Solution

Let $x=0$ and $x=l$ be the positions of the left and the right sources, respectively. Then, the superposition of the waves generated by both sources is given by

$$A[\cos(\omega t-k|x|)+\sin(\omega t-k|x-l|)],$$

where $k=\omega/c$ is the wave number and $c$ is the speed of the wave. Clearly, a standing wave is formed for $0\le x\le l$. In order that left traveling wave for $x<0$ vanishes and only right traveling wave for $x>l$ remains, we must have

$$\cos(\omega t+kx)+\sin(\omega t+k(x-l))=0\qquad\Rightarrow\qquad kl=n*2\pi+\frac{\pi}{2},$$

where $n$ is an integer. Consequently, $l=(n+1/4)\lambda$, where $\lambda=2\pi c/\omega$.

On the other hand, in order that right traveling wave for $x>l$ vanishes and only left traveling wave for $x<0$ remains, we must have

$$\cos(\omega t-kx)+\sin(\omega t-k(x-l))=0\qquad\Rightarrow\qquad kl=n*2\pi+\frac{3\pi}{2}.$$

Thus, $l=(n+3/4)\lambda$.