# 2005 HiMCM A:Modeling Ocean Bottom Topography

Background:

A marine survey ship maps ocean depth by using sonar to reflect a sound pulse off the ocean floor. Figure A shows the ship’s location at B on the surface of the ocean. The sonar apparatus aboard the ship is capable of emitting sound pulses in an arc measuring from 2 to 30 degrees. In two dimensions this arc is shown within Figure A by$\angle ABC$, and the emanating sound pulses are displayed by the dashed lines and the solid lines $BA$ and $BC$.

When a sonar sound pulse hits the bottom of the ocean, the pulse is reflected off the ocean bottom the same way a billiard ball is reflected off a pool table; that is, the angle of incidence $\alpha$equals the angle of reflection $\beta$ as illustrated in Figure B. Since the ship is moving when the sound pulse is emitted, it will pick up a reflected sound pulse at location F in this picture. The actual depth of the water is the length of BD in Figure A

Figure A

Figure B

Useful Information:

Oceanography vessels usually travel at a speed of $2m/s$ while Navy vessels travel at $20m/s$. The sonar apparatus aboard these ships is capable of emitting sound pulses in an arc measuring from 2 to 30 degrees. The typical speed at which a sonar sound pulse is emitted is $1500m/s$.

Devise a model for mapping the topography of the ocean bottom. Write a letter to the science editor of your local paper summarizing your findings.