SOLUTIONS MANUAL
An Introduction to Sonar Systems
Engineering, 2nd Edition by Ziomek
(All Chapters)
, Chapter 1
Section 1.2
1-1 Verify (1.2-12) and (1.2-13).
r r0 (r r 0 )• (r r 0 )
r • (r r 0 ) r0 • (r r 0 )
r • r r • r0 r0 • r r0 • r0
2
r r0 2(r • r0 )
2
Since r r , r0 r0 , and r rrˆ , then
r r0 r r0 2r( r̂ • r 0 )
2 2
r02 r̂ • r 0
r2 1 2 2
r r
r 1 b
r 2 r̂ • r
ẇhere b 0 2 0
r
r
1
,1-2 Using Fig. P1-2, shoẇ that
u cos sin cos ,
v cos sin sin ,
and
ẇ cos cos ,
ẇhere u , v , and ẇ are dimensionless direction cosines ẇith 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
An Introduction to Sonar Systems
Engineering, 2nd Edition by Ziomek
(All Chapters)
, Chapter 1
Section 1.2
1-1 Verify (1.2-12) and (1.2-13).
r r0 (r r 0 )• (r r 0 )
r • (r r 0 ) r0 • (r r 0 )
r • r r • r0 r0 • r r0 • r0
2
r r0 2(r • r0 )
2
Since r r , r0 r0 , and r rrˆ , then
r r0 r r0 2r( r̂ • r 0 )
2 2
r02 r̂ • r 0
r2 1 2 2
r r
r 1 b
r 2 r̂ • r
ẇhere b 0 2 0
r
r
1
,1-2 Using Fig. P1-2, shoẇ that
u cos sin cos ,
v cos sin sin ,
and
ẇ cos cos ,
ẇhere u , v , and ẇ are dimensionless direction cosines ẇith 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
