Simple Nature - Light and Matter

Our final result for the speed of the pulses isv = w/t√2T=µ.The remarkable feature of this result is that the velocity of thepulses does not depend at all on w or h, i.e., any triangular pulsehas the same speed. It is an experimental fact (and we will alsoprove rigorously below) that any pulse of any kind, triangular orotherwise, travels along the string at the same speed. Of course,after so many approximations we cannot expect to have gotten allthe numerical factors right. The correct result for the speed of thepulses is√Tv = .µThe importance of the above derivation lies in the insight itbrings —that all pulses move with the same speed — rather than inthe details of the numerical result. The reason for our too-high valuefor the velocity is not hard to guess. It comes from the assumptionthat the acceleration was constant, when actually the total force onthe segment would diminish as it flattened out.Treatment using calculusAfter expending considerable effort for an approximate solution,we now display the power of calculus with a rigorous and completelygeneral treatment that is nevertheless much shorter and easier. Letthe flat position of the string define the x axis, so that y measureshow far a point on the string is from equilibrium. The motion ofthe string is characterized by y(x, t), a function of two variables.Knowing that the force on any small segment of string dependson the curvature of the string in that area, and that the secondderivative is a measure of curvature, it is not surprising to find thatthe infinitesimal force dF acting on an infinitesimal segment dx isgiven bydF = T ∂2 y∂x 2 dx .(This can be proved by vector addition of the two infinitesimal forcesacting on either side.) The symbol ∂ stands for a partial derivative,e.g., ∂/∂x means a derivative with respect to x that is evaluatedwhile treating t as a constant. The acceleration is then a = dF/ dm,or, substituting dm = µ dx,∂ 2 y∂t 2 = T µ∂ 2 y∂x 2 .The second derivative with respect to time is related to the secondderivative with respect to position. This is no more than a fancy348 Chapter 6 Waves