Solutions Manual – An Introduction to Sonar Systems Engineering, 2nd Edition by Ziomek (All Chapters)

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