L. Cremer· M. Heckl

Structure-Borne Sound Structural Vibrations and Sound Radiation at Audio Frequencies

Translated and revised by E. E. Ungar

Second Edition

With 210 Figures

Springer-Verlag Berlin Heidelberg GmbH

Dr.-lng. L. Cremer em. o. Professor Direktor des Instituts flirTechnische Akustik derTechnischen Universitiit Berlin Einsteinufer 27 1000 Berlin 10

Dr. (Eng. Sc. D.) E. E. Ungar BBN Laboratories Incorporated 10 Moulton Street Cambridge, MA 02238 USA

Title of the Original Edition

Korperschall

Dr. rer. nat. M. Heckl Professor Institut flirTechnische Akustik derTechnischen Universitiit Berlin Sekretariat TA 7 Einsteinufer 27 1000 Berlin 10

Physikalische Grundlagen und technische Anwendungen

ISBN 978-3-662-10123-0

Library of Congress Cataloging in Publication Data Cremer, Lothar Structure-borne sound. Rev. translation of: Korperschall. Includes index. 1. Noise. 2. Noise control. 3. Vibration. I. Heckl, M. (Manfred) II. Title. TA365.C713 1987 620.2'3 87-28451 ISBN 978-3-662-10123-0 ISBN 978-3-662-10121-6 (eBook) DOI 10.1007/978-3-662-10121-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, repro­duction on microfilms orin other ways, and storage in data banks. Duplication of this publication or parts thereofis only permitted underthe provisions of the German CopyrightLawofSeptember9, 1965, in its version ofJune 24, 1985,and a copyright fee must always be paid. Violations fall underthe prosecu­tion act of the German Copyright Law.

© Springer-Verlag Berlin, Heidelberg 1973 and 1988 Originally published by Springer-Verlag Berlin Heidelberg New York in 1988 Softcover reprint of the hardcover 2nd edition 1988 The use of registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

2161/3020-5 43 210

Foreword to Second Edition

The authors and the translator have been most gratified by the inter­est and enthusiasm with which the technical community met the first edition of this book, which appeared nearly fifteen years ago. Both the German and English versions have served as textbooks for numerous students, and both versions appear also to have found wide acceptance by researches and practicing engineers.

In view of the continuing relevance of the book's topics to under­standing of the applicable phenomena and to a broad range of technolo­gical applications, the book has been written with the aim of making the subject matter available in a coherent, accessible form. Like the first edition, the second edition is intended to serve not as a handbook, but essentially as a textbook which focuses on discussions of relatively basic problems in great depth in order to facilitate the reader's solving other problems himself.

Because much of this book deals with physical fundamentals, and because basic concepts tend not to change, a large portion of this second edition is identical to the first edition - except for corrections, clarifi­cations, and updated references. However, there have been added a major section in which the plate and shell equations of motion are derived simultaneously from the general field equations, a section that addresses the impedances of orthotropic plates and tubes, and also a section deal­ing with sound radiation from cylindrical shells.

The authors are grateful to all who collaborated on the first edition, as well as to Mrs. C. Grotz, B. Topfer and K. Westphal for their help in preparing the manuscript of the second edition.

The translator is pleased to acknowledge the meticulous typing efforts of Ms. C. Prybylo and Ms. G. A. Cianci, as well as the encou­ragement of his colleagues at BBN Laboratories Incorporated.

Berlin and Munich; West Germany Cambridge, MA; USA

L. Cremer, M. Heckl E. E. Ungar

Foreword to Original Edition

Over fifteen years ago, the senior author, L. Cremer, on the basis of arrangements made by Professor Dr. E. Meyer, was commissioned by the Department of Scientific and Industrial Research, London, to write a monograph entitled "Propagation of Structure-Borne Sound" (Sponsored Research (Germany) No.1 (Series B)), dealing with analysis of the propagation of structure-borne sound in vehicles and buildings.

Since the initial 60 copies of the monograph were rapidly exhausted and photocopied many times, the idea of making the monograph available to a wider audience (and in German) suggested itself soon after this first publication.

However, at the time the author found it necessary to devote himself to the development of an Institute Professorial Chair and to other tasks, so that he was able to return to this monograph idea only in the last few years. Because a long time had passed since the initial publication, he felt it advisable to broaden the content considerably, particularly to include experimental considerations and new appli­cations in the field of noise control.

