You would not asked the above question if you have understood the meaning of wave.

Wave is the same change of some physics quantity, however, there is a time delay due to distance in space.

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The source determine the frequency of the wave. The wave propagate with the same frequency.
The same phase will produce the same amplitude. $\phi=\vec{k}\cdot\vec{r}-\omega t$
when wave move from $\vec{r}$ to $\vec{r}+d\vec{r}$, there is a time delay $dt=\frac{\vec{k}\cdot d\vec{r}}{\omega}$.
Since it is always the same change in time (however, different time delay due to distance from the source),
wave always keep the same frequency when propagate. (otherwise, it is not a wave).

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