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Engineering, 2e by Lawrence J. ziomek Ch
2Page
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MANUAL for An Introduction
MANUALtoforSonar
An Introduction
Systems Engineering,
to Sonar Systems
2e by Lawrence
Engineering,
J. ziomek
2e by Lawrence
Chapter 1-1510/12/20
J. ziomek Ch
,ONS
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2026-10-12Page
Chapter
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Engineering,
PM10/12/2026SOLUTIONS
2e by Lawrence J.MANUAL
ziomek Chapter
for An Introduction
1-1510/12/2026,
to Sonar
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Systems
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Engineering, 2e by Lawrence J. ziomek Ch
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
2Page
ce J. ziomek
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17910/12/2026,
Chapter 1-1510/12/2026,
21:16:47
of 17910/12/2026,
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1:16:47 PM10/12/2026SOLUTIONS
MANUAL for An Introduction
MANUALtoforSonar
An Introduction
Systems Engineering,
to Sonar Systems
2e by Lawrence
Engineering,
J. ziomek
2e by Lawrence
Chapter 1-1510/12/20
J. ziomek Ch
,ONS
ce J. MANUAL
ziomek
2026-10-12Page
Chapter
for An Introduction
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Engineering,
PM10/12/2026SOLUTIONS
2e by Lawrence J.MANUAL
ziomek Chapter
for An Introduction
1-1510/12/2026,
to Sonar
1:15:42
Systems
PMhttps://www.stuvia.com/user/aaaplus_)W()*E
Engineering, 2e by Lawrence J. ziomek Ch
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
2Page
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17910/12/2026,
Chapter 1-1510/12/2026,
31:16:47
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1:16:47 PM10/12/2026SOLUTIONS
MANUAL for An Introduction
MANUALtoforSonar
An Introduction
Systems Engineering,
to Sonar Systems
2e by Lawrence
Engineering,
J. ziomek
2e by Lawrence
Chapter 1-1510/12/20
J. ziomek Ch
,ONS
ce J. MANUAL
ziomek
2026-10-12Page
Chapter
for An Introduction
1-1510/12/2026,
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Engineering,
PM10/12/2026SOLUTIONS
2e by Lawrence J.MANUAL
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for An Introduction
1-1510/12/2026,
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1:15:42
Systems
PMhttps://www.stuvia.com/user/aaaplus_)W()*E
Engineering, 2e by Lawrence J. ziomek Ch
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
2Page
ce J. ziomek
4 of2026-10-12Page
17910/12/2026,
Chapter 1-1510/12/2026,
41:16:47
of 17910/12/2026,
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1:16:47 PM10/12/2026SOLUTIONS
MANUAL for An Introduction
MANUALtoforSonar
An Introduction
Systems Engineering,
to Sonar Systems
2e by Lawrence
Engineering,
J. ziomek
2e by Lawrence
Chapter 1-1510/12/20
J. ziomek Ch
ce J. MANUAL
ziomek
2026-10-12Page
Chapter
for An Introduction
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Engineering,
PM10/12/2026SOLUTIONS
2e by Lawrence J.MANUAL
ziomek Chapter
for An Introduction
1-1510/12/2026,
to Sonar
1:15:42
Systems
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Engineering, 2e by Lawrence J. ziomek Ch
2Page
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Chapter 1-1510/12/2026,
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1:16:47 PM10/12/2026SOLUTIONS
MANUAL for An Introduction
MANUALtoforSonar
An Introduction
Systems Engineering,
to Sonar Systems
2e by Lawrence
Engineering,
J. ziomek
2e by Lawrence
Chapter 1-1510/12/20
J. ziomek Ch
,ONS
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ziomek
2026-10-12Page
Chapter
for An Introduction
1-1510/12/2026,
2 of 17910/12/2026,
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Systems
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Engineering,
PM10/12/2026SOLUTIONS
2e by Lawrence J.MANUAL
ziomek Chapter
for An Introduction
1-1510/12/2026,
to Sonar
1:15:42
Systems
PMhttps://www.stuvia.com/user/aaaplus_)W()*E
Engineering, 2e by Lawrence J. ziomek Ch
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
2Page
ce J. ziomek
2 of2026-10-12Page
17910/12/2026,
Chapter 1-1510/12/2026,
21:16:47
of 17910/12/2026,
PM10/12/2026SOLUTIONS
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1:16:47 PM10/12/2026SOLUTIONS
MANUAL for An Introduction
MANUALtoforSonar
An Introduction
Systems Engineering,
to Sonar Systems
2e by Lawrence
Engineering,
J. ziomek
2e by Lawrence
Chapter 1-1510/12/20
J. ziomek Ch
,ONS
ce J. MANUAL
ziomek
2026-10-12Page
Chapter
for An Introduction
1-1510/12/2026,
3 of 17910/12/2026,
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1:15:42
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1:16:47
Engineering,
PM10/12/2026SOLUTIONS
2e by Lawrence J.MANUAL
ziomek Chapter
for An Introduction
1-1510/12/2026,
to Sonar
1:15:42
Systems
PMhttps://www.stuvia.com/user/aaaplus_)W()*E
Engineering, 2e by Lawrence J. ziomek Ch
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
2Page
ce J. ziomek
3 of2026-10-12Page
17910/12/2026,
Chapter 1-1510/12/2026,
31:16:47
of 17910/12/2026,
PM10/12/2026SOLUTIONS
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1:16:47 PM10/12/2026SOLUTIONS
MANUAL for An Introduction
MANUALtoforSonar
An Introduction
Systems Engineering,
to Sonar Systems
2e by Lawrence
Engineering,
J. ziomek
2e by Lawrence
Chapter 1-1510/12/20
J. ziomek Ch
,ONS
ce J. MANUAL
ziomek
2026-10-12Page
Chapter
for An Introduction
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to Sonar
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Systems
PMhttps://www.stuvia.com/user/aaaplus_)W()*E
Engineering, 2e by Lawrence J. ziomek Ch
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
2Page
ce J. ziomek
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Chapter 1-1510/12/2026,
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MANUAL for An Introduction
MANUALtoforSonar
An Introduction
Systems Engineering,
to Sonar Systems
2e by Lawrence
Engineering,
J. ziomek
2e by Lawrence
Chapter 1-1510/12/20
J. ziomek Ch
