Let $v_1$ and $v_2$ be the speeds of the waves on the two strings, and let $t_1$ and $t_2$ be the times taken for the wave pulses to travel from one end to the other.

The speed of a wave on a string is given by $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension and $\mu$ is the linear mass density.

Therefore, $v_1 = \sqrt{\frac{T}{\mu_1}}$ and $v_2 = \sqrt{\frac{T}{\mu_2}}$.

The time taken for a wave to travel a distance $L$ is given by $t = \frac{L}{v}$.

So, $t_1 = \frac{L}{v_1} = \frac{L}{\sqrt{\frac{T}{\mu_1}}} = L\sqrt{\frac{\mu_1}{T}}$ and $t_2 = \frac{L}{v_2} = \frac{L}{\sqrt{\frac{T}{\mu_2}}} = L\sqrt{\frac{\mu_2}{T}}$.

We want to find the ratio $\frac{t_1}{t_2}$.

$\frac{t_1}{t_2} = \frac{L\sqrt{\frac{\mu_1}{T}}}{L\sqrt{\frac{\mu_2}{T}}} = \sqrt{\frac{\mu_1}{\mu_2}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$.