Answer
a) $25\;\rm cm$
b) $25\;\rm cm$
Work Step by Step
a) The path length difference for destructive interference for two in-phase waves is given by
$$d= \left(m+\frac{1}{2}\right)\lambda$$
where $m=0,1,2,3,...$,
And for the smallest $d$, $m=0$
Thus
$$d= \frac{1}{2} \lambda$$
where $v=\lambda f$ and hence, $\lambda= v/f$
$$d= \frac{v}{2f} $$
Plugging the known;
$$d= \frac{343}{2(686)}=0.25\;\rm m $$
$$d=\color{red}{\bf 25}\;\rm cm$$
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b) The path length difference for constructive interference for two out-of-phase waves is given by
$$d=\left( m-\frac{1}{2}\right)\lambda$$
where $m=0,1,2,3,...$,
And for the smallest $d$, $m=1$
Thus,
$$d=\dfrac{\lambda}{2}=\dfrac{v}{2f}=\dfrac{343}{2(686)}$$
$$d=\color{red}{\bf 25}\;\rm cm$$
