HUYGEN'S SECONDARY SOURCE

Huygen's secondary source is the concept that a plane wave can be regarded as broken into many point sources. Each point makes a circular wave in (x,z)-space or equivalently a hyperbola in (x,t)-space. The hyperbolas shown in Figure 2 are all tangent to the horizontal line. Observe that the line density on the plot is great near the line but weaker away from it. We begin analysis from the assumption that each hyperbola carries a simple impulse function of time. We will add these hyperbolas together and discover that the sum is a plane wave, but the waveform is not an impulse function of time, but another function with a tail in the time domain and an $\omega^{-1/2}$ amplitude spectrum. From this we conclude that if the original waveform on the hyperbola had a $\omega^{1/2}$ amplitude spectrum (instead of being an impulse with a constant spectrum) then we would have an improved representation of the plane wave as a sum of points.