A listener is stationary with respect to a source of sound. A wind...

Let $v$ be the velocity of sound in still air, $v_w$ be the velocity of wind, and $f_0$ be the frequency of the source.

Since the source is stationary, the effective velocity of sound relative to the ground becomes $v' = v \pm v_w$.

The wavelength of the sound waves is given by $\lambda' = \dfrac{v'}{f_0} = \dfrac{v \pm v_w}{f_0}$. Thus, the wavelength changes.

The listener is also stationary, so the speed of sound relative to the listener is $v'$.

The frequency observed by the listener is $f' = \dfrac{v'}{\lambda'} = \dfrac{v \pm v_w}{\dfrac{v \pm v_w}{f_0}} = f_0$.

Thus, the frequency remains unchanged.

Since the time period is the reciprocal of frequency ($T = \dfrac{1}{f}$), the time period also remains unchanged.