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General Properties of Waves Reflection Seismology Geol 4068 Questions and answers to first lecture homework September 8, 2005 Homework 1-due September 8, 2005 at 9.30 a.m. Q. 1 What is the P-wave velocity of the following earth materials measured near the surface of the earth: basalt, granite, peridotite, gabbro and iron Answer. +/- 10% basalt-- 5 km/s granite --- 6.5 km/s peridotite -- 8.1 km/s gabbro---7.2 km/s SOURCES must be referenced!!! Q. 2 If following a surface explosion, a “ray” of sound enters the blue synform, what will the angle of refraction at point A for the following interface geometry? Apply Snell’s Law with the values provided. Explain your work clearly and succinctly. Hint: simplify the geometry of the geology We can assume that the explosion is at the center of a circle and that the synform describes, approximately the shape of a circle. With this assumption, we start by noting that lines of radius intersect the circumference of a circle at right angles. If so then the incident angle of refraction is 90 degrees. At this normal incidence, reflected rays will return along the same path as the incident ray. Not all the energy of the ray is is returned entirely to the place where of the explosion took place. Some energy may be transmitted through the boundary. Q. 2 Q. 3 If the lowest frequency your body size can register is about 8 times your greatest dimension, does this value change whether you are in water or in air? What are these values? Will your body “feel” or register these waves? Or, is your body too small? Assume your are 2 m tall ! Assume sound travels at (I) 1500 m/s in water and (II) 330 m/s in air. Assume you are 2 m high. Consider that the dominant frequency in your signal is (a) 10 Hz, and (b) 400 Hz , Let us start with the following equation: VP frequency ( f ) wavelength ( ) VP f (1) If we rearrange (1) we obtain the wavelength as a function of velocity and frequency we are better ready to answer the question: VP f Note that based on the information provided in the question you would have to grasp that if you are smaller than 8 times the dominant wavelength your body will not "register" or "feel" the wave. Or, in other words you will be too small to be "seen" by the wave. (a) STEP 1: Substitute values for part a into (1) 1500 10 150 m 1 m s s The wavelength is 150m. BUT, you are looking to see if you will be able to feel the wave through your body. You would then have to be no shorter than 1/8 of 150 m, i.e. ~18.75 m We perform similar calculations for a velocity of 330 m/s, 330 m/s = 10 Hz x 33 m, and 1/8 of 33 is about 4 m which is taller than the tallest human and therefore undetectable. (b) Similarly, if the frequency is 400 Hz, the minimum height you would have to be is inversely proportional to the frequency. From the calculations you should obtain 1500 m/s = 400 Hz x 3.75 m, and 1/8 of 3.75 m divided by 8 is ~0.5 m, which is much smaller than anyone. We perform similar calculations for a velocity of 330 m/s, 330 m/s = 400 Hz x 0.825 m, and 1/8 of 0.825 is about 10 cm. It is best sometimes to graph the results and see what we can infer. I have done this in MATLAB. You could also do it by hand. Matlab code Q. 4 What is the critical angle between water and basalt? This is a typical scenario in oceanic spreading ridges. Assume a P-wave velocity of 1500 m/s First, let’s study Snell’s Law for this case at critical conditions, i.e., when the angle of refraction 90 degrees and Snell’s Law reduces to: sin(incident angle) / V in first medium = sin(90deg.) / V in second medium (1) sin(critical angle)/V in first medium = 1/V in second medium (2) sin(critical angle) = V in first medium/V in second medium critical angle = angle whose sine is: V in first medium/V in second medium (3) In other words, critical angle = arcsin(V1/V2) Note carefully, that your calculator may either return arcsin in radians or degrees. Always consider you units, Now, taking V1=1500 m/s and V2=5000 m/s we obtain that arcsin(1500/5000) = 17.45 degrees We can also plot the data out for the general case to understand the behavior better: Matlab code