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Unformatted text preview: Phys374, Spring 2008, Prof. Ted Jacobson Department of Physics, University of Maryland Waves on a stretched string A stretched string will vibrate when plucked. If the string is finite it oscil- lates, while if the string is infinite it supports traveling waves. The physical parameters defining the problem are the mass per unit length and tension of the string. Dimensional analysis Exercise a : Show that the combination p / has dimensions of veloc- ity, and that one cannot make a dimensionless quantity using , , and the wavelength . This allows us to infer that the wave speed is independent of wavelength and is proportional to p / . Exercise b : If the string has fixed endpoints separated by a length ` then it can vibrate at a particular set of normal mode frequencies. The lowest frequency must be proportional to some combination of the available constants , , and ` . Find this combination. Wave equation Here we use Newtons law to derive a partial differential equation describing the motion of the string. We suppose the equilibrium configuration of the string lies along the x axis, and we let y ( x, t ) denote the perpendicular displacement of the string from its equilibrium at position x and time t . We assume that the displacement of the string is very small, in the sense that y x 1 (1) which means that the slope of the string is everywhere very small compared to one. Equivalently the angle between the string and the horizontal is small. Since different parts of the string have different motions, we need to apply Newtons law F = m a to each infinitesimal bit of the string separately. To this end, consider the bit of string that runs from x to x + dx . Since we 1 assume the slope is very small, the length of this bit of string is nearly just dx . The correction is of order ( dx...
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This note was uploaded on 12/29/2011 for the course PHYSICS 374 taught by Professor Jacobson during the Fall '10 term at Maryland.
- Fall '10