EARTH 520
Problem Set: “Snell’s Law” and “Constructing a P Wave Travel Time Curve”

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Part 1: Snell’s Law

1. Calculate the angle of refraction for a ray of light passing from air to water with an incident angle of 45°. Assume the index of refraction of water (nwater)is 1.33 and nair is 1.

2. Calculate the angle of refraction for a ray of light passing from water to air with an incident angle of 45°.

3. Suppose you have a ray of light that passes through three layers: air - water - air. The angle of incidence at the first air - water boundary is 30°. (This angle is marked in red in the diagram below. Calculate the angle of refraction at the first boundary (orange angle in the diagram below) and calculate both the angle of incidence (green angle) and angle of refraction (blue angle) at the second (water - air) boundary. See the diagram below, which is not drawn to scale!

4. Now suppose you have a ray of light that passes through the three following layers: air - water - crown glass (ncrown glass = 1.52). The angle of incidence of the light ray at the air-water boundary is 30°. Calculate the speed of light in crown glass.

5. Now suppose you have a ray of light that passes from air into a material through which light travels at a velocity of 1.5 x 108 m/s. If the angle of incidence is 45°, calculate the angle of refraction.

6. Which words make this a true sentence (more than one possibility is correct)?

"When a ray of light passes from a fast/slow material to a fast/slow material, the ray is bent towards/away from the normal."

7. Now let's consider the raypath taken by a seismic wave instead of light. For the purposes of this calculation, we'll pretend the Earth is flat. (On the next page in this lesson we'll see what happens for a spherical Earth). An earthquake happens at the surface of a series of layers as pictured below. Consider a P wave that leaves the source along the raypath as shown in the cartoon and hits the boundary between the upper layer and the second layer with an angle of incidence of 30°. Given the transmitting velocities for a P wave in all the subsequent layers, sketch the path of the ray until it hits the bottom, and find all the angles of incidence and refraction along the way. The velocities in this problem are some typical mantle velocities (Dziewonski and Anderson, 1981). You can make your sketch using my java-based drawing tool that is linked from the web page with the problem set on it, or do it by hand or with the program of your choice. If you use my tool, insert your halfspace.jpg file here.

This is the end of Part 1 of this problem set. You'll want to read through the webpage "Raypaths Through the Earth" in the course content before plowing ahead with Part 2 of this problem set.

Part 2: Constructing a P Wave Travel Time Curve

The seismograms you need to look at for this activity are linked from the "Constructing a P Wave Travel Time Curve" course page.

For #s 1, 2, and 3 I find it easiest to make a table in which I write down the station name in column 1, The P wave arrival time at that station in column 2, the travel time in column 3 and the distance in degrees in column 4. Doing so will make it a lot easier to make your plots. I am leaving it up to you to construct such a table, or if you hate tables, to ignore this advice.

1. For each seismogram, pick the arrival time of the P wave. The P wave is the first impulsive arrival that rises appreciably higher than the background noise level.

2. Calculate the time it took the P wave to get to each station.

  1. Calculate the distance between each station and the event in degrees.

4. Make a plot of distance in degrees vs. P wave travel-time. Note that it is common practice to put distance on the x axis and travel time on the y axis because travel time is the quantity you are measuring. The distance is something you are calculating but it isn't the quantity you are measuring here (does that make sense?).

  1. Which station was closest and which station was farthest away? What were the distances between the earthquake and each of these two stations?
  2. This event was large enough that it was recorded by stations even farther away than the farthest station you worked with. Why didn't I make you pick P waves for farther away stations?
  3. Choose any station in this exercise and make the following two sketches. I'm not looking forskilled artwork here. I am looking for two things in each sketch: (1) a more or less accurate representation of the angular distance separating your chosen station and the event, and (2) a P-wave path that has about the right shape. To make your sketches, use the java-based drawing tools linked from the course web page, then insert your jpg file into your problem set worksheet. (or do it with your own program if you would rather).
  4. Draw the event-station path taken by the P wave for the case of a homogeneous mantle
  5. Draw the event-station path taken by the P wave for a mantle whose velocity increases smoothly with depth.
  1. Assume a constant mantle velocity of 11 km/sec. Draw the travel-time curve for this velocity on your plot using a different symbol / linestyle than your original line that you made for your own observations. There is a screencast hint regarding this problem on the course website where you downloaded this worksheet. Check it out.
  2. How does our assumed constant velocity of 11 km/sec compare to your actual observations?

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