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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Resolve.mws  20 Aug 95   \+
Note: This problem should be done only after the 'detect' problem has \+
been done." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 294 "Two sources of radio waves in the sky are located at theta =0.
600 radians and at theta = 0.67 radians, each at a distance of 10^9 m \+
from a line of N equally-spaced detectors along the y-axis.  The detec
tor spacing is 0.10 m, the wavelength is 0.21 m, and the initial numbe
r of detectors is N=7." }}{PARA 0 "" 0 "" {TEXT -1 269 "a) Using what \+
you learned from the 'detect' problem, create two source response func
tions s1 and s2.  Plot both of these together, and also the sum of the
 two responses, all on the same graph of response vs phase delay D. Th
is is the same type of plot done in 'detect'." }}{PARA 0 "" 0 "" 
{TEXT -1 186 "b) By plotting just where the responses are large, deter
mine an accurate value for the angular distance between the peak of on
e response, and the nearby angle where the response is zero." }}{PARA 
0 "" 0 "" {TEXT -1 73 "c) Draw a sketch of 7 phasors such that their r
esultant is exactly zero. " }}{PARA 0 "" 0 "" {TEXT -1 106 "d) What an
gle between adjacent phasors is there in your sketch?  Compare this to
 what you found in part a)" }}{PARA 0 "" 0 "" {TEXT -1 220 "e) Determi
ne the minimum number of detectors N in  the array for you to be 'just
 able' to tell that there are two nearby sources in the sky, and not j
ust one. [Keep the spacing and wavelength and source angles the same.]
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "assume(k>0,wave
length>0,t>0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 342 "The number of \+
digits is set to 14 because of the 10^9 m distance used later in the p
roblem. It turns out you need at least 4 or 5 more digits available so
 that the calculated values of kL are accurate.  There is a compromise
 between accuracy and speed.  Using more digits improves accuracy, but
 calculations (and plots) can take a lot longer." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=14:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "k:=2*Pi/wavelength:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "X:=R*cos(Theta):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Y:=R*sin(Theta):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 5 "N:=7:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "m:=trunc(N/2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
69 "Determine the distances L[i] from the source to each detector elem
ent" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "for i from 1 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "L[i]:=s
qrt( (Y-spacing*(N-i-m))^2 + X^2);  od:" }}}{EXCHG {PARA 2 "" 0 "" 
{TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "R_x:=0:  R_y:=0:
" }}{PARA 0 "" 0 "" {TEXT -1 102 "Let t=0 and get magnitude of resulta
nt; first obtain x and y components of the resultant, then square." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for
 i from 1 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "R_x:=R_x+A*cos
(k*L[i]-D*(i-1)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "R_y:=R_y+A*sin
(k*L[i]-D*(i-1)):       od:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 1 "\n
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Rsquared:=R_x^2+R_y^2:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 52 "values:=\{R=1*10^9,A=1,spacing=0.10,wavelength=0.21
\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 83 "Substitute problem values into the resultant vector squared, ex
cept for theta value" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "g:=subs(values,Rsquared): " }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 65 "Now create the response function from source #1 a
nd #2 in the sky" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "s1:=subs(Theta=0.600,g):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "s2:=subs(Theta=0.670,g):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 44 "plot(\{s1,s2,s1+s2\},D=0..2*Pi,numpoints=100);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Remarks:" }}{PARA 0 "" 0 "" {TEXT 
-1 63 "Put in semicolons instead of colons to see intermediate output.
" }}{PARA 0 "" 0 "" {TEXT -1 76 "The plot range wants to be cut down s
o one gets just the region of interest." }}{PARA 0 "" 0 "" {TEXT -1 
92 "To get mouse coordinates in the plot, click the left button. Coord
s are given at lower left." }}{PARA 0 "" 0 "" {TEXT -1 102 "The angula
r distance in the initial plot is around 0.9 rad, and it should be 2 P
i/ 7 rad (very close!)" }}{PARA 0 "" 0 "" {TEXT -1 71 "Around 40 detec
tors are needed in the array to resolve the two sources." }{MPLTEXT 1 
0 0 "" }}}}{MARK "17 5 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }