The Principle of Superposition for Waves — Question 8

Question 8

Two pulse supports determine the complete overlap interval

Two compact pulses have centers initially at $-d$ and $+d$, and move toward each other at equal speed $v$. Their half-widths are $a$ and $b$, with $2d>a+b$.

1. Find the first and last times their supports overlap.

2. For $d=3\,\mathrm{m}$, $a=.40\,\mathrm{m}$, $b=.60\,\mathrm{m}$ and $v=2\,\mathrm{m/s}$, find those times. Does overlap imply permanent alteration of either pulse in the linear model?

Question 8 - Solution

Two pulse supports determine the complete overlap interval

Method and mathematical working

Overlap starts when center separation equals the sum of half-widths and ends when the centers have crossed by that same separation. Keep the relative speed $2v$.$$\begin{aligned}|2d-2vt|&\le a+b,\\\frac{2d-a-b}{2v}&\le t\le\frac{2d+a+b}{2v}.\\[4pt]t_{\mathrm{first}}&=\frac{6-0.4-0.6}{4}=1.25\,\mathrm s,\\t_{\mathrm{last}}&=\frac{6+0.4+0.6}{4}=1.75\,\mathrm s.\end{aligned}$$

(a) Calculation and conclusion

The center separation is $|2d-2vt|$. Supports overlap when this is at most $a+b$, so $t_{\mathrm{first}}=(2d-a-b)/(2v)$ and $t_{\mathrm{last}}=(2d+a+b)/(2v)$.

(b) Interpretation and limits

The interval is $1.25\le t\le1.75\,\mathrm{s}$. During overlap the observed displacement is the sum. In the ideal linear uniform medium the component pulses continue translating unchanged afterward; the temporary combined shape is not a permanent collision product.

Physical check

The overlap duration is $(a+b)/v$ and does not depend on the initial center separation, provided the pulses start apart. The initial separation determines when the interval begins.