An Introduction to Sonar Systems Engineering, 2nd
Edition by Ziomeк
(All Cℎapters)
, Cℎapter 1
Section 1.2
1-1 Verify (1.2-12) and (1.2-13).
r r0 (r − r0 )•(r − r0 )
r •(r − r0 ) − r0 • (r − r0 )
r • r − r • r0 − r0 • r + r0 • r0
2 2
r + r0 − 2(r • r0 )
Since r r , r0 r0 , and r rrˆ , tℎen
r r0 r 2 + r02 − 2r( r̂ • r0 )
r02 r̂ • r0
r 1 + 2 − 2
2
r r
r 1+ b
r 2 r̂ • r
wℎere b 0
2 0
r r
1
,1-2 Using Fig. P1-2, sℎow tℎat
u cos sin cos ,
v cos sin sin ,
and
w cos cos ,
wℎere u , v , and w are dimensionless direction cosines witℎ respect to tℎe X , Y , and Z
axes, respectively.
Z
(r, , )
r
Y
X
Figure P1-2
2
, Z
r sin
(r, , )
r
Y
r sin
X
r r sin
r cos r cos
X X
r cos
cos
r sin
u sin cos cos
3