利用费马原理证明反射定律和透射定律.pdf

1 Objective To understand the importance of Fermat’s principle in deriving Snell’s law in the reflection and refraction cases. Introduction Fermat’s principle states that a wave will take that path which will make the traveltime stationary (i. e., maximum or minimum). Mathematically: dT / dX = 0, where T is the total traveltime along the wave path and X is the distance from the source to the point where the wave changes its direction (e.g., point of reflection or refraction). In most situations in the earth, the stationary path is the minimum-time path. In this exercise, we will use Fermat’s principle to derive Snell’s law in the following cases: 1. Reflection. 2. Refraction.( Exercises 1. Given Figure 1: a. Use Fermat’s principle to derive Snell’s law in terms of the angles of incidence and reflection. b. Find at what value of X (as a function of D) does Fermat’s principle hold? c. Verify that this value of X refers to a minimum-time path. d. Verify that the ray will always take a minimum-time path. 2. Given Figure 2: a. Use Fermat’s principle to derive Snell’s law in terms of the angles of incidence and refraction. b. Verify that the ray will always refer to a minimum-time path. Exercise 1 S: Source R: Receiver C: Reflection point ES: Earth’s surface SR: Subsurface reflector H: Layer thickness V: Layer velocity D: Source-receiver offset X: Distance to reflection point Θ: Incidence angle i Θ: Reflection angle Exercise 2 Θ 2 Objective To calculate the reflection coefficients between different lithologies and determine the effect of ignoring the density in calculating the reflection coefficient.