A sound wave travels from a region of air at 20^ ^ C into a region...

The speed of sound in an ideal gas is given by $v = \sqrt{\dfrac{\gamma R T}{M}}$, which implies $v \propto \sqrt{T}$, where $T$ is the absolute temperature.

The absolute temperatures are $T_{20} = 20 + 273 = 293 \text{ K}$ and $T_{40} = 40 + 273 = 313 \text{ K}$.

Since the frequency $f$ remains constant and $v = f \lambda$, the wavelength $\lambda$ is directly proportional to the speed of sound $v$.

Therefore, $\lambda \propto \sqrt{T}$.

The ratio of the wavelengths is $\dfrac{\lambda_{40}}{\lambda_{20}} = \sqrt{\dfrac{T_{40}}{T_{20}}} = \sqrt{\dfrac{313}{293}}$.

Answer: $\sqrt{\dfrac{313}{293}}$