Consider a single pulse traveling down an infinitely long string. Assume that at t = 0, th...

Consider a single pulse traveling down an infinitely long string. Assume that at t = 0, the shape of the pulse, or the vertical displacement of the string, is

Analogous to the discussion of Fourier series in Section 3.9, this pulse can be thought of as a superposition of harmonic waves of differing wave numbers k. The infinite sum of Section 3.9, however, that approximates a repetitive function needs to be replaced here by an integral over an infinite number of harmonic waves, each one weighted by an appro­priate amplitude function, that is,

We use cosine functions because y(x) is an even function of x. The amplitude function a(k) is given by

(a) Calculate a(k) using Equation 3.

(b) Substitute a(k) into Equation 2, and show that it yields y(x).

(c) Integrate Equation 2 numerically for values of x ranging from 0 to 3, and show that the results agree with the exact values of y(x). Assume that die speed of the pulse is given by v = ω/k = 1. In such a case, the shape of die pulse is preserved as it travels down the string.

(d) Write down an exact expression for y(x, t) assuming that at t = 0, y(x,0) is given by Equation 1.

(e) Write down the appropriate integral expression for y(x, t) using Equation 2.

(f) Now assume that die pulse is traveling down a “dispersive” string, for which die wave velocity is not a constant but depends on die wave number of the wave. Assume that ω/k = 1 + 0.25 k2. The many waves of differing k that are superimposed to make the traveling pulse change their phase relationship as each moves down die string. Thus, die shape of the pulse changes. To see this effect, modify the integral expression for y(x, t) obtained in part (e) using the “dispersive” value of ω/k given above. Numerically integrate the resulting expression to obtain y(x, t) for t = 2.5,5.0, and 10.0 s. Pick a broad range of x about the location of the peak of the pulse at each of these times.

(g) Plot these resultant waveforms, and compare diem with y(x,0). Comment on the result.