compliant water wave absorbers

263 Pages·2007·10.02 MB·English

COMPLIANT WATER WAVE ABSORBERS
by
JER(cid:127)ME H. MILGRAM
S.B., Massachusetts Institute of Technology
(1961)
M.S., Massachusetts Institute of Technology
(1962)
3UBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF
PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF
TECHNOLOGY
August, 1965
Signature of Author.
Dprtment of Naval Architecture and
Afii ZEngineering, August 20, 1965
Certified by.....
Thesis Supervisor /
Accepted by .....
Chairman, Departmental Committee
on Graduate Students
-i-
ABSTRACT
COMPLIANT WATER WAVE ABSORBERS
by
Jerome H. Milgram
Submitted to the Department of Naval Architecture and Marine
Engineering on August 20, 1965 in partial fulfillment of the requirement
for the degree of Doctor of Philosophy.
This report comprises a detailed theoretical and experimental study
of the problem of absorbing plane water waves by means of a moving boundary
at one end of a channel. The non-linear problem is formulated as a sequence
of linear problems by means of perturbation techniques. This formulation is
carried out first for a general moving boundary and then for the specific
case where the boundary is a hinged paddle above a solid wall. In order to
avoid the parameter of the channel length in the theoretical work, this work
is carried out for a semi-infinite tank. Solutions for the necessary wave
absorber characteristics are determined by the first order (linear) theory.
Second order solutions are determined when the incident wave is a plane,
periodic, progressive wave. The theoretical developments are done with the
neglect of surface tension, but these effects are considered in a separate
chapter and they are accounted for in the computer programs used for the
design of a wave absorbing system. The problem of synthesizing a wave ab-
sorbing system whose characteristics closely approximate an ideal absorber
and which can be constructed readily is solved. The solution of this problem
requires a computer-aided design procedure for electric filters which may
be of general interest for its own sake, apart from. the remainder of this
work. The stability of wave absorbers is examined by a utilization of the
theory of waves with complex wave numbers. It is shown that such waves can
be constructed as combinations of waves with real wave numbers travelling
in skew directions in the vertical plane of the channel. The absorption
of wave pulses is considered. The velocity potential for a wave pulse can
be represented as an integral over the normal modes of the absorbing channel
if certain restrictions on the characteristics of the absorber are met. Ex-
periments on the absorption of periodic waves and wave pulses were carried
out. In addition an experiment was performed which confirms the theoretical
relationship between pressure and surface elevation. This was done as part
of an examination of the possibility of activating a wave absorber with a
pressure signal in the future.
Thesis Supervisor: Martin A. Abkowitz
Title: Professor of Naval Architecture
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ACKNOWLEDGEMENTS
The author wishes to express his appreciation to Professor Martin
A. Abkowitz, project supervisor; and Professors Erik L. Mollo-Christensen
and Justin E. Kerwin, project committee members; as well as to the M.I.T.
Computation Center which provided the Time-Sharing System on which the
computer work was done. The author is also grateful to Block Associates,
Inc. whose kind loan of some electronic apparatus facilitated the
experiments, and to Mr. Raymond Johnson whose Shop at M.I.T. fabricated
parts of the apparatus.
Special credit is due to Mr. Stanley Chang who did the equation
lettering, and to Mr. Donald Yansen who was of great assistance in
constructing the experimental apparatus and in proofreading parts of the
theoretical work.
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TABLE OF CONTENTS
Subject Page
TITLE PAGE iv
ABSTRACT ii
ACKNOWLEDGEMENTS iii
TABLE OF CONTENTS iv
LIST OF FIGURES vi
LIST OF SYMBOLS ix
CHAPTER 1. Introduction
CHAPTER 2. General Formulation of the Problem
CHAPTER 3. Formulation of the Boundary Condition at the
Wave Absorber and the Hydrodynamic Moment on
the Absorber When the Absorber is a Hinged
Paddle
CHAPTER 4. Solution of the Equations for the First-Order
Waves When the Incident Wave is a Plane,
Periodic, Progressive Wave
CHAPTER 5. Solution of the Equations for the Second-Order
Waves When the Incident Wave is a Plane, Periodic
Progressive Wave
CHAPTER 6. The Effect of Surface Tension
CHAPTER 7. Synthesis of a Linear Wave Absorbing
System Function
CHAPTER 8. Obilque Waves 96
CHAPTER 9. The Stability of Linear Active Wave Absorbers 112
CHAPTER 10. Determination of the Reflection Coefficient 116
CHAPTER 11. Absorption of Wave Pulses 130
CHAPTER 12. Discussion of Theoretical and Experimental 142
Wave Absorber Results
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Subject Page
CHAPTER 13. Investigation of the Relationship Between 147
Pressure and Surface elevation
CHAPTER 14. Conclusions 163
REFERENCES 165
166
APPENDIX A. Design and Construction of the
Experimental Wave Tank
APPENDIX B. Design of the Wave blsorbing Filter 168
APPENDIX C. Wave Height to Voltage Transducer 187
APPENDIX D. Alterations to the Chart Recorder 191
APPENDIX E. Integral Tables 194
APPENDIX F. Descriptions and Listings of Computer Programs 200
APPENDIX G. Computer-Aided Design Procedure for the synthesis 235
of a Rational System Function Which Approximates
Prescribed Frequency Characteristics
LIST OF FIGURES
Figure Page
1-1 Picture of experimental apparatus 7
1-2 Picture of details of the absorbing 9
end of the tank
2-1 Diagram of the geometry for a general absorber 20
3-1 Diagram of the geometry for a hinged paddle absorber 28
3-2 The change in paddle immersion due to a change 28
in paddle angle
6-1 Computer printout showing the effect of
surface tension on the eigenvalues for 79
the wave absorber problem
7-1 Computer printout for the synthesis of a comparatively 88
simple wave absorbing system function
7-2 Computer printout for the stability study of the system 90
function synthesized in figure 7-1
7-3 Computer printout for the theoretical determination
of the reflection coefficients for the system function 91
synthesized in figure 7-1.
