1, so y varies between −a and +a. A crest has y = +a and a trough has y = −a. The phase θ = kx − ωt + φ identifies the stage of the local oscillation. At x = 0 and t = 0 it equals φ. Changing the coordinate origin or the instant called zero changes φ without changing the physical wave. But once initial conditions and origins are specified, φ cannot simply be dropped without changing those conditions.

At a fixed time t₀, the phase is kx plus a constant, so a photograph shows a sine curve. At a fixed location x₀, the phase is a constant minus ωt, so that particular element performs SHM. For example, with φ = 0, the element at the origin has y(0,t) = −a sin(ωt). The minus sign only sets the initial direction of its oscillation; it does not make its amplitude negative.

Combining sine and cosine terms

Expand sin(θ + φ) as sin θ cos φ + cos θ sin φ. Matching coefficients gives A = a cos φ and B = a sin φ. Squaring and adding removes φ because sin²φ + cos²φ = 1. This is why a combination of same-frequency, same-wave-number sine and cosine terms is still one harmonic travelling wave. If both A and B are zero there is no oscillation and its phase is undefined.

Wavelength is a full spatial repeat

At the same instant, neighbouring crests are separated by the smallest positive distance that increases phase by 2π. Write k(x + λ) − kx = 2π. This gives kλ = 2π. Points separated by λ have the same displacement and the same stage of motion. Merely sharing an instantaneous displacement is insufficient: two points on opposite sides of one crest can have the same height and different phases. A zero crossing can repeat after λ/2 with the opposite spatial slope.

Period is a full time repeat

Hold the element coordinate fixed and let one period T pass. Its phase changes by −ωT. A complete oscillation repeats when the magnitude of this phase change is 2π, so ωT = 2π. The frequency f, also written ν, counts complete oscillations per second and is 1/T. All elements in an ideal uniform harmonic progressive wave have the same amplitude and frequency. Their phases at one time differ by kΔx.

For a longitudinal wave, use s(x,t) for the signed particle displacement along the propagation direction. The same functional form s = a sin(kx − ωt + φ) applies. A graph of s versus x shows forward or backward displacement of air elements, not a literal up-and-down profile of a sound wave. Pressure and density waves accompany that displacement, but their peaks need not coincide with displacement peaks.

Following phase gives the propagation speed

Follow a crest, or any selected fixed phase, rather than a material element. At two nearby events on this moving feature, kx − ωt + φ = k(x + Δx) − ω(t + Δt) + φ. Subtracting the common terms gives kΔx − ωΔt = 0. Hence Δx/Δt = ω/k. In differential notation dx/dt is the instantaneous velocity of this selected phase point; no new physical assumption is hidden in taking the limit of a small interval.

For a sin(kx + ωt + φ), holding phase constant gives kΔx + ωΔt = 0 and dx/dt = −ω/k: it moves towards decreasing x. Multiplying the entire phase by −1, or converting a cosine to a sine, can change the appearance of an equation without changing its propagation direction. Determine the direction from the relative signs and a constant-phase calculation, not from the sign of the displacement.

Particle velocity is a different derivative. At a fixed x, the rate of change of y = a sin(kx − ωt + φ) is uᵧ = −aω cos(kx − ωt + φ). The factor −ω appears because the phase decreases at that rate. This velocity changes during each cycle and is transverse for a rope; its maximum magnitude is aω. Wave speed ω/k describes the advancing pattern along x. A particle can momentarily be at rest at a crest even while the crest itself continues moving.

A translating mathematical shape still needs a physical domain

A shape F(x − vt) translates to the right without changing form: when t increases by Δt, increasing x by vΔt keeps its argument fixed. F(x + vt) translates left. The function need not be sinusoidal. For a physical small-displacement wave, also check that its argument and output have proper units, that it is defined on the required domain, and that displacement and strain stay within the model. A formal solution can fail as a global physical disturbance.

For example, (x − vt)² is a translating parabola after a coefficient with suitable units is supplied. It is finite at every finite coordinate, but grows without bound over an infinite medium, so it is not a globally bounded small-amplitude disturbance. log[(x + vt)/x₀] requires x₀ to be a positive reference length and x + vt > 0; it is singular as that positive argument tends to zero. 1/(x + vt) has a pole at x + vt = 0. None is a globally bounded, regular small-amplitude wave on the entire real line. Restricted-domain descriptions need explicit boundaries and limits. Similarly, 2√(x − vt) is real only for x − vt ≥ 0 and has a singular slope at the edge; it is not an acceptable smooth wave over the whole infinite medium.

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