Predicting the future with strings

(Be sure to ask in section if you have difficulties with this problem!)

In this question, ignore reflections from the ends of the strings (assume the ends are at $x = \pm\infty$).

(a)

At t = 0, a string is plucked. This means the string is initially motionless $(v_y(x, 0) = 0)$ and has a given initial shape--shown on Figure 1. Consider the form of the general solution of the wave equation $y(x,t) = f(x - ct) + g(x + ct)$. What must the relationship between $f(u)$ and $g(u)$ be, given that the string is initially motionless?

Figure 1: Plucked string.

(b)

Sketch what the string looks like at $t = d/c$ and $t = 3d/c$.

(c)

At $t=0$, the hammer of a piano hits a piano string centered at point $x = 0$. In a simplified model, the string is perfectly flat at $t=0$, but the hammer has given it an initial velocity distribution $v_y(x, 0)$. Consider the form of the general solution of the wave equation $y(x,t) = f(x - ct) + g(x + ct)$. What must the relationship between $f(u)$ and $g(u)$ be given that the string is initially flat?

(d)

The function $f(u)$ for the piano string is sketched on Figure 2. Use it to sketch the initial velocity distribution $v_y(x, 0)$.

Figure 2: The function $f(u)$ for a piano string.

(e)

Sketch what the piano string looks like at $t = d/c$ and at $t = 3d/c$.