N Element Array
d
d
θ
v1  Mp1  t   Mp max cos  k  0   t 
v 2  Mp 2  t   Mp max cos  k  x   t 
v3  Mp3  t   Mp max cos  k  2x   t 
v 4  Mp 4  t   Mp max cos  k  3x   t 
v N  Mp N  t   Mp max cos  k  (N  1)x   t 

output   v1  v 2   Mp max cos  t   cos    t   cos  2  t   cos  3  t  
2
where   kx  kd sin 

 cos  (N  1)  t  
2
Phasor Addition
δ=kdsinθ


cos(k  0  t)  cos(k  d sin   t) 

v TOT  Mp 0 

cos(k

2d
sin



t)

...cos(k

(n

1)d
sin



t)








cos(t)  cos(t  )  cos(t  2)  ...
v TOT  v 0 

cos(t  (n  1))

v TOT  A cos(t  )
nδ
nδ/2
G
M
δ/2
δ
A
a
B
V()
 n 
 R sin   Where R is distance AP
2
 2 


sin
n

2
 V()  nV0 

 n sin 

2
  nd

 sin   sin   

 b()   
 n sin  d sin   



 

2
Directivity Index for an
n-Element Array








n

DI  10 log 
 2d  
n    sin 


n 1 
2



1  
2d
 n 1




Linear Array
2
2
  nd
    L
    L

sin
sin

sin
sin

sin
sin

  
  

  








 
 
b()  
 n sin  d sin     n d sin    L sin  

 

 



 
 
Nulls:
Lsin
 n, n  1, 2,3,...

DI  10 log 
1
2
 b   cos d
0
 2L 
 10 log 

  
2
Piston Array
  D sin   

 2J1 



b    
 D sin  




D
0.6
J1(Dsin/)
2
Bessel Function
0.4
0.2
5
10
15
20
-0.2
Dsin/
J1(Dsin/) has zero crossings (nulls) at Dsin/ = 3.83, 7.02, 10.17, 13.32, 16.47, ....
J1(Dsin/) has extremes (side lobes) at Dsin/ = 1.84, 5.33, 8.54, 11.71, 14.86, ....
defining parameters
beam pattern function
b() =
directivity index
DI
null angles
b() = 0
null
side lobes
b()=1
max
half power angles
b()=0.5
hp
BW=2hp
(only for beam about array axis)
2-element array
continuous line array
circular piston
element separation distance – d
array length – L
array diameter - D
 L

 sin  sin   

 
 L

sin  




 d

cos 2  sin  
 




2
10 log 
  sin 2d

1  
  2d
 


sin   m 

2d
m  1, 3, 5, ...
sin   m

d
m  0, 1, 2, 3 ....
n
4d
n  1,3,5, 7,...
sin  hp 







 
 
10 log
2L

2
for L  
sin   m 
 D 
10 log 

  
sin    z 

2
2
for D  

D
z  1.22, 2.23, 3.24, 4.24, ...
L
m  1, 2, 3, ...
 D

roots of J 1 
sin    0
 

 L sin    L sin  
tan 


     

sin   y 
L
where y  1.43, 2.46, 3.47, 4.48, ...
sin  hp  0.442

 D

 2 J 1   sin   




D

sin  





L
sin   w

D
where w  1.64, 2.68, 3.70, ....
sin  hp  0.51

D
Beam width of a piston array
1
b
2
  D sin   

 2J1 


   0.5
b    
 D sin  



0.8
0.6
0.4
0.2
2
D sin 
 1.6

6
4
sin  
D sin 

1.6 

 .51
 D
D
8