Wave speed is set by the medium, not the source

01
The tempting model
Assuming shaking a rope faster makes the wave travel faster.
02
Why it feels right
More vigorous shaking feels like it should send the wave faster.
03
Where it breaks · how to catch it
You changed v when only the frequency changed.
04
The correct model
v = √(T/μ) for a string — fixed by tension and density. Raising frequency shortens wavelength (v = fλ), not the speed.
05
Repair · see it, then test it
worked example You shake a rope twice as fast. What happens to wave speed?
Tempting move
It doubles.
Correct move
Speed is unchanged; wavelength halves.
Test the repair SHM-02 · transfer

The tension in a string is quadrupled while its linear density is unchanged. The wave speed changes by a factor of:
1 2 4 16

Make a fresh prediction before referring to the worked example.