He therefore asked Dr. M. Heckl to take on a part of the required effort for the six chapters of this book, which deal with

I. Transducers

II. Wave Types

III. Damping

IV. Impedances

V. Attenuation

VI. Radiation.

Chapters I, II, and V were written by L. Cremer, and Chaps. III, IV, and VI by M. Heckl. However, both authors feel responsible for the entire book and hope that their use of similar approaches and of connecting references, and their avoidance of repetition, have enabled them to succeed in producing a coherent entity.

Foreword VII

Both authors aimed more at a textbook-which, on the basis of thorough discussions of relatively simple problems, enables the reader to solve other problems himself-than at a handbook, which attempts to summarize everything available up to the time of submission of the manuscript. References to existing literature have been included to the degree it seemed necessary; no attempt was made to achieve completeness. In order to limit this book to reasonable size, discussion of such important structures as shells and anisotropic plates also had to be omitted.

The authors are sincerely grateful to Dr. G. Boerger, Mr. M. Hu­bert, Mr. U. Kurze, Mr. H. Lazarus, Mr. H. Mueller, Mr. J. Nutsch, Dr. L. Schreiber and Mrs. Anna Heckl for their help in proofreading, as well as to all others who had a part in the completion of the manu­script and drawings.

The authors are particularly indebted to the publisher for his excellent production of the book, in regard both to the text and to the illustrations, and for his care and patience in meeting the authors' many requests.

Berlin and Munich October 1966

L. Cremer 11. Heckl

Authors' Preface to the Translation

First of all, the authors wish to express their sincere thanks to their colleague, Dr. Eric E. Ungar, not only for his careful translation, but also for his many suggestions for improvements.

The publication of this translated version enabled the authors to introduce some corrections and clarifications, as well as some new material dealing with recent developments. In particular, an extended treatment of Statistical Energy Analysis has been included (Section 8 of Chapter V), whereas the original made only brief mention of this topic. Thus, the translation in fact constitutes a revised second edi­tion.

The authors also are grateful to the publisher for his expeditious and meticulous efforts in the execution of this new edition.

Berlin and Munich October 1972

L. Cremer M. Heckl

Translator">s Preface

When the original German edition of this book first appeared in print, the undersigned was asked to review it for an English-language journal. In the course of this review process he was so struck by the book's unusual approach, as well as by the realization that no similar collection of information was available in English, that he conceived the idea of undertaking this translation. Arrangements with the original authors and with the publisher were completed late in 1969; the translation task was begun at that time and occupied nearly two and one-half years-largely because the undersigned's other professional activities permitted him to devote only liis spare time to the translation effort.

Because this passage of time also brought with it some advances in the state of the art, an attempt was made to include the most significant of these advances, as well as newer references and some minor correc­tions, in the translated version. Except for these changes, which were made with the enthusiastic endorsement and collaboration of the origi­nal authors, the translated version remains very close to the original. In fact, the translator took special care to preserve not only the mean­ing, but also the "flavor", of the original text.

The translator is most grateful to Professor L. Cremer and Dr. M. Heckl for their continuous cooperation, and to several of his colleagues at Bolt Beranek and Newman, Inc. for their comments and corrections. He is also indebted to Mrs. P. G. Abadzoglou and Miss C. Prybylo, as well as to other members of the secretarial and publica­tions staffs of Bolt Beranek and Newman, Inc., for their careful typing of the several drafts of the manuscript. Last, but by no means least, he is sincerely thankful for his wife's and daughters' patience, under­standing and encouragement, without which this task would not have been accomplished.

Newton and Cambridge, Mass. November 19-'t2

E. E. Ungar

Contents

Chapter I

Delinition, Measurement, and Generation of Structure-Borne Sound

1. Defin.ition . . . . . . . . . . . . . . . . . . . . . . .

2. Mechanical Measurement Methods and Related Considerations

3. Sensors that Control Electric Circuits . . . . . . . . . .

4. Electromechanical Transducers for Airborne Sound . . . . a) Application to Measurement of Structure-Borne Sound b) Electrodynamic Transducers . c) Electrostatic Transducers . . d) Electromagnetic Transducers . e) Piezoelectric Transducers . .

