Continuous Systems, Applied Forces, Interacting Systems
Having discussed the behavior of a homogeneous, undamped, undriven string, subject to no external forces except those that hold it fixed at certain points, let us now study the effect of a discontinuity of the string’s parameters. We suppose that the string at x < 0 has tension J and mass density µ, and this string is attached at x = 0 to another which extends to x = ∞, having tension J′, and mass density µ′. There is no practical difficulty in the way of joining strings with different mass densities; however, the difference of tensions will require a force in the x-direction applied at the junction in order to compensate for the net F x that would otherwise be acting at that point. We do not claim that such a mechanism would be easy to design and apply frictionlessly to the junction of two strings (perhaps a frictionless rod parallel to the y-axis and piercing the string at the junction) but we ask the reader’s indulgence because such discontinuities of media commonly affect waves of other types, and we wish to simulate such situations in our study of waves on a string.
KeywordsMass Density Apply Force Interact System Force Density Series Capacitance
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- 1.When h = 0, the co —integral fort t’ reduces to an integral around a pair of branch points connected by a cut,i.e., a segment that one may not cross. This integral in turn reduces to the integral along this segment of the difference between the values assumed by the integrand on the two sides of the cut. This integral involves functions that we do not have time and space to deal with here. On the general subject of integrals of multiple-valued functions, see the works previously cited by Churchill, R.V., J.W. Brown, and R.F. Verhey. Complex Variables and Applications. 3d ed. New York: McGraw Hill, 1974; also Franklin, P. Functions of Complex Variables. Englewood Cliffs, NJ: Prentice-Hall, 1947; and Smith, L.P.Mathematical Methods for Scientists and Engineers. Englewood Cliffs, NJ: Prentice Hall, Dover Reprint, 1961.Google Scholar
- 2.For a calculation of L and C per unit length of a parallel-wire line, and also a calculation of wave propagation guided by such wires, see: Jackson, J.D.,Classical Electrodynamics. 2d ed. New York: Wiley, 1975.Google Scholar