The Principle of Superposition for Waves — Question 4

Question 4

Initial displacement and velocity separate the directional components

Write an infinite-string solution as $y=R(x-vt)+L(x+vt)$. At $t=0$, the measured fields are $h(x)=y(x,0)$ and $g(x)=y_t(x,0)$.

1. Find $R^{\prime}(x)$ and $L^{\prime}(x)$ in terms of $h^{\prime}$ and $g$.

2. If $g=0$ everywhere and the localized fields vanish at infinity, determine the two component profiles. What fraction of the initial displacement travels each way?

Question 4 - Solution

Initial displacement and velocity separate the directional components

Method and mathematical working

Write the initial displacement and initial velocity as two equations for the derivatives of the rightward and leftward profiles. Add and subtract to isolate them.$$\begin{aligned}h^{\prime}&=R^{\prime}+L^{\prime},\qquad g=-vR^{\prime}+vL^{\prime},\\R^{\prime}&=\tfrac12(h^{\prime}-g/v),\quad L^{\prime}=\tfrac12(h^{\prime}+g/v).\\[4pt]g=0&\Rightarrow R^{\prime}=L^{\prime}=h^{\prime}/2,\\h,R,L\to0\ \text{at infinity}&\Rightarrow R=L=h/2.\end{aligned}$$

(a) Calculation and conclusion

$h^{\prime}=R^{\prime}+L^{\prime}$ and $g=-vR^{\prime}+vL^{\prime}$. Solving gives $R^{\prime}=(h^{\prime}-g/v)/2$ and $L^{\prime}=(h^{\prime}+g/v)/2$. Integration constants are fixed by the localization conditions.

(b) Interpretation and limits

For initial rest, $R=L=h/2$, so $y=[h(x-vt)+h(x+vt)]/2$. Half the displacement profile travels each way. This statement concerns amplitude; the total energy must be calculated from the initial and later derivatives rather than assigning energy in direct proportion to displacement.

Physical check

If the initial velocity vanishes, the two profiles have equal derivatives and each contributes half of a localized initial displacement. A pure rightward pulse instead requires $g=-vh^{\prime}$.