SOLUTIONS MANUAL: An Introduction to Sonar Systems Engineering, 2nd Edition by John L. Ziomek | All Chapters | Verified Answers

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 − r0 )•(r − r0 )
r •(r − r0 ) − r0 • (r − r0 )
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 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 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