Two vibrating strings of the same material but length $\mathrm{L}$ and $2\mathrm{L}$ have radii $2\mathrm{r}$ and $\mathrm{r}$ respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length $\mathrm{L}$ with frequency ${\mathrm{\upsilon }}_1$ and the other with frequency ${\mathrm{\upsilon }}_2$. The ratio $\frac{{\mathrm{\upsilon }}_1}{{\mathrm{\upsilon }}_2}$ is given by