Displacement relation of a progressive wave

A wave needs a displacement that depends on position and time together. For a harmonic transverse wave on a string moving along +x: y(x, t) = a sin(kx − ωt + φ).

A general form A sin(kx − ωt) + B cos(kx − ωt) is the same wave with a = √(A² + B²) and φ = tan⁻¹(B/A).

Changing the sign to y = a sin(kx + ωt + φ) makes the wave travel towards −x instead.

Amplitude a is the largest displacement of a particle from its mean position; it is taken as positive. A crest is a point of largest positive displacement, a trough a point of largest negative displacement.

The phase is the whole argument (kx − ωt + φ); it fixes the displacement at any x and t. φ is the initial phase angle, the phase at x = 0 and t = 0.

Wavelength λ is the least distance between two points in the same phase, for example crest to crest. The angular wave number (propagation constant) k = 2π/λ, in rad m⁻¹.

Every particle does SHM of angular frequency ω, so the period is T = 2π/ω and the frequency ν = 1/T = ω/2π.

A longitudinal wave has the same form, s(x, t) = a sin(kx − ωt + φ), where s is the particle's displacement along the direction of travel.

Worked example (NCERT): y = 0.005 sin(80.0x − 3.0t) in SI units. a = 0.005 m = 5 mm; k = 80.0 rad m⁻¹, so λ = 2π/80.0 = 7.85 cm; ω = 3.0 rad s⁻¹, so T = 2π/3.0 = 2.09 s and ν = 0.48 Hz.

At x = 30.0 cm and t = 20 s the phase is 80.0 × 0.3 − 3.0 × 20 = −36 rad, which is the same as −36 + 12π = 1.699 rad (about 97°); the displacement is 0.005 sin(1.699) ≈ 5 mm.