** Degrees of Freedom**

In the previous case, the target object was known to be on the axis of the speaker and could be located by just calculating the distance (1 degree of freedom). In most SONAR applications the direction to the object will be unknown (3 degrees of freedom). For this lab, the target object will be at approximately the same height as the speaker, and therefore only two spatial coordinates must be solved (2 degrees of freedom). Since we can only measure time delay in our signals with our system, two different reflection measurements must be made to solve for the target coordinates. This will be accomplished by moving the microphone to known directions from the speaker axis (±30⁰).

6. In the figure below, draw the path of the sound (speaker-object-microphone) on the figure above, for both microphones.

7. If the object is located at and the speaker is located at origin, what is the equation for the distance that the sound traveled from the speaker to the object?

8. If the object is located at and the microphone at position A is located at , what is the equation for the distance that the sound traveled from the object to position A?

9. What is the total distance the sound traveled from the origin to the object and then to the microphone at position A?

10. Similarly, what is the total distance the sound traveled from the origin to the object and then to the microphone at position B, ?

“SONAR2DOF.mat” contains a sample set of 2 DOF SONAR data with the input signal to the speaker (*input_V*), the baseline signal at microphone position A without the object (*baselineA_V*), the target signal at microphone position A with the object *(targetA_V)*, the baseline signal at microphone position B without the object (*baselineB_V*), the target signal at microphone position B with the object *(targetB_V)*, and the time for each signal (*time_s*). Using analysis of the signals similar to the 1 DOF case, this data can to be used to determine the distances traveled by the sound to the microphone in positions A and B. For your calculations assume that the measured speed of sound is 344 m/s.

11. What are the distances to the microphone in positions A and B using the respective baseline data in **meters**? *(HINT: Both are between 0.25 and 0.35 meters.)*

12. What are the coordinates of the microphone at position A and B in **meters** if and?

13. What is the total distance that the reflective sound traveled from the origin to the object and then to the microphone at positions A and B in meters? *(HINT: Both are between 1.4 and 1.7 meters.)*

14. Using the calculated microphone coordinates and distances from problems 12 and 13, the only unknowns now in the distance equations from problems 9 and 10 are the object coordinates, . What are the coordinates of the target object in **meters**? *(HINT: Try using the vpasolve() function in MATLAB to numerically solve the problem. Use the online help resources if you are unfamiliar with the function. There are multiple numerical solutions to this problem, but only one physical one. You may need to supply a reasonable guess to the solver and should verify the answer it returns makes physical sense.)*

15. If the actual speed of sound was 1 m/s faster than the measured speed of sound, what is the error in the calculated coordinates in **meters**?