Answer
$\dfrac{1}{3}$
Work Step by Step
We know that the two loudspeakers emit identical in-phase waves.
We also know that the net amplitude is given by
$$A=2a\cos\left( \dfrac{\Delta \phi }{2}\right)$$
where $\Delta \phi=\dfrac{2\pi\Delta x}{\lambda}$ where $\Delta \phi_0=0$
Hence,
$$A=2a\cos\left( \dfrac{ \pi\Delta x}{\lambda}\right)$$
When $A=a$,
$$\color{red}{\bf\not} a=2\color{red}{\bf\not} a\cos\left( \dfrac{ \pi\Delta x}{\lambda}\right)$$
$$ \cos\left( \dfrac{ \pi\Delta x}{\lambda}\right)=\frac{1}{2}$$
$$ \dfrac{ \pi\Delta x}{\lambda}= \cos^{-1}\left(\frac{1}{2}\right)$$
$$ \dfrac{ \Delta x}{\lambda}= \dfrac{\cos^{-1}\left(\frac{1}{2}\right)}{\pi}=\frac{\frac{\pi }{3}}{\pi}$$
$$ \dfrac{ \Delta x}{\lambda}=\color{red}{\bf\frac{1}{3}}$$
