This was a tricky one and also something that every serious JEE aspirant must be aware of. The velocity of a wave v travelling in a string of tension T and Mass per unit length ? is given by the equation,
V= ?(T/?) .
As can be easily see from the equation, the velocity is completely a function of the properties of the medium and independent of source properties. This implies that any change in source properties won’t affect the wave velocity.
A very common mistake made by students is that they look at the equation v=f x ? and conclude that since the frequency is doubling, the wave velocity must double as there appears to be a direct proportionality between these two quantities.
However, what happens in reality is that when the medium remains the same and frequency changes (due to changes in source properties), the wavelength (?) adjusts itself in a way that its product with the frequency (which gives us the wave velocity) remains unchanged and equals ?(T/?) (as was the case previously). It goes without saying that in this case, when the frequency doubles, the wavelength becomes half keeping the product v constant. Therefore, the velocity remains the same that is, v.