Answer:
[tex]2\; {\rm s}[/tex].
Explanation:
The frequency [tex]f[/tex] of a wave is the number of cycles completed in unit time.
In this question, it is given that the frequency of the wave is [tex]f = 0.5\; {\rm Hz} = 0.5\; {\rm s^{-1}}[/tex]. Thus, on average, [tex]0.5[/tex] cycles of this wave are completed in every second.
The period [tex]T[/tex] of a wave is the time it takes to complete one cycle of the wave.
For example, if [tex](\text{number of cycles})[/tex] of cycles are completed in a certain [tex](\text{duration})[/tex], the time it takes to complete each cycle would be:
[tex]\displaystyle T = \frac{(\text{duration})}{(\text{number of cycles})}[/tex].
From the frequency of this wave, [tex]0.5[/tex] cycles are completed on average in [tex]1\; {\rm s}[/tex]. Thus, [tex](\text{duration}) = 1\; {\rm s}[/tex] while [tex](\text{number of cycles}) = 0.5[/tex]. The time it takes to complete each cycle would be:
[tex]\begin{aligned}T &= \frac{(\text{duration})}{(\text{number of cycles})} \\ &= \frac{1\; {\rm s}}{0.5} \\ &= 2\; {\rm s}\end{aligned}[/tex].
In general, the period of a wave is the reciprocal of frequency:
[tex]\begin{aligned} T = \frac{1}{f}\end{aligned}[/tex].
For the wave in this question:
[tex]\begin{aligned} T &= \frac{1}{f} \\ &= \frac{1}{0.5\; {\rm Hz}} \\ &= \frac{1}{0.5\; {\rm s^{-1}}} \\ &= 2\; {\rm s}\end{aligned}[/tex].
