with a higher velocity – a common situation – is refracted away from the normal, according to Snell’s Law (Eq. 4.5). Consider rays progressively more oblique to the interface (Fig. 6.1): There must come an angle, the critical angle, ic, where the refracted angle is exactly 90°; that is, the refracted ray travels along the interface (if the ray is yet more oblique, all the seismic energy is reflected, called total internal reflection). The value of ic depends on the ratio of
velocities either side of the interface:
Eq. 6.1 What does a ray travelling along the interface mean: Is it in layer 1 or 2? This is best answered by considering wave fronts. As rays and wave fronts always intersect at right angles, a ray along the interface means waves perpendicular to the interface (Fig. 6.2a). A wave is due to oscillations of the rock, and as the layers are in contact, the rocks just above the interface must oscillate too. Therefore, as a wave front travels along just below the interface a disturbance matches it just above the interface. This in turn propagates waves – and therefore rays – up to the surface at the critical angle (Fig. 6.2b).
To understand why this is so, we use the concept of Huygens’s wavelets. sini v sin v v v c= ° = 1 2 1 2 90 65 S C ic v1 v2 > v1 = sin ic v1 sin 90° v2 i1 i2
Figure 6.1 Critical angle of refraction.
(b) (a) E F G H F ' head w aves wavelets C ic direct ray v2
>
v1 v1 wa ve fro nts ic D E refracted r ay S6.1.1 Huygens’s wavelets
Huygens (a 17th-century scientist) realised that when a particle of a material oscillates it can be thought of as a tiny source of waves in its own right. Thus, every point on a single wavefront acts as a small source, generating waves, or wavelets as they are called. In Figure 6.3 the left-hand line represents a planar wave front travelling to the right at some instant. Wavelets are generated from all points along its length, and a few are shown. The solid half-circles are the crests of the wavelets after they have travelled for one wavelength (which is the same as that of the wave). At C1, C2, C3, . . . these overlapping crests
add, called reinforcement, so that they form a new crest. The succeeding troughs will have advanced half as far, shown by the dashed half-circles. At each of T1, T2, . . . there are two troughs and these will
add, giving a trough between the two crests. But at Z1, Z2, . . . there is a crest and a trough, and these
cancel perfectly. As there will be just as many troughs as crests along the left-hand line – assuming it is indefinitely long – then there will be perfect can- cellation all along the line. Therefore, the crest has advanced from left to right. This is just what we
have been using since waves were introduced in Sec- tion 4.1, which is why wavelets have not been intro- duced before. However, wavelets are helpful in some special cases, such as critical refraction and diffrac- tion, the latter being considered first.
Diffraction. If waves pass through a gap, such as water waves entering a harbour mouth, it might be expected that once inside they would continue with the width of the gap, with a sharp-edged shadow to either side. But if this were so, at the crest ends there would be stationary particles next to ones moving with the full amplitude of the wave, which is not feasible. Con- sidering wavelets originating in the gap shows that there is no such abrupt cutoff (Fig. 6.4). At C1, C2, . . .
the crests will add as before, so the wave will continue as expected. But at Z1there is no trough to cancel the crest and so there is a crest there, though weaker because it is due to only one wavelet. The resulting wave crest follows the envelope of the curves.
This bending of waves into places that would be a shadow according to ray theory is called diffraction. It helps explain how we can hear around corners (there is often reflection as well). It will be met again in Sec- tion 6.10.1, and in some later chapters. Next, we return to the critically refracted wave of Figure 6.2.
wavelength
original wave crest
T1 T2 C1 C2 C3 Z1 Z2 wavelet crest trough new position of wave crest
Figure 6.3 Huygens’s wavelets.
Z1 Z2 C1 C2 C3 C4 C5 C6 C7 new position of wave crest
6.1.2 Head waves
As a wave travels below the interface it generates wavelets (Fig. 6.2b). By the time a wavelet in the lower layer has travelled from E to F, a wavelet in the upper layer – travelling at velocity v1– has gone
only a distance EF′. If we draw the wavelets pro- duced by successive waves below the interface, at E, F, . . . , and add up their effects, the result is waves travelling up to the right (Fig. 6.2b). These are called head waves, and their angle ihead (which equals ⬔EFF′) depends on the ratio of EF′ to EF:
Eq. 6.2
This is the ratio of the velocities, and so is the value of the critical angle (Eq. 6.1): Therefore, head rays leave the interface at the critical angle.
The seismic refraction method depends upon
timing the arrivals of head waves at receivers on the surface; the corresponding rays are usually called refracted rays.