5. Electromechanical Transducers for Structure-Borne Sound . a) Sensors ........... . b) Exciters of Structure-Borne Sound ........ .

Chapter II

Survey of Wave Types and Characteristics

1. Longitudinal Waves ............. . a) Pure Longitudinal Waves ......... . b) Quasi-Longitudinal Waves on Beams and Plates

2. Transverse Waves ..... a) Transverse Plane Waves . b) Torsional Waves . .

3. Bending Waves a) Pure Bending Waves b) Corrected Bending Waves

4. Wave Motions on Beams of Finite Length a) Longitudinal Natural Vibrations b) Natural Vibrations in Bending

5. The General Field Equations

1

3

24

32 32 36 45 54 58

62 62 69

75 75 81

87 87 90

95 95

109

115 116 121

130

Contents XI

60 Wave Fields at Free Surfaces 137

a) Reflection of Plane Waves 137 <X) Some Simple Special Cases o 137 {J) Velocity Potential and Stream Function 138 y) Equality of Trace Velocities o 0 o o o 140 o) Determination of Reflection Coefficients and Reflection

Efficiencies

b) Surface Waves 0 0 0 0 0 0 0 0 0 o 0 0 0 <X) Forced Surface Waves o o o o 0 o o o {J) Free Surface Waves (Rayleigh Waves) o

143

147 147 150

7o Free Plate Waves 0 0 0 0 0 0 0 o o o 0 o o 152

a) Boundary Conditions and Types of Solutions o 152 b) Waves with Displacements only Parallel to the Surface o 155 c) Waves which also have Displacement Components Perpendicular

to the Surface 0 0 0 0 0 o o o o o o o o o o o o o o 0 0 o o 158

80 Simultaneous Derivation of Plate and Shell Equations from the General Field Equations 0 0 0 0 0 0 o 0 0 o o o o 168

a) Extensional and Shear Waves in Flat Plates 0 0 0 0 o o o o o o o o 168 b) Bending waves in Flat Plates 0 0 0 0 0 0 0 0 o o o o o o 0 o o o o 173 c) Structure-home Sound in Circular Cylindrical Shells 176 <X) Basic Equations 176 {J) Membrane Equations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 181 y) Consideration of Flexure 0 0 0 0 0 0 o 0 0 o o o 185 o) Resonance Frequencies of Rings, Tubes, and Cylindrical and Spherical

Shells 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o 0 0 o o o o o o o o o o o 189

Chapter III

Damping

1. Damping Mechanisms and their Mathematical Description 0

20 The Complex Modulus of Elasticity o 0 0 o o o o o

3o Resonant Vibrations of Damped Beams o o o o o o a) Quasi-Longitudinal Waves and Torsional Waves b) Bending Waves 0 0 0 0 0 0 0 0

4o Measurement of Complex Moduli o o a) Measurements on Small Samples

<X) Stress-Strain Curve o o

{J) Mechanical Impedance y) Vibration Decay o o o o) Resonance Frequency and Half-Value Bandwidth o

b) Measurements on Beams <X) Half-Value Bandwidth o o o o o o 0 o {J) Decay Time 0 0 0 0 0 0 0 o o 0 0 0 0 y) Attenuation of Vibrations with Distance 0

o) Other Methods o o o o o o o o o o o o c) Measurements on Other than Beam-Like Samples 0

5o Experimental Data

a) Metals

195

199

204

204 211

215

216 216 218 221 223 224 225 227 228 230 230

231

231

XII Contents

b) Plastics . . . . . . . . c) Building Materials . . .

6. Plates with Attached Layers . a) Plates with Simple, Extensionally Loaded Layers . b) Plates with Multi-Layer Treatments

~X) Stiff Base Plate with Thin Cover Plate {3) Two Equal Plates with Thin Interlayer y) General Composite Plate or Beam Configurations .

c) Damping by Means of Resonant Systems .

7. Other Damping Mechanisms ........ . a) Damping at Metal Interfaces; Air Pumping b) Damping by Granular Materials ..

Chapter IV

Impedances

1. Definition of Point-Impedance . . . . .

2. Measurement of Mechanical Impedances . a) Measurement of Force and Velocity . b) Comparison with Known Impedance

3. Input Impedances of Infinite Beams and Plates a) Quasi-Longitudinal Waves in Beams .... b) Bending Waves in Beams . . . . . . . . c) Bending-Wave Equation for Homogeneous Thin Plates d) Driving-Point Impedance of Homogeneous Plates in Flexure .