7-4, 7-5, and 7-6 These figures correspond to figures 7-1, 92
7-2, and 7-3 respectively, where in this 94
case a more complicated system function 95
is considered.
10-1 Diagram of the experiment to measure the 125
reflection coefficient
10-2 Sample record from the experiment to measure 126
the reflection coefficient
10-3 Graph of theoretical and experimental reflection 127
coefficients
10-4 Graphs of the experimental and theoretical 128
system functions
10-5 Diagram of the experiment to measure the 129
system function
11-1 Surface elevation vrs. time for a wave pulse in 140
a channel with an absorber
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Figure Page
11-2 Surface elevation vrs. time for a wave pulse in 141
in a channel with a 10 degree sloping beach
11-3 Surface elevation vrs. time for a wave pulse in 141
a channel terminated with a solid wall
13-1 Diagram of the experiment to measure the relationship
between pressure and wave height 152
13-2 to 13-10. Graphs of the results of the experiment to 154
measure the relationship between pressure to
and wave height 162
A-i Scale drawing of the wave tank 167
B-1 Representation of an operational amplifier 168
B-2 Simple amplifier circuit 169
B-3 General linear circuit activated by 171
an operational amplifier
B-4 Diagram of the wave absorbing system 184
B-5 Picture.of the gearing between motor,paddle, 185
and feedback potentiometer
B-6 Circuitry around the feedback amplifier 173
B-7 Pole-zero plot for the system function used in 186
the experiments
B-8 e.i4.1 B-99 .Parts of the circuitry used to achieve 177
the desired system function
B-10 through B-13 Parts of the circuitry needed to achieve 181
a more complicated system function than 182
the one which was constructed 183
C-1 Schematic diagram of the circuit for the Wave Height to 190
Voltage Transducer
D-1 Schematic diagram showing alterations to the paper
chart recorder used to attain a high frequency 192
rolloff in the response of the recorder
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Figure Page
D-2 Amplitude frequency response for the altered 193
paper chart recorder
G-1 and G-2 Sample computer outputs for the computer-aided 249
design scheme for synthesizing electric filters to
253
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LIST OF SYMBOLS
S- Gradient
-V - Laplacian
x,y - rectangular axis
h - depth of water channel
g - acceleration due to gravity
0 - density of the fluid
1 - surface elevation
V - velocity
- velocity potential
.
t - time
T - surface tension
- horizontal position of the wave absorber
V - component of velocity normal to the absorber at the absorber
n
- perturbation parameter; also used as a small positive number in Chapter 11
.
p - depth of the pivot for a hinged paddle absorber
P - pressure
A - paddle angle, also used to represent wave obliqueness angle in Chapter 8
A - change in paddle imnersion
Mh- hydrodynamic moment on the absorber
B - amplitude of sinusoidal paddle motion
a) - radian frequency
A - distance from absorber to end wall of the channel
a,v,f - eigenvalues (In Chapter 8, c=fh)
A,B,N,D,G(with subscripts only),R(with subscripts only) - coefficients
Hm- ratio of first order moment to first order angle of the paddle
m
Sh- ratio of first order complex paddle angle to first order complex
amplitude of the wave height a distance d from the paddle
-ix-
rnh(as subscripts) -non-homogenous
nh(as subscript) - homogenous
L - linear operator as defined
K - curvature
R - radius of curvature
S=-icito - type of complex frequency usually used by electrical engineers
I - phase angle