Sinusoidal waves

wave traveling in the positive x-direction:  y(x,t) = Asin(kx - ωt + φ)  or   y(x,t) = Acos(kx - ωt + φ)
wave traveling in the negative x-direction:  y(x,t) = Asin(kx + ωt + φ)  or   y(x,t) = Acos(kx + ωt + φ)

A = amplitude, k = wave number = 2π/λ,  ω = angular frequency = 2πf,  φ = phase constant
v = λf = ω/k = speed, E = energy carried = proportional to A²

Wave packets

Wave packets can be constructed by adding sinusoidal waves with different wave numbers or wavelengths.  Narrower wave packets require a wider range of wave numbers.  For wave pulses we generally find the product Δx Δk and Δt Δω to be on the order of one or greater.