238 241

243 243 247 249 252 255 255

261 261 263

266

268 268 271

275 275 278 283 286

e) Analysis of Plate Flexural Vibrations by Means of Fourier Trans-forms ..................... .

f) Driving-Point Impedance of Thick Plates in Flexure g) Driving Point Impedance of Orthotropic Plates in Bending h) Impedances of Plate Strips and Tubes . i) Moment Impedances ..... . j) Summary of Impedance Formulas

4. Point-Excitation of Finite Systems a) General Properties . b) Some Applications . . c) Power Considerations .

5. Some Specific Applications a) Footfall Noise . . . . b) Isolation of Machinery

Chapter V

Attenuation of Structure-Borne Sound

1. Changes in Material and Cross-Section a) Attenuation of Longitudinal Waves . b) Attenuation of Bending Waves ..

2. Corners and Branches at Right Angles

295 298

. 301

. 304

. 311

. 315

318 318 323 327

333 333 339

342 343 347

352

Contents

3. Elastic Interlayers . . . . . . . . . . a) Attenuation of Longitudinal Waves . b) Attenuation o~ Bending Waves ...

4. Blocking Masses . . . . . . . . . . . a) Attenuation of Longitudinal Waves . b) Attenuation of Bending Waves ... c) Coupling of Longitudinal and Bending Waves

5. Spatially Periodic Structures • . . . . . . . . . a) Transmission and Attenuation of Longitudinal Waves . b) Cascades of Flexural Elements

6. Oblique Incidence . . . . . . . a) General Considerations b) General Consequences of the Boundary Conditions c) Two.Dimensional Analysis of Walls Joined at Right Angles d) Plate with Reinforcing Beam . . . . . . . . . . . . .

7. Parallel Plates . . . . . . . . . . . . . . . . . . . . . . a) Continuous Coupling by Elastic Interlayer (Floating Floor). b) Point-Acting Sound Bridges

8. Statistical Energy Analysis a) Introduction . . . . . . . b) Power Flow between Linearly Coupled Oscillators c) Coupled Multimodal Systems . d) Applications . . . . . . . . . . . . . . . . .

Chapter VI

Sound Radiation from Structures

1. Measurement of Radiated Power . . . .

2. Definition and Measurement of Radiation Efficiency

3. Radiation Loss Factor

4. Elementary Radiators a) Spherical Radiators . b) Infinite Plates . . . c) Cylindrical Radiators . . . . . . . .

5. Plane Radiator as Array of Point Sources a) Rectangular Array of Point Sources . b) Membrane with Axially Symmetric Velocity Distribution

6. Radiation from Bending Waves .. a) Critical Frequency . . . . . . . . . . . . . . . . b) Bending Waves on Finite Plates . . . . . . . . . . c) Radiation from Flexural Nearfield at Excitation Point d) Some Experimental Results . . . . . . . . . . . . e) Radiation from Point-Excited Tubes ........ .

7. Plate Response to Excitation by Airborne Sound (Attenuation of Airborne Sound) ........... .

XIII

370 370 375

385 386 388 402

405 405 415

425 425 431 436 437

450 450 462

474 474 478 482 485

492

495

497

499 499 502 505 510 512 519

523 523 526 534 537 540

543

XIV Contents

a) Response of Homogeneous Plates . . b) Double-Walls with Sound Bridges

544 548

8. Relation between Radiation and Response • 550

a) Reciprocity . . . . . . . . . . . . 550 b) Response and Radiation in a Reverberant Room 551 c) Effect of Radiation Loading . . . . . . . . . 554

d) Attenuation and Flanking Transmission Above the Critical Fre· quency . . . . . . . . . . . . . . . . . . . . . . . 556

e) Relation between Airborne and Impact Sound Transmission 560 f) Application of Statistical Energy Analysis 563

Index . . . . . . . . . . . . . . . . . . . . 565

Symbols

General Remarks

The notation used here is ba&ed on that recommended in the German stand­ards DIN 1302, 1304, and 1332 1•

However, some deviations could not be avoided for the particular subject at hand. For example, the symbol w is used to represent angular velocity in order to avoid possible confusion with the radian frequency ro.

