Book contents
- Frontmatter
- Contents
- ERRATA
- ART. 273 Interference of Sound
- ART. 274 Some General Theorems concerning Forced Vibrations and Resonance
- ART. 275 On the Law of the Pressure of Gases between 75 and 150 millimetres of Mercury
- ART. 276 On the Pressure of Vibrations
- ART. 277 On the Question of Hydrogen in the Atmosphere
- ART. 278 Does Chemical Transformation influence Weight?
- ART. 279 Is Rotatory Polarization influenced by the Earth's Motion?
- ART. 280 Does Motion through the Æther cause Double Refraction?
- ART. 281 On the Distillation of Binary Mixtures
- ART. 282 Note on the Theory of the Fortnightly Tide
- ART. 283 On the Free Vibrations of Systems affected with Small Rotatory Terms
- ART. 284 On the Vibrations of a Rectangular Sheet of Rotating Liquid
- ART. 285 On the Spectrum of an Irregular Disturbance
- ART. 286 Considerations respecting the Combustion of Modern Propellants in Closed Vessels and in Guns
- ART. 287 On the Bending of Waves round a Spherical Obstacle
- ART. 288 On the Proportion of Argon in the Vapour rising from Liquid Air
- ART. 289 On the Theory of Optical Images, with Special Reference to the Microscope
- ART. 290 On the Production and Distribution of Sound
- ART. 291 On the Work done by Forces operative at one or more Points of an Elastic Solid
- ART. 292 On the Acoustic Shadow of a Sphere
- ART. 293 Shadows
- ART. 294 Sir George Gabriel Stokes, Bart., 1819—1903
- ART. 295 On the Measurement of certain very short Intervals of Time
- ART. 296 Note on the Application of Poisson's Formula to Discontinuous Disturbances
- ART. 297 Fluid Friction on Even Surfaces
- ART. 298 On the Electrical Vibrations associated with thin terminated Conducting Rods
- ART. 299 On the Density of Nitrous Oxide
- ART. 300 Note to a Paper by Prof. Wood on the Achromatization of approximately Monochromatic Interference Fringes by a highly dispersive Medium
- ART. 301 On the Open Organ-Pipe Problem in Two Dimensions
- ART. 302 Extracts from Nobel Lecture
- ART. 303 On the Compressibility of Gases between One Atmosphere and Half an Atmosphere of Pressure
- ART. 304 On the Pressure of Gases and the Equation of Virial
- ART. 305 The Dynamical Theory of Gases and of Radiation
- ART. 306 An Optical Paradox
- ART. 307 The Problem of the Random Walk
- ART. 308 On the Influence of Collisions and of the Motion of Molecules in the Line of Sight, upon the Constitution of a Spectrum Line
- ART. 309 On the Momentum and Pressure of Gaseous Vibrations, and on the Connexion with the Virial Theorem
- ART. 310 The Origin of the Prismatic Colours
- ART. 311 On the Constitution of Natural Radiation
- ART. 312 On an Instrument for compounding Vibrations, with Application to the drawing of Curves such as might represent White Light
- ART. 313 On Electrical Vibrations and the Constitution of the Atom
- ART. 314 On the Production of Vibrations by Forces of Relatively Long Duration, with Application to the Theory of Collisions
- ART. 315 On the Dilatational Stability of the Earth
- ART. 316 Some Measurements of Wave-Lengths with a Modified Apparatus
- ART. 317 On the Experimental Determination of the Ratio of the Electrical Units
- ART. 318 On the Interference-Rings, described by Haidinger, observable by means of Plates whose Surfaces are absolutely Parallel
- ART. 319 On our Preception of Sound Direction
- ART. 320 Acoustical Notes
- ART. 321 On the Passage of Sound through Narrow Slits Appendix
- ART. 322 On the Dynamical Theory of Gratings
- ART. 323 Note on the remarkable case of Diffraction Spectra described by Prof. Wood
- ART. 324 On the Light dispersed from Fine Lines ruled upon Reflecting Surfaces or transmitted by very Narrow Slits
- ART. 325 On the Relation of the Sensitiveness of the Ear to Pitch, investigated by a New Method
- ART. 326 Effect of a Prism on Newton's Rings
- ART. 327 Further Measurements of Wave-Lengths, and Miscellaneous Notes on Fabry and Perot's Apparatus
- ART. 328 On the Aberration of Sloped Lenses and on their Adaptation to Telescopes of Unequal Magnifying Power in Perpendicular Directions
- ART. 329 Hamilton's Principle and the Five Aberrations of von Seidel
- ART. 330 Vortices in Oscillating Liquid
- ART. 331 Acoustical Notes.—VIII
