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Near-field and far-field

Eq. 4 decomposes the wavefield into a near-field term decaying with r-2 and a far-field term decaying with r-1:

 \begin{displaymath}u_r=u_n+u_f, \qquad u_n=\frac{1}{r^2} \, f(t-\frac{r}{c}), \qquad u_f= \frac{1}{rc} \,f'(t-\frac{r}{c})
\end{displaymath} (5)

un and uf determine each other:

 \begin{displaymath}u_f=\frac{r}{c} \,\dot{u}_n,\qquad u_n=\frac{c}{r} \, \int_{-\infty}^t{u_f}(t') dt'
\end{displaymath} (6)

In case of a harmonic time dependence $u_n=u_0 \, e^{j \omega t}$ we have

\begin{displaymath}u_f=\frac{r}{c} \, j \omega u_n
\end{displaymath} (7)


\begin{displaymath}\frac{\vert u_f\vert}{\vert u_n\vert}=\omega r/c = 2 \pi r / \lambda
\end{displaymath} (8)

where $\lambda$ is the wavelength. The far-field term thus dominates at radii greater than $1/2 \pi$ wavelengths. When the deformation is quasistatic (i.e. sufficiently slow), only the near-field term is relevant.

Erhard Wielandt