If you have a string of 2m in length, and the wave speed on the string is 2m/s. and when then string vibrates at fundamental frequency the wavelength of the wave would then be 4m.
However, the sound created by the vibrating string would have a speed of around 340m/s (speed of sound depending on temperature). Since the velocity increased this means either the wavelength or the frequency must change because:
$$v = \lambda f$$
My question is, would the wavelength increase, or the frequency, or both? If you can explain a bit that would be even better.
Best Answer
The frequency of a wave cannot easily change: to do so, you would have to somehow "store the phase difference" between points. What I mean is: if one point is generating 10 wave cycles per second, and another point observes only 5, there must be five wave cycles "not yet observed". This is exactly what happens in the Doppler effect: if you move away from a sound source, there are "waves on their way" that haven't reached you yet, which is why you can observe a different frequency than the one emitted.
But back to your string: if both source and observer are stationary, they must observe the same frequency. And so, if the velocity of the sound wave is different in the string and in the air, then the wave length in the air will also be different.