$4\mathrm{.}21.$ For a triangular pulse of duration $t_1,$ show that when $\frac{t_1}{}=\frac{1}{2},$ the peak response occurs at

$t=t_1,$ which can be established from the equation

$2cos(\frac{2t_1}{})(\frac{t_p}{t_1}-0\mathrm{.}5)-cos2\frac{t_1}{}(\frac{t_p}{t_1}-1)-cos(\frac{2t_1}{})\frac{t_p}{t_1}=0$

found by differentiating the equation for the displacement for $t>t_1.$ The response spec$-$

trum for the triangular pulse is shown in Fig. P$4\mathrm{.}21.$