1. Superposition

2. Fourier Series

3. Constructive Interference of a pulse

4. Destructive Interference of a pulse

5. Constructive Interference of Harmonic Waves

6. Destructive Interference of Harmonic Waves

7. 2 Dimensional Example

8. Single Hydrophone Display 1 “omni-directional” hydrophone processor

9. Two Hydrophones Beam Former 1 2 hydrophonesprocessor Display Why not ???

10. Incident Wave Beam Former 1 2 xx d  Display

11. Identities

12. Power Output from the Processor

13. Beam Pattern Function Trig identity

14. Example =.5 d  =0 

15. Maximum Power Directions  =0 

16. Null Angles  =0 

17. Beam Width The beamwidth of a beam is the angular displacement between the angles where the beam pattern function, b(  ), is greater than 0.5. 3 dB down points The beamwidth is important because it is proportional to the bearing accuracy of the specific beam.  =0 

18. 3-d Beam Pattern

19. Effect of Increasing Frequency