Diffracted waves

The idea of the Fresnel zone is useful in understanding when the the waves diffracted from scattering grains add constructively and what their relative importance is. Figure (8) shows two scattering responses of a spherical grain to different frequencies of the arriving wave at a given time t. The velocity of the grain, v_g, is assumed to be higher than the velocity of the host medium (water), v_w. The left drawing represents the high-frequency response and shows that the transmitted plane wave (slightly curved in the proximity of the grain because of the diffraction effect) is far from the purely diffracted spherical wave centered on the grain. At a lower frequency (right drawing), the planar and spherical waves add constructively in the region around point $\bf A$. In the high-frequency case, the point of highest amplitude behind the grain on the axis of propagation is located on the diffracted spherical wave, although this is not a zone of constructive interference (point $\bf a$ on the drawing). The distance d_s from point $\bf S$ (the first point met by the planar wave incident to the grain) to point $\bf a$ is  

$$d_s = 2r + \left( t - \frac{2r}{v_g} \right) v_w ,$$ (4)

where r is the radius of the grain. In the low-frequency case, the point of highest amplitude behind the grain is not located exactly on the spherically diffracted wave but is closer to the center of the grain: the constructive interferences have moved the point of highest amplitude backward. As a consequence, the velocity appears higher in the high-frequency measurement than in the low-frequency measurement.

 

Freqvel
Figure 8 The scattering response of a spherical grain to different frequencies of the arriving wave.