CMSC AMSC 460 Fall 2007 Homework 7 Due Tuesday December 11 before class begins 25 points The Problem we discussed on the first day of class Kahaner Moler and Nash Problem 8 15 The speed of sound in ocean water depends on pressure temperature salinity These vary with depth in complicated ways Let z be the depth in feet under the ocean surface The z axis points downward Someone has measured these values for c z the speed of sound in ft sec at depth z c 0 c 1500 c 3000 c 5000 c 8000 c 11000 5042 c 500 4887 c 2000 4865 c 3500 4875 c 6000 4918 c 9000 4973 c 12000 4995 c 1000 4868 c 2500 4869 c 4000 4887 c 7000 4933 c 10000 4991 4948 4863 4875 4905 4949 Since the speed of sound varies with depth sound rays travel in curved paths It s like the refraction of light when you look into a fishbowl We make a mathematical model for the paths using Snell s Law Suppose we broadcast a sound from a point x 0 z z0 and let z x be the depth of the ray when it is a horizontal distance x from the source Let x denote the angle between the tangent to z x and the horizontal axis dz tan x dx 1 Snell s Law says cos a c z where a is a constant Putting all of this information together by differentiating these equations we can obtain the model d2 z c0 z dx2 a2 c z 3 2 cos 0 2 a c z0 z 0 z0 dz 0 tan 0 dx a 10 Use ode45 to trace z for the ray beginning at z0 2000 ft and 0 5 4o Use a spline to evaluate c z and c0 z wherever they are needed Follow the ray for 24 nautical miles 1 nautical mile 6076 ft and plot your solution z x x 0 24mi Remember that Matlab trig functions use radians not degrees Your value for z at 24mi should be close to 3000 b 5 Now suppose that a sound source at a depth of 2000 ft transmits to a receiver x 24 miles away at a depth of 3000 ft Write a Matlab function depth theta that traces the ray with initial angle theta and z0 2000 ft and returns the value z x 3000 Print a table of values of this function for theta in the range 10 to 10 degrees c 10 Use fzero with starting values obtained from part b to find 4 rays with angles between 10o and 10o that pass through the receiver 2 What to submit Hand in a listing of your main program and any Matlab functions you wrote Each of these should have some documentation name date purpose description of parameters for any functions a few words about how it works The output your program produces designed to be easy to read and understand the plot for part a well labeled by Matlab 3