263 Pages·2007·10.02 MB·English

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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 -ii - 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. -iii- 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 -iv- 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 -vi - 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 -vii- 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 -viii- 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

The stability of wave absorbers is examined by a utilization of the theory of of an examination of the possibility of activating a wave absorber with a.

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