SOLUTIONS MANUAL AN Introduction To Sonar Systems Engineering 2nd Edition By Lawrence J. Ziomek |All chapters |Latest Version 2025 A+

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