. The Bell System technical journal . ) COS (f+ {popn-l + i!l^»-2 + /2/n-3 + + pn-lpo) ■ Since / ^ sin ede = --- I $ J(/?,Jo P^ J vi z /(,« — 1; (59) we can write/ $ sin Bid = —. 2pl sin [n — l)<p , 2{pQpi + ^i/o) sin {n — 2)ip n — 1 + n — 2 + • • • + (pOpn-l + plp«-2 + • • • + pn-lpo)ip •Pi (60) For an end-on array with nulls equispaced in the range of z, (60) becomes2 / Jo 0 I3(i n - 1 2(-)^o . , .,., (n- l)/3^ ^ ^ -^ sm (w — l)j3c cos n — 1 2(-y-\p,p, + ^,^o) . (/^ - 2)^<^ + -^— ^^--^ sm (w — 2)pf cos — + • • • (ol) n — 2 n — 1 + (Popn-l + plpn-2 + • • • + pn-lpo)^t . Substituting in
. The Bell System technical journal . ) COS (f+ {popn-l + i!l^»-2 + /2/n-3 + + pn-lpo) ■ Since / ^ sin ede = --- I $ J(/?,Jo P^ J vi z /(,« — 1; (59) we can write/ $ sin Bid = —. 2pl sin [n — l)<p , 2{pQpi + ^i/o) sin {n — 2)ip n — 1 + n — 2 + • • • + (pOpn-l + plp«-2 + • • • + pn-lpo)ip •Pi (60) For an end-on array with nulls equispaced in the range of z, (60) becomes2 / Jo 0 I3(i n - 1 2(-)^o . , .,., (n- l)/3^ ^ ^ -^ sm (w — l)j3c cos n — 1 2(-y-\p,p, + ^,^o) . (/^ - 2)^<^ + -^— ^^--^ sm (w — 2)pf cos — + • • • (ol) n — 2 n — 1 + (Popn-l + plpn-2 + • • • + pn-lpo)^t . Substituting in (54), we shall obtain the gain of the array. Similar expressions can be obtained for an end-on array in which theamplitudes of the individual elements are equal. Thus we have $ = i- [e(-^)^ + e-^ + •••+/ + !] + e -i(n—1)^1 (62) = — [2 cos (n — \)\l/ -\- 4: cos (n — 2)\p -\- 6 cos (n — S)\p + • • • + 2{n — 1) cos \p + n], 106 BELL SYSTEM TECHNICAL JOURNAL. Fig. 21—The directive gain in decibels of a pair of sources with equal amplitudes. (A),the phase delay between the sources is lirt/X; (B), the phase delay between thesources is tt — lirl/X. n-3 015 SEPARATION IN WAVELENGTHS Fig. 22—The gain as a function of separation in wavelengths: n is the number of elements. A MATHEMATICAL THEORY OF LINEAR ARRAYS 107 where \f/ = ^f (cos 6 — \). In this case D — 1 and r^ • «^« 2 r ... sin 2{n - IW , 2 sin 2(« - 2)^^ / $ sni ddd = -—- n^l + + —^— Jo fir^^ L w — 1 n — 2 ^ (63)^...^3sin2(.-3W^...^^^_^^^.^ n^ w — o J When the separation between the elements is exactly an integral numberof quarter wavelengths, (63) becomes I 2^sinddd = - (64) 0 n and consequently the gain is G = 10 Logio n. (65) Figure 21 contrasts the directive gain of a pair of sources of equal strengthwith the phase delay ItC/X (Curve A) with a directive gain of another pairof sources of equal strength but with the
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