利用费马原理证明反射定律和透射定律.pdf

作业作业 1 利用费马原理证明反射定律和透射定律利用费马原理证明反射定律和透射定律 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.(在此处，我们指透射） 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 1S: Source R: Receiver C: Reflection point ES: Earth’s surface SR: Subsurface reflector H: Layer thickne ss V: Layer velocity D: Source-receiver offset X: Distance to refle tion point c Θ Θi: Incidence angle Θ Θ: Reflection angle Exercise 2S: Source R: Receiver C: Refraction point ES: Earth’s surface SR: Subsurface reflector Z1: Depth from source to reflector Z2: Depth from receiver to reflector V1: Velocity in Layer 1 V2: Velocity in Layer 2 D: Source-receiver offset X: Distance to refraction point Qi: Incidence angleΘt: Refraction angle 作业作业 2 计算反射系数计算反射系数 Objective To calculate the reflection coefficients between different lithologies and determine the effect of ignoring the density in calculating the reflection coefficient. IntroductionIntroduction The reflection and transmission coefficients (R, T) are defined as: R = (Z2 – Z1) / (Z2 + Z1) T = 1 – |R| = 2Z1 / (Z2 + Z1); where Z = V is the acoustic impedance, r and V are the density and velocity, respectivel y. Exercises Exercises Given the attached velocity-density model, use an Fig 1 : 1. Calculate R = R(V, ) at each interface. 2. Calculate T = T(V, ) at each interface. 3. Calculate R = R(V) at each interface using only velocities (i.e., drop r from the formula). 4. Calculate the absolute error between R found in steps 1 layer 2 has a velocity of 2000 m/s and a density of 1800 kg/m3. The reflection coefficient for a seismic wave incident on layer 2 from layer 1 is: (i) 0.1 (ii) -0.1 (iii) 0.25 (iv) -0.25 ( v) 0.5 (11) If we sample a seismic trace every 1 ms, the Nyquist frequency is: (i) 1000 Hz (ii) 100 Hz (iii) 500 Hz (iv) 1 Hz (v) 2000 Hz (12) If a seismic wave has a wavelength of 100 m and a frequency of 50 Hz, its velocity is: (i) 50 m/s (ii) 1000 m/s (iii) 2500 m/s (iv) 5000 m/s (v) none of the above (13) Young’s modulus in a uniaxial stress experiment is: (i) stress/strain (ii) strain/stress (iii) Vp/Vs (iv) 0.5 (v) ratio of axial contraction to lateral expansion. What is critical refraction ? If V1 = 1500 m/s and V2 = 2500 m/s, calculate the critical angle, ic. 。