SOLUTIONS MANUAL for An Introduction to Sonar Systems Engineering 2nd Edition By Lawrence J. Ziomek (All Chapters) Complete Guide A+

SOLUTIONS
MANUAL for An
Introduction to
Sonar Systems
Engineering, 2e
by Lawrence J.
Ziomek
(All Chapters)




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, Chapter 1

Section 1.2

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

r − r0 = (r − r0 )i(r − r0 )
= r i(r − r0 ) − r0 i (r − r0 )
= r i r − r i r0 − r0 i r + r0 i r0
2 2
= r + r0 − 2(r i r0 )


Since r = r , r0 = r0 , and r = r r̂ , then


r − r0 = r 2 + r02 − 2r( r̂ i r0 )

⎛ r02 r̂ i r0 ⎞
= r ⎜ 1+ 2 − 2
2

⎝ r r ⎟⎠
= r 1+ b

2
⎛ r0 ⎞ r̂ i r0
where b = ⎜ ⎟ − 2
⎝r⎠ r




1




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

u = cos α = sin θ cosψ ,

v = cos β = sin θ sinψ ,

and
w = cos γ = cosθ ,

where u , v , and w are dimensionless direction cosines with respect to the X , Y , and Z
axes, respectively.








  













Figure P1-2




2




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, 



 




  







 










  


 








r cos α
cosψ =
r sin θ
u = sin θ cosψ = cos α




3




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