The Theory of Sound, Vol. II

CONTENTS.

PAGE

of wave-lengths. Relative pitch of various tones. General motion expressible by simple vibrations. Case of uniform initial velocity. Vibrations of gas included between concentric spherical surfaces. Spherical sheet of gas. Investigation of the disturbance produced when plane waves of sound impinge upon a spherical obstacle. Expansion of the velocity-potential of plane waves. Sphere fixed and rigid. Intensity of secondary waves. Primary waves originating in a source at a finite distance. Symmetrical expression for secondary waves. Case of a gaseous obstacle. Equal compressibilities.

CHAPTER XVIII. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

§§ 336–343 .
Problem of a spherical layer of air. Expansion of velocity-potential in Fourier’s series. Differential equation satisfied by each term. Expressed in terms of μ and of ν. Solution for the case of symmetry. Condition to be satisfied when the poles are not sources. Reduction to Legendre’s functions. Conjugate property. Transition from spherical to plane layer. Bessel’s function of zero order. Spherical layer bounded by parallels of latitude. Solution for spherical layer bounded by a small circle. Particular cases solvable by Legendre’s functions. General problem for unsymmetrical motion. Transition to two dimensions. Complete solution for an entire sphere in terms of Laplace’s functions. Expansion of an arbitrary function. Formula of derivation. Corresponding formula in Bessel’s functions for two dimensions. Independent investigation of plane problem. Transverse vibrations in a cylindrical envelope. Case of uniform initial velocity. Sector bounded by radial walls. Application to water waves. Vibrations, not necessarily transverse, within a circular cylinder with plane ends. Complete solution of differential equation without restriction as to absence of polar source. Formula of derivation. Expression of velocity-potential by descending semi-convergent series. Case of purely divergent wave. Stokes’ application to vibrating strings. Importance of sounding-boards. Prevention of lateral motion. Velocity-potential of a linear source. Significance of retardation of ½λ. Problem of plane waves impinging upon a cylindrical obstacle. Fixed and rigid cylinder. Mathematically analogous problem relating to the transverse vibrations of an elastic solid. Application to theory of light. Tyndall’s experiments showing the smallness of the obstruction to sound offered by fabrics, whose pores are open. [Reflection from series of equidistant and parallel sheets.]

CHAPTER XIX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

§§ 344–352 .
Fluid Friction. Nature of viscosity. Coefficient of viscosity. Independent of the density of the gas. Maxwell’s experiments. Comparison of equations of viscous motion with those applicable to an elastic solid. Assumption that a motion of uniform dilatation or contraction is not opposed by viscous forces. Stokes’ expression for dissipation function.