- ART. 332 On Reflexion from Glass at the Polarizing Angle
- ART. 333 Note on Tidal Bores
- ART. 334 Notes concerning Tidal Oscillations upon a Rotating Globe
- ART. 335 On the Instantaneous Propagation of Disturbance in a Dispersive Medium, exemplified by Waves on Water Deep and Shallow
- ART. 336 On the Resistance due to Obliquely Moving Waves and its Dependence upon the Particular Form of the Fore-part of a Ship
- ART. 337 On the Perception of the Direction of Sound
- ART. 338 The Theory of Crookes's Radiometer
- ART. 339 To determine the Refractivity of Gases available only in Minute Quantities
- ART. 340 Note as to the Application of the Principle of Dynamical Similarity
- ART. 341 The Principle of Dynamical Similarity in Reference to the Results of Experiments on the Resistance of Square Plates Normal to a Current of Air
- ART. 342 Note on the Regularity of Structure of Actual Crystals
- ART. 343 Colours of Sea and Sky
- ART. 344 The Incidence of Light upon a Transparent Sphere of Dimensions comparable with the Wave-Length
- ART. 345 On Colour Vision at the ends of the Spectrum
- ART. 346 Aerial Plane Waves of Finite Amplitude
- ART. 347 Note on the Finite Vibrations of a System about a Configuration of Equilibrium
- ART. 348 The Problem of the Whispering Gallery
- ART. 349 On the Sensibility of the Eye to Variations of Wave-Length in the Yellow Region of the Spectrum
ART. 347 - Note on the Finite Vibrations of a System about a Configuration of Equilibrium
Published online by Cambridge University Press: 05 December 2011
Book contents
- Frontmatter
- Contents
- ERRATA
- ART. 273 Interference of Sound
- ART. 274 Some General Theorems concerning Forced Vibrations and Resonance
- ART. 275 On the Law of the Pressure of Gases between 75 and 150 millimetres of Mercury
- ART. 276 On the Pressure of Vibrations
- ART. 277 On the Question of Hydrogen in the Atmosphere
- ART. 278 Does Chemical Transformation influence Weight?
- ART. 279 Is Rotatory Polarization influenced by the Earth's Motion?
- ART. 280 Does Motion through the Æther cause Double Refraction?
- ART. 281 On the Distillation of Binary Mixtures
- ART. 282 Note on the Theory of the Fortnightly Tide
- ART. 283 On the Free Vibrations of Systems affected with Small Rotatory Terms
- ART. 284 On the Vibrations of a Rectangular Sheet of Rotating Liquid
- ART. 285 On the Spectrum of an Irregular Disturbance
- ART. 286 Considerations respecting the Combustion of Modern Propellants in Closed Vessels and in Guns
- ART. 287 On the Bending of Waves round a Spherical Obstacle
- ART. 288 On the Proportion of Argon in the Vapour rising from Liquid Air
- ART. 289 On the Theory of Optical Images, with Special Reference to the Microscope
- ART. 290 On the Production and Distribution of Sound
- ART. 291 On the Work done by Forces operative at one or more Points of an Elastic Solid
- ART. 292 On the Acoustic Shadow of a Sphere
- ART. 293 Shadows
- ART. 294 Sir George Gabriel Stokes, Bart., 1819—1903
- ART. 295 On the Measurement of certain very short Intervals of Time
- ART. 296 Note on the Application of Poisson's Formula to Discontinuous Disturbances
- ART. 297 Fluid Friction on Even Surfaces
- ART. 298 On the Electrical Vibrations associated with thin terminated Conducting Rods
- ART. 299 On the Density of Nitrous Oxide
- ART. 300 Note to a Paper by Prof. Wood on the Achromatization of approximately Monochromatic Interference Fringes by a highly dispersive Medium
- ART. 301 On the Open Organ-Pipe Problem in Two Dimensions
- ART. 302 Extracts from Nobel Lecture
- ART. 303 On the Compressibility of Gases between One Atmosphere and Half an Atmosphere of Pressure
- ART. 304 On the Pressure of Gases and the Equation of Virial
- ART. 305 The Dynamical Theory of Gases and of Radiation
- ART. 306 An Optical Paradox
- ART. 307 The Problem of the Random Walk
- ART. 308 On the Influence of Collisions and of the Motion of Molecules in the Line of Sight, upon the Constitution of a Spectrum Line
- ART. 309 On the Momentum and Pressure of Gaseous Vibrations, and on the Connexion with the Virial Theorem
- ART. 310 The Origin of the Prismatic Colours
- ART. 311 On the Constitution of Natural Radiation
- ART. 312 On an Instrument for compounding Vibrations, with Application to the drawing of Curves such as might represent White Light
