SOLUTIONS MANUAL for An Introduction to Sonar Systems Engineering, 2e by Lawrence J. ziomek Chapter 1-15.

, Chapter 1

Section 1.2

1-1 Verify (1.2-12) and (1.2-13).

r  r0 


2 2
 r  r0  2(r r0 )

Since r  r , r0  r0 , and r  rrˆ , then


r  r0  r 2 r 02 2r( rˆ• r0 )
  r02 
 r 1  2  2
2 0
 r r 

 r 1 b

 r 2 rˆ• r
where b  0
2 0
 r 
 r

,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

, 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