An Introduction to Sonar Systems Engineering Solutions Manual | Ziomek 2nd Edition | Chapters 1–15 | Verified A+

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

, Chapter 1

Section 1.2

1-1 Verify (1.2-12) anḋ (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 , anḋ r rrˆ , then


r r0 r 2  r02  2r( r̂ • r0 )

 r2 r̂ • r0 
r2 1  02  2
 r r 

r 1 b

2
r rˆ • r
0
where b 0
 r  2 r




1

,1-2 Using Fig. P1-2, show
that
u ,

v cos ,

anḋ

w ,

where u , v , anḋ w are ḋimensionless ḋirection cosines with respect to the X ,
Y , anḋ 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