- ART. 313 On Electrical Vibrations and the Constitution of the Atom
- ART. 314 On the Production of Vibrations by Forces of Relatively Long Duration, with Application to the Theory of Collisions
- ART. 315 On the Dilatational Stability of the Earth
- ART. 316 Some Measurements of Wave-Lengths with a Modified Apparatus
- ART. 317 On the Experimental Determination of the Ratio of the Electrical Units
- ART. 318 On the Interference-Rings, described by Haidinger, observable by means of Plates whose Surfaces are absolutely Parallel
- ART. 319 On our Preception of Sound Direction
- ART. 320 Acoustical Notes
- ART. 321 On the Passage of Sound through Narrow Slits Appendix
- ART. 322 On the Dynamical Theory of Gratings
- ART. 323 Note on the remarkable case of Diffraction Spectra described by Prof. Wood
- ART. 324 On the Light dispersed from Fine Lines ruled upon Reflecting Surfaces or transmitted by very Narrow Slits
- ART. 325 On the Relation of the Sensitiveness of the Ear to Pitch, investigated by a New Method
- ART. 326 Effect of a Prism on Newton's Rings
- ART. 327 Further Measurements of Wave-Lengths, and Miscellaneous Notes on Fabry and Perot's Apparatus
- ART. 328 On the Aberration of Sloped Lenses and on their Adaptation to Telescopes of Unequal Magnifying Power in Perpendicular Directions
- ART. 329 Hamilton's Principle and the Five Aberrations of von Seidel
- ART. 330 Vortices in Oscillating Liquid
- ART. 331 Acoustical Notes.—VIII
- ART. 332 On Reflexion from Glass at the Polarizing Angle
- ART. 333 Note on Tidal Bores
- ART. 334 Notes concerning Tidal Oscillations upon a Rotating Globe
- ART. 335 On the Instantaneous Propagation of Disturbance in a Dispersive Medium, exemplified by Waves on Water Deep and Shallow
- ART. 336 On the Resistance due to Obliquely Moving Waves and its Dependence upon the Particular Form of the Fore-part of a Ship
- ART. 337 On the Perception of the Direction of Sound
- ART. 338 The Theory of Crookes's Radiometer
- ART. 339 To determine the Refractivity of Gases available only in Minute Quantities
- ART. 340 Note as to the Application of the Principle of Dynamical Similarity
- ART. 341 The Principle of Dynamical Similarity in Reference to the Results of Experiments on the Resistance of Square Plates Normal to a Current of Air
- ART. 342 Note on the Regularity of Structure of Actual Crystals
- ART. 343 Colours of Sea and Sky
- ART. 344 The Incidence of Light upon a Transparent Sphere of Dimensions comparable with the Wave-Length
- ART. 345 On Colour Vision at the ends of the Spectrum
- ART. 346 Aerial Plane Waves of Finite Amplitude
- ART. 347 Note on the Finite Vibrations of a System about a Configuration of Equilibrium
- ART. 348 The Problem of the Whispering Gallery
- ART. 349 On the Sensibility of the Eye to Variations of Wave-Length in the Yellow Region of the Spectrum
Summary
The theory of the infinitesimal free vibrations of a system, depending on any number of independent coordinates, about a position of stable equilibrium has long been familiar. In my book on the Theory of Sound (2nd ed. Vol. II. p. 480) I have shown how to continue the approximation when the motion can no longer be regarded as extremely small, and the following conclusions were arrived at:—
(a) The solution obtained by this process is periodic, and the frequency is an even function of the amplitude (H1) of the principal term (H1 cos nt).
(b) The Fourier series expressive of each coordinate contains cosines only, without sines, of the multiples of nt. Thus the whole system comes to rest at the same moment of time, e.g. t = 0, and then retraces its course.
(c) The coefficient of cos rnt in the series for any coordinate is of the rth order (at least) in the amplitude (H1) of the principal term. For example, the series of the third approximation, in which higher powers of H13 than are neglected, stop at cos 3nt.
(d) There are as many types of solution as degrees of freedom; but, it need hardly be said, the various solutions are not superposable.
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- Information
- Scientific Papers , pp. 611 - 616Publisher: Cambridge University PressPrint publication year: 2009First published in: 1912