A MATHEMATICAL THEORY OF LINEAR ARRAYS 81. 



Radiation Intensity and Field Strength 



Consider a linear array of n equispaced nondirective sources (Fig. 1). 

 Apart from the inverse distance factor, the instantaneous field strength of 

 the array in the direction making an angle 6 with the line of sources may be 

 expressed as follows 



V^- = ^p cosicoi + ??o) + .li cos{ojt + ;A + ^i) + A2 cos{wt -\- 2yp + di) 



+ •••-[- .■I„_2 cosiwt + n-2\P -\- t?„_2) + cos{(jit + n-\ \}/), (1) 



yp ^ ^( cos 6 - -&, |8 = ^. 



A 



Fig. 1 — A linear array of equispaced non-directive sources. If two sources are of equal 

 intensity and in phase, their fields at a distant point are substantially equal in 

 intensity but differ in phase by /3^ cos d where ( cos 6 is the projection of the distance 

 between the sources upon the particular spatial direction under consideration. If 

 the sources are unequal, an allowance must be made for the relative field inten- 

 sities in proportion to magnitudes of the sources and the phases must be adjusted 

 for the phase difference between the sources. 



In this equation: Aq , Ax , • • • An-i = 1 are the relative amplitudes of the 

 elements of the array; ?? is a progressive phase delay, from left to right, be- 

 tween the successive elements of the array; t?i , t?2 , • • • «?»-2 , ^n-\ = repre- 

 sent the phase deviations from the above progressive phase delay; ^ = lir/X 

 is the phase constant, where X is the wavelength. The radiation intensity, 

 that is the power radiated per unit solid angle, is proportional to the square 

 of the amplitude of ^/^^ . 



Forming another expression similar to (1) but with sines in the place of 

 cosines, multiplying the result by i = \/— 1 and adding it to (1), we have 



V^i = [Aoe''' 4- Aie'"^"-''' + Aii'"^^'"' + • • • 



(2) 



+ An-J''-^'^^''''-' + e""'*] e"'. 



The true instantaneous value of the field strength is the real part of (2). 



