SOLUTIONS ṀANUAL
An Introduction to Sonar Systeṁs
Engineering, 2nd Edition by Zioṁeк
(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 • r 0 r 0 • r r 0 • r0
2
r r0 2(r • r0 )
2
Since r r , r0 r0 , and r rrˆ , then
r r0 r 2 r02 2r( r̂ • r0 )
r2 r̂ • r0
r2 1 02 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 ,
v cos ,
and
w ,
where u , v , and w are diṁensionless 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
r
u
3
An Introduction to Sonar Systeṁs
Engineering, 2nd Edition by Zioṁeк
(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 • r 0 r 0 • r r 0 • r0
2
r r0 2(r • r0 )
2
Since r r , r0 r0 , and r rrˆ , then
r r0 r 2 r02 2r( r̂ • r0 )
r2 r̂ • r0
r2 1 02 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 ,
v cos ,
and
w ,
where u , v , and w are diṁensionless 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
r
u
3
