Superposition. Fourier Series Constructive Interference of a pulse

Superposition

Fourier Series

Constructive Interference of a pulse

Destructive Interference of a pulse

Constructive Interference of Harmonic Waves

Destructive Interference of Harmonic Waves

2 Dimensional Example

Single Hydrophone

Display1

tpMtv

“omni-directional”

hydrophoneprocessor

2output v 2

2 Mp tvPowerR R

Two Hydrophones

Beam Former

1

2 21 2output v v

hydrophones processor

Display

2 21 2output v v Why not ???

tpMtv

Incident Wave

Beam Former

1

2xd

21 2output v v

Display

1 1 max

2 2 max

221 2 max

v Mp t Mp cos k 0 t

v Mp t Mp cos k x t

output v v Mp cos t cos t

where k x kdsin

Identities

cos cos 2cos sin2 2

2 2sin cos 1

cos cos cos sin sin

2 1 1cos cos 22 2

ie cos i sin

1cos cos cos cos2

Power Output from the Processor 2

2max coscos ttR

MpP

cos1

cos2cos21

21

cos2cos212coscos

coscos2coscos

2max

2max

222

max

222

max

RMpP

tR

MpP

tttR

MpP

ttttR

MpP

R

MpP2

max20

k x kdsin

Beam Pattern Function

2max

2max

Mp1 cosP Rb

P 0 Mp1 cos 0

R1 cos kdsin

b2

sincos

or 2sincos

2

2

db

kdb

21 cos 2 cos2

Trig identity

k x kdsin

Example = .5 d

null

max

BW

Beam Pattern Function (/d = 0.5)

array elements

=0

Maximum Power Directions

dn

dn

nd

d

db

1max

max

max

max

max2max

sin

sin

.0,1,2,3,..n wheresin

1sin

cos

sincos1

null

max

BW

Beam Pattern Function (/d = 0.5)

array elements

=0

Null Angles

dndn

nd

d

db

null

null

null

nullnull

2sin

2sin

.1,3,5,7,..n where2

sin

0sin

cos

sincos0

1

null

2

null

max

BW

Beam Pattern Function (/d = 0.5)

array elements

=0

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.

null

max

BW

Beam Pattern Function (/d = 0.5)

array elements

=0

3-d Beam Pattern

Effect of Increasing Frequency Frequency = 750 Hz Frequency = 1500 Hz

Frequency = 3000 Hz Frequency = 6000 Hz