THREE DIMENSIONS

Interestingly, the Huygen's wavelet is different in three dimensions than in two dimensions. In three dimensions, there is no Hankel tail. In three dimensions we have the definition x²+y²=r² and from t²v²=x²+y²+z² we find on a plane of constant z, that the equation for a circle expanding with time is $r=\sqrt{t^2v^2-z^2}$.Between time t and $t+\Delta t$ is a ring with an area $2\pi r \Delta r$.Taking the signal amplitude in the ring to be $1/\Delta t$,analogous to equation (8) the amplitude at time t is

$$\begin{eqnarray} A(t) &=& {\rm step}(t-z/v) \ 2\pi r \ {\Delta r \over \Delta ... ...t}\ \sqrt{t^2v^2-z^2} \\ A(t) &=& {\rm step}(t-z/v) \ 2\pi v^2 \ t\end{eqnarray}$$ (9) (10) (11)

As before, in seismology we are interested in the high frequency behavior so the scaling t in equation (11) is not nearly so important as is the step function. By equation (1) the step function causes the spectrum to decay as $\omega^{-1}$.Our original erroneous assumption that Huygen's hyperbola of revolution should carry a positive impulse leads to the contradiction that an impulsive plane wave decomposed into Huygen's sources and added together again does not preserve the constant spectrum of the original impulsive waveform. We can get the missing $\omega$ back into the spectrum by having the hyperbola of revolution carry a time-derivative filter instead of an impulse. Thus in three dimensions, a plane wave can be regarded, approximately, as the superposition of many hyperboloidal responses, each carrying a d/dt waveform.