Continuous Systems, Applied Forces, Interacting Systems

Abstract

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.