For wave pulses we generally find the product Δx Δk to be on the order of one or greater.  If we want to build a wave pulse by superimposing harmonic waves, we need waves with many different wave numbers k = 2π/λ or wavelength λ.  The narrower we make the width of the pulse, the greater is the range of wavelength of harmonic waves needed to synthesize the pulse.  A wave pulse does not have a well defined wavelength, but is made up from harmonic waves with a range of wave numbers or wavelengths.

Instead of looking at a wave at t = 0 as a function of x, we can also look at it at x = 0 as a function of t.  A harmonic wave then is of the form

y(t) = Asin(ωt + φ),

and wave pulses can be viewed as the superposition of many sine and cosine  waves with different angular frequencies ω.
If we replace x by t and k by ω, everything we learned above about Fourier's theorem and the Fourier transform applies to periodic waves and wave pulsed y(t).  For a wave pulse y(t) the pulse width Δt has units of s.  The width of the Fourier transform Δω  is measured  in units of 1/s.  For wave pulses we generally find the product Δt Δω to be on the order of one or greater.  If we want to build a wave pulse by superimposing harmonic waves, we need waves with many different angular frequencies ω or frequencies f.  The narrower we make the width of the pulse Δt, the greater is the range of frequencies of harmonic waves needed to synthesize the pulse.  A wave pulse does not have a well defined frequency, but is made up from harmonic waves with a range of frequencies.