Solution Manual for
An Introduction to Sonar Systems Engineering, 2nd Edition by
Lawrence J. Ziomek
All Chapters 1-15
Chapter 1: Complex Aperture Theory – Volume Apertures – General Results
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
ẇhere b =
r − 2 r
0 0
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
, r cos Y
r r sin
r cos
r cos Y X r cos
r cos
sin =
r sin
v = sin sin = cos
From Fig. P1-2, = . Therefore,
u = cos = sin cos
v = cos = sin sin
ẇ = cos = cos
Note:
x = r cos = r sin cos = ru
y = r cos = r sin sin = rv
z = r cos = r cos = rẇ
4
An Introduction to Sonar Systems Engineering, 2nd Edition by
Lawrence J. Ziomek
All Chapters 1-15
Chapter 1: Complex Aperture Theory – Volume Apertures – General Results
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
ẇhere b =
r − 2 r
0 0
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
, r cos Y
r r sin
r cos
r cos Y X r cos
r cos
sin =
r sin
v = sin sin = cos
From Fig. P1-2, = . Therefore,
u = cos = sin cos
v = cos = sin sin
ẇ = cos = cos
Note:
x = r cos = r sin cos = ru
y = r cos = r sin sin = rv
z = r cos = r cos = rẇ
4
