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absolute pitch amplitude application approximately arbitrary axis boundary calculated circle of latitude circuit circular clamped coefficients coil component condition considered constant constraint corresponding cosh curvature curve cylinder denote density depends determined diameters differential equation displacement effect equal equal temperament equation of motion equilibrium expressed finite forced vibrations fork free vibrations frequency friction give given gravest tone harmonic increases independent induction inertia infinite initial integral interval kinetic energy length load longitudinal membrane modes of vibration mutual induction negative nodal lines nodal system nodes normal co-ordinates normal functions obtained octave period phase Phil pitch plane plate positive potential energy principal curvature problem proportional quadratic function quantities quarter period radius ratio represented resistance result roots satisfied shews solution sound square string suppose surface symmetrical tension theorem theory tion vanish velocity waves wire Young's modulus zero
Page 272 - ... and m is an abstract number. Hence for a given material and mode of vibration the frequency varies directly as 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 70 - 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 108 - Routh2.] 88. The interpretation of the equations of motion leads to a theorem of considerable importance, which may be thus stated4. 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...
Page 181 - ... (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 475 - 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 (k'V-kV)t-(k'-k)x=0, which shews that the velocity of the group is (KV - kV) -f- (k
Page 198 - L a 4a a which becomes great, but not infinite, when sin — = 0, or the point of application is a node. If the imposed force, or motion, be not expressed by a single harmonic term, it must first be resolved into such. The preceding solution may then be applied to each component separately, and the results added together. The extension to the case of more than one point of application of the impressed forces is also obvious. To obtain the most general solution satisfying the conditions, the expression...
Page 128 - 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...
Page 87 - HARMONIUM^THE methods described depend upon the principle that the absolute frequencies of vibration of two musical notes can be deduced from the interval between them, ie, the ratio of their frequencies, and the number of beats which they occasion in a given time when sounded together. For example, if x and y denote the frequencies of two notes whose interval is an equal temperament major third, we know that y — i '25992 x.
Page 72 - To suppose, as is sometimes done in optical speculations, that a train of simple waves may begin at a given epoch, continue for a certain time involving it may be a large number of periods, and ultimately cease, is a contradiction in terms.
Page 108 - ... essentially real. It is remarkable that both Lagrange and Laplace fell into the error of supposing that equality among roots necessarily implies terms in the solution of the form te xt (or tcospt), and therefore that for stability the roots must be all unequal.
From Google Scholar
GE Backus, JF Gilbert - 1967 - Geophysical Journal International
Stephen O Rice - 1951 - Communications on Pure and Applied Mathematics
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KA O’Donriell, ER Mendez - 1987 - Journal of the Optical Society of America A