Wave functions, speed & dispersion

Take any smooth function f and write y(x, t) = f(x − vt). At t = 0 you see the shape f(x); at a later t the identical shape sits a distance vt further along +x, because the value at x₁ + vt equals the old value at x₁. Nothing about f matters — the travelling is in the argument. The sinusoid is this with f a sine: factor k out of A sin(kx − ωt) to get A sin[k(x − (ω/k)t)], so v = ω/k. You can also ride a crest directly. Hold the phase φ = kx − ωt constant and differentiate: k(dx/dt) − ω = 0, so dx/dt = ω/k. A plus sign, A sin(kx + ωt), gives dx/dt = −ω/k, the same shape running the other way. Any f(x − vt) also satisfies ∂²y/∂x² = (1/v²)(∂²y/∂t²), the wave equation, and v is the only constant in it.

∂y/∂x at fixed t is the slope of the snapshot, a pure number. ∂y/∂t at fixed x is the transverse velocity of one piece of the cord, in m s⁻¹, at right angles to the direction the wave is going. For y = A sin(kx − ωt): ∂y/∂x = kA cos(kx − ωt) and ∂y/∂t = −ωA cos(kx − ωt). Divide one by the other and the cosines cancel: ∂y/∂t = −(ω/k)(∂y/∂x) = −vₚ (∂y/∂x). The medium's speed is the wave's speed times the local slope, so the two are linked but never equal. On a cord carrying 180 Hz at amplitude 2.5 mm, ω = 1.13 × 10³ rad s⁻¹ and the fastest any point ever moves is ωA = 2.83 m s⁻¹, while the wave itself runs at 1.20 × 10² m s⁻¹. The steepest slope is kA = 0.0236, and 2.83 ÷ 120 is that same 0.0236.

On a stretched cord v = √(F_T/μ): a restoring tension over an inertia per unit length. Sound in a fluid gives √(B/ρ), the same structure of a stiffness over an inertia, and neither expression contains the frequency. Light in glass gives c/n, that structure once more, except that n itself drifts a little with wavelength — the section after next takes that up. Meanwhile the oscillator at the end of the cord sets f, and the cord has no say, because the point where the two meet has to move as one. That is also why f is continuous across a junction between media. What is left to adjust is λ = v/f. Numbers: F_T = 92.0 N on a cord of μ = 6.40 g m⁻¹ gives v = √(92.0 ÷ 6.40 × 10⁻³) = 1.20 × 10² m s⁻¹, and at 180 Hz, λ = 0.666 m. Splice on a cord four times as heavy per metre at the same tension: v halves to 59.9 m s⁻¹, f stays 180 Hz, and λ halves to 0.333 m.

Plot ω against k for the medium. A straight line through the origin, ω = vk, means every wavelength moves at the same speed: the medium is non-dispersive and a pulse keeps its shape indefinitely. An ideal cord and light in vacuum behave this way. Bend that line and the phase speed ω/k — the slope of the chord from the origin out to the point (k, ω) — changes from one wavelength to the next. Deep water obeys ω = √(gk), so vₚ = √(g/k) = √(gλ/2π): a 180 m swell runs at 16.8 m s⁻¹ and a 120 m swell at only 13.7 m s⁻¹, the longer wave being the faster one. Glass disperses in the same sense, and far more weakly. Borosilicate crown has n = 1.5224 at 486 nm and 1.5143 at 656 nm, so blue light travels at 1.969 × 10⁸ m s⁻¹ and red, the longer wavelength, at 1.980 × 10⁸ m s⁻¹ — a gap of half a percent where the two swells differ by about a fifth. Splitting those two speeds apart is the whole job of a prism.

Add two equal-amplitude sinusoids whose wavenumbers differ a little, k₀ ± Δk/2, with frequencies ω₀ ± Δω/2. The sum factorises: y = 2A cos[(Δk/2)(x − (Δω/Δk)t)] sin[k₀(x − (ω₀/k₀)t)]. That is a fast carrier inside a slow envelope. The carrier's crests move at vₚ = ω₀/k₀; the envelope moves at Δω/Δk, which for a narrow spread of wavenumbers becomes v_g = dω/dk. Energy and information ride the envelope, so v_g is the speed at which a pulse actually arrives. Deep water once more: ω = √g √k, so v_g = dω/dk = ½√(g/k) = ½ vₚ. You can watch this from a pier. Crests appear at the back of a group of swell, run forward through it and vanish at its front, because over the ground each crest travels at twice the speed of the group it is crossing. A 180 m group advances at 8.38 m s⁻¹ while its crests move at 16.8 m s⁻¹.

Everything above is linear and small-amplitude: displacements small enough that the restoring force stays proportional to them, so two waves pass through each other without altering each other's speed. Three failures are worth naming. In a dispersive medium a pulse has no single speed at all — its sinusoidal components separate, and a sonar chirp or a fibre-optic pulse arrives longer than it left; that spreading, not absorption, is what caps the bit rate of a long fibre. At large amplitude the speed depends on amplitude: a shallow-water wave of height h on depth d travels at roughly √(g(d + h)), so the crest overtakes the trough ahead of it and the front steepens into a breaker. And where a medium absorbs strongly, dω/dk can exceed c or turn negative, at which point it no longer means the speed of a signal.