In structure-borne sound, various quantities occur sometimes per unit length (for beams) and sometimes per unit area (for plates). In order to avoid use of different letters for these quantities, the convention is employed that each prime corresponds to division by a length dimension.

Example: , mass

m = mass, m = length, , mass

m =-­area

This notation has the advantage that it permits one easily to recognize inter­relations between these quantities.

This use of primes is also applied to the symbols for the various impedances and has made it possible to avoid cumbersome subscripts to indicate the different units corresponding to these impedances.

Example: z _ Force Z' _ Forceflength Z" _ Pressure

- Velocity ' - Velocity ' - Velocity ·

In structure-borne sound, there also occur the analogous ratios of moment (or torque) to angular velocity. A separate letter, W, is used here for these mo­ment impedances.

Since primes are used to indicate division by a length dimension, they cannot be employed also to designate real and imaginary parts. Instead, these are indicated by Re {} and Im {}, or by 1 and ll (instead of the often used ' and "). Since these symbols resemble the inverted letters T and II, one may call them "et" and "ip".

Example: !! = kl +jkll.

Complex quantities are indicated by underlining. However, this is omitted in the ·later part of this book; in general, special notational characterizations are omitted throughout this book for the sake of simplicity, wherever no confusion is likely.

1 DIN is an abbreviation for "German Eng~neering Standard".

XVI Symbols

In Sec. 6 of Chapter V there are introduced doubly underlined letters. These represent the phasor (rotating vector) corresponding to a field variable for which the time-dependence is given by the factor e;wt and for which also the spatial dependence in one direction (e.g., the x-direction) is given by a factor efkflf&. Here kz is called the trace-wavenumber.

In Chapter I, however, complex electrical quantities are represented by script letters, whereas the German standard recommends Gothic letters for this purpose. The complex transducer constants that relate mechanical and electrical quantities are represented by script letters that also are underlined.

In order to facilitate differentiation between electrical and mechanical ele­ments, the circuit symbols employed here to represent electrical resistances and inductances differ from those in common use in the United States. However, the "square wave" symbol used for resistances is reminiscent of the usual zig­zag, as well as of printed circuit resistors, so that the reader should have no difficulty recalling its meaning; similarly, the abbreviated-loop coil symbol used for inductances may be interpreted easily, because it is closely related to the common full-loop symbol.

The symbol definitions used in this book are tabulated below, arranged alphabetically, by type of letter. Symbols that are used only in passing, in the course of a derivation, for the sake of brevity or clarity, are not included here. Mter each definition there is indicated the equation in which the symbol first appears. The meanings of subscripts are given for only a few exceptional cases.

English Capitals

A amplitude I (68), equivalent absorption area VI (6) B inductance I (77), flexural rigidity II (71), gyroscopic coefficient V (471) C capacitance I (69), group velocity II (88), C(x) = (cosh x +cos x)/2 III (46),

specific heat III (72a), flexural rigidity of elastic layer V (158) D longitudinal stiffness II (2) D' level difference per unit length III (71) E energy density II (5), modulus of elasticity II (23) F force G shear modulus II ( 42) H height, Hankel function IV (54) I current I (138), moment of inertia of area about centroidal axis II (75),

impulse IV (3) J intensity II (8), Bessel function VI (68) K bulk modulus I (75), transducer constant I (156), shear stiffness of beam

or rod II (104), shear stiffness of elastic layer V (157) L length, level (L11 velocity level) I (38), inductance I (69) M mass I (39), absolute value of transducer constant I (120), force-moment or

torque II (55) N absolute value of transducer constant I (131), number of natural (resonance)

frequencies IV (4) P power Q charge I (101), quality of resonance, Table III, 1, heat III (72a) R frictional resistance I (43), electrical resistance I (64), distance from origin

IV (65), transmission loss V (13) S stiffness of spring I (39), area I (72), S(x) = (sinh x + sin x)/2 III (46)

T period of one cycle T = ~ II (19), torsional stiffness II (56), reverbera­

tion time III (66)

Symbols

U voltage I (70), circumference VI (70) V volume W energy I (110), moment impedance IV (102) Y mechanical admittance (1/Z) I (19) Z mechanical impedance Zr transmission impedance II (266)

