Given that this wave was sampled at a sampling frequency f:


Why does the wave sampled at a sampling frequency 3f/2 look like this?


What does 3f/2 mean? Does it mean that we sample every 2 waves 3 times?


The wave is the whole curve: you shouldn't think of your first diagram as showing "four waves" as if you were counting waves from the sea hitting the beach. Although your graphs aren't labelled, the horizontal axis is likely intended to be time, not distance, and the graph shows the oscillation of a single particle over time. So the dashed line shows the movement of the particle over time: think of it as moving up and down.

The sampling frequency is the number of times per second that you measure the particle's position. In the first diagram, the measurements all happen when the particle is in the same place so sampling at that particular frequency causes you to believe that the particle isn't moving at all (the orange line). In the second diagram, we're sampling $3/2=1.5$ times more often so, now we see the particle in different positions and, in particular, we see that it's not stationary.

Note that the two diagrams are drawn at different scales, which isn't very helpful.


Both $f$ and $3f/2$ are frequencies, measured in units of $\mathit{time}^{-1}$. Sampling at frequency $f$ means sampling every $1/f$ time units. If you're sampling with frequency $3f/2$, you're sampling $3/2$ times more than when sampling with frequency $f$ for the same time duration.


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