INTRODUCTION

Let’s examine a cross-section under a typical seismic refraction experiment. Here we have a layer of low P-wave velocity lying over an "infinite half-space" of higher P-wave velocity. As in Mohorovicic’s classic work the top layer could represent the crust and the deeper one, the mantle. The interface is the "Moho" or M discontinuity. On a smaller scale the top layer could be the vadose zone which is separated from the deeper phreatic zone by the water table. The seismic source and receivers are located on the ground surface.

At zero time the source is activated (earth quake occurs, ground struck by hammer) and body waves radiate in all directions. Some of the energy travels in the upper medium just at the ground surface. These waves form the "direct" ray or direct wave. The arrival time of the direct wave equals the distance traveled divided by the wave velocity in the upper layer.

Some of the energy travels obliquely downward, bends at the interface (Moho, water table, etc.), travels along the interface and then returns obliquely to the ground surface where it is picked up by our detectors (seismographs, geophones, etc.). Why does the ray bend (refract) at the interface and why does it reradiate back to the ground surface?

SNELL’S LAW

Willibrod Snell was a prominent Dutch scientist of the seventeenth century. He is famous for his laws of reflection and refraction of waves. His lesser known accomplishments include the first (?) modern triangulation survey and the first use of contours to represent elevations (actually water depths),

The law of reflection states that the angle of incidence of a ray is equal to the angle of reflection. This is a key concept supporting the seismic reflection method that we’ll study in a few weeks.

The law of refraction states that a ray will refract (bend, change direction) at an interface where the velocity changes. Stated another way we can say that the ratio of the sine of the angle of incidence to the velocity stays constant along a ray.

Physical derivations of Snell’s Laws can be done using the fundamentals of stress and strain coupled to Newton’s Second Law (force equals mass times acceleration). The derivations are beyond the scope of this course.

Using elementary calculus we can show that Snell’s Laws are consistent with Fermat’s Principle that the ray path from one place to another is always an extreme time path. In other words the time is always less (or greater) than any other path connecting the same two points.

EFFECT OF ANGLE OF INCIDENCE

There are four possible situations when a plane wave impinges on an interface with higher velocity material. For the moment we’ll only consider P-waves or S-waves in both media, not the more general case where both may occur together.

In the first case the angle of incidence is zero; that is, the incident ray is normal to the interface and the incident wavefront is parallel to it. In this case some of the wave energy is reflected and some is transmitted (refracted). Except for the 180 degree turn of the reflected wave the directions of travel are not changed.

The second case is the most general. Here the incident ray approaches the interface obliquely. Some of the energy is reflected according the law of reflection. The rest continues into the faster medium at an angle determined by the law of refraction. The ray "bends away" from the normal to the interface.

For today’s class the most important case is the third. Here we have a reflected ray as before but the refracted ray travels along the interface between the two media. In other words the angle of refraction is ninety degrees, the biggest value it can have. This refracted ray travels at the velocity of the higher velocity medium. The sine of the incident angle ("critical angle") equals the ratio of the lower velocity to the higher. Again the refracted ray bends away from the normal.

The fourth and final case is when the angle of incidence exceeds the critical angle. In this case all the energy goes into the reflected wave and there is no refracted one.

HEAD WAVE

For a ray incident on the interface at the critical angle the refracted ray travels along the interface at the velocity of the faster medium. The wavefronts are normal to the interface.

These wavefronts act as traveling seismic sources and reradiate energy to form a "head wave" that obliquely travels up to the ground surface where geophones can detect their passage. The head waves travel at the velocity of the upper layer material and in a direction equal to the critical angle. How does this happen?

According to Huygen’s Principle any point on a wavefront acts as a source of waves. Using a simple geometrical construction we can draw a sequence of wavefronts and thus determine where a wave will go. Your physics book will give details. When we apply Huygen’s Principle to the refracted wave going along the interface we see that there must be a head wave ascending to the ground surface at the critical angle.

A simple analogy is the V-shaped shock wave produced by a jet airplane flying faster than the speed of sound. The plane is analogous to the refracted wavefront racing along the interface. The shock wave traveling at the speed of sound is analogous to the head wave traveling at the velocity in the upper layer. Arrival of the head wave is analogous to a sonic boom.

In the next class we’ll examine the same seismic case in detail and learn how we can determine the two wave velocities and also the thickness of the upper layer using the direct wave and the head wave.