English Lower-Oa&e Letters

xvn

a amplitude I (68), separation distance III {78), attenuation coefficient V (258), radius II (235Z)

b width, half-value bandwidth III (50)££. c propagation velocity

cL of longitudinal waves II {13) cu same, in plates II {34) CLJJ same, in beams II (32) cp of torsional waves II (47) cB phase velocity of flexural waves II (85) c(x) = {cosh x - cos x)/2 ill (46) Cz phase velocity in axial direction Fig. II/31

d separation distance, thickness e base of natural logarithms, auxiliary length I (74), rotational energy in

flexural waves, after II (92k) · f frequency [fc critical frequency V {369)] g acceleration of gravity I {13), complex propagation coefficient V {253) h plate thickness II (35), cylinder wall thickness II (235) Z i current I (70)

J v-1 k wave-number, strain gage constant I {67)

kp shear wave number II (244) lcLz, lcLII extensional wave number length

m mass n integer II (118), index of refraction II (155a), number of reflections V (314) p sound pressure [p. static atmospheric pressure I (75a)] q charge I {101), volume velocity or souroe strength VI {27) r mechanical frictional resistance I (21), reflection coefficient II {161), distance

from point source IV {54) a spring stiffness I {14), reduced trace-wavelength II {171),

a{x) = {sinh x -sin x)/2 III {46) time, transfer coefficient V (20)

u voltage v velocity (particle velocity) I {10) w angular velocity II {57) x spatial coordinate y spatial coordinate z spatial coordinate

Script Capitals

J phasor of alternating current I (77 a) ..1 transducer constant (voltage/velocity or force/current) I (114) .A'" transducer constant {current/velocity or force/voltage) I (129)

XVlll Symbols

1ft phasor of alternating voltage I (80) 1l' electrical impedance I (80)

Lower-Caae Semi-Bold Lettera

i spatial unit vector in x-direction II (131) j spatial unit vector in y-direction II (131) k spatial unit vector in z-direction II (131) s displacement vector II (131) v velocity vector II (136e) w angular velocity vector II n40a)

Greek Capitala

.1 Laplace operator II (136a}, incremental quantity, e.g., .1w I (35) @ mass moment of inertia V (193), temperature change III (72) A logarithmic decrement III (57), thermal conductivity III (72a}, norm of

eigenfunction (characteristic function) IV (114) .:, displacement I (1), flow resistance per unit length V (388) II function of point-excited flexural wave-field IV (59) 4> velocity potential II (148) 'l' vector potential (stream function) II (149) {J radian frequency I (68)

Lower-CaBe Greek Lettera

~ material constant III (72), absorption efficiency III (108) p slope of beam or plate II (68), frequency parameter V (50) 'Y shear angle II (41}, phase change II (116c) «5 decay constant I (29), dilatation II (133) s strain I (66), configuration-dependent parameter V (165), complex amplitude

ratio V ( 403) C displacement in z-direction II (129b); efficiency VI (13) 1J displacement in y-direction II (22), loss factor III (7b) i1 angle of incidence II (152a), damping coefficient III (1), particular time

constant IV (39}, configuration-dependent parameter V (200) " ratio of. specific heats I (75a}, shear distribution parameter II (100}, ratio

of two wavenumbers V (26) ,1. wavelength p maBB ratio I (39), Poisson's ratio I (67 a), permeability I (137), frequency

parameter V (199) , frequency parameter II (198), V (163), II (250Z) ~ displacement, particularly in x-direction n 3.14 •.. (] electrical resistivity I (64}, density I (71), reflection efficiency II (170a),

polar coordinate IV (61:!) a tensile stress II (2}, ratio of cross-sectional areas ,or thicknesses V (11},

porosity V (387}, radiation efficiency VI (8) T shear streBB II (136), rela.,xation time III (6), (72), transmiBBion efficiency V (12) ·~p phase angle II (51}, auxiliary phase angle II (213). memory or after-effect

function III (4}, mode shape or eigen-function (characteristic function) IV (108)

x twist angle II (51}, matching parameter V (111) tp auxiliary phase angle II (213}, curvature· III (73), ratio of two impedances

v (27) w radian frequency