The Theory of Sound, Volume 1 |
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A₁ absolute pitch amplitude application approximately arbitrary axis B₁ calculated circle of latitude circuit circular clamped coefficients component condition constant corresponding cos² curvature curve cylinder denote depends determined differential equation displacement dx² effect equal equal temperament equilibrium expressed finite forced vibrations fork free vibrations frequency friction given gravest tone harmonic inductance inertia infinite initial integral kinetic energy length membrane modes negative nodal nodal lines nodes normal co-ordinates normal functions obtained octave period phase pitch plane plate positive potential energy principal curvature quadratic function quantities quarter period ratio represented resistance result roots shews sin² solution sound square string suppose T₁ tension theorem theory vanish velocity waves wire Y₁ Young's modulus zero
Popular passages
Page 273 - K — the radius of gyration of the section about an axis perpendicular to the plane of bending — and inversely as the square of the length. These results might have been anticipated by the argument from dimensions, if it were considered that the frequency is necessarily determined by the value of...
Page 475 - ... and velocities of propagation are so related in each case that there is no change of position relatively to the boat. The mode of composition will be best understood by drawing on paper two sets of parallel and equidistant lines, subject to the above condition, to represent the crests of the component trains.
Page 389 - If there be a small load at any point of the circumference, a slight augmentation of period ensues, which is different according as the loaded point coincides with a node of the normal or of the tangential motion, being greater in the latter case than in the former. The sound produced depends therefore on the place of excitation; in general both tones are heard, and by interference give rise to beats, whose frequency is equal to the difference between the frequencies of the two tones.
Page 71 - TT (n2 — n2) t. In passing through zero the amplitude changes sign, which is equivalent to a change of phase of 180', if the amplitude be regarded as always positive. This change of phase is readily detected by measurement in drawings traced by machines for compounding vibrations, and it is a feature of great importance. If a force of this character act upon a system whose natural frequency is $ (n, + n^), the effect produced is comparatively small.
Page 109 - The period of a conservative system vibrating in a constrained type about a position of stable equilibrium is stationary in value when the type is normal. We might prove this from the original equations of vibration, but it will be more convenient to employ the normal co-ordinates. The constraint, which may be supposed to be of such a character as to leave only one degree of freedom, is represented by taking the quantities <£ in given ratios. If we put ^ = 4,0, $, = AJ,*K ................... (1),...
Page 182 - ... (1) For a given string and a given tension, the time varies as the length. This is the fundamental principle of the monochord, and appears to have been understood by the ancients1. (2) When the length of the string is given, the time varies inversely as the square root of the tension. (3) Strings of the same length and tension vibrate in times, which are proportional to the square roots of the linear density. These important results may all be obtained by the method of dimensions, if it be assumed...
Page 129 - PROPERTY. 129 to deduce the motion depending on initial displacements from that depending on the initial velocities, it is only necessary to differentiate with respect to the time, and to replace the arbitrary constants (or functions) which express the initial velocities by those which express the corresponding initial displacements. Thus, if <£ be any normal co-ordinate satisfying the equation...
Page 72 - Zir(n — m)t (3); which shews that the intermittent vibration in question is equivalent to three simple vibrations of frequencies n, n + m, n — m. This is the explanation of the secondary sounds observed by Mayer.
Page 177 - A, in terms of the initial displacements. 122. We will now investigate independently the partial differential equation governing the transverse motion of a perfectly flexible string, on the suppositions (1) that the magnitude of the tension may be considered constant, (2) that the square of the inclination of any part of the string to its initial direction may be neglected. As before, p denotes the linear density at any point, and T, is the constant tension.
Page 476 - If k' — k, V - V be small, we have a train of waves whose amplitude varies slowly from one point to another between the limits 0 and 2, forming a series of groups separated from one another by regions comparatively free from disturbance. The position at time t of the middle of that group, which was initially at the origin, is given by...