A treatise on dynamics. Containing a considerable collection of mechanical problems / By William Whewell.

  • William Whewell
Date:
1823
Catalogue details

Licence: Public Domain Mark

Credit: A treatise on dynamics. Containing a considerable collection of mechanical problems / By William Whewell. Source: Wellcome Collection.

    419/444 (page 383)
    twice as fast as another A, the note produced by a is said to be an octave above that produced by A. If a string e vibrate three times, while a vibrates twice, the note of e is a Jifth above that of a; and so on. And notes where numbers of vibrations are to each other in ratios so simple as these, are found when combined, to be agreeable to the ear. It appears from what has been said, that a string may vibrate so that no point of it is at rest, except its two extremities, and thus give the fundamental note; or it may vibrate so that its middle point is at rest, in which case it will produce the octave to the fundamental note. Or it may vibrate so as to have two points at rest, dividing it into three equal parts, and it will then give the fifth above the octave; and so on. It appears by experiment, that a wire which performs 240 vi¬ brations in a second, sounds the note called C in the middle of the scale of a harpsichord. It has been proposed however to con¬ sider 256, or 28 as the number of vibrations determining this note, in which case every other note C would also have a power of 2 for the number of its vibrations. Hence, we may find the note pro¬ duced by any string; by finding the number of its vibrations, and comparing them with the number just mentioned. Prob. II. To find the form of a string, that it may oscillate symmetrically, that is, that all its points may cbme to the axis at the same time. In this case, the force which accelerates each particle, must be as its distance from the axis; that is, as y. Hence, taking the expression for the force obtained in p. 378, we have Fag cty ~W~'d? — my, m being a constant quantity. d2y Wm or ~T® + -— dx -C ag y = 0. Of this the integral is, if W m Fag y — C sin. kx + C cos. kx; ... & - i dx = kC cos. kx — kC' sin. kx.