In a standing wave, we have the equation:
$$v = \sqrt{\frac{T}{\mu}}$$
where $T$ is string tension and $\mu$ is linear mass density. By this, if we increase linear mass density, wave velocity will decrease. But how does this affect wavelength? Velocity is the product of wavelength and frequency, but what happens when velocity changes?
Best Answer
A standing wave is set up because you have a source that is creating periodic movements of a certain frequency that creates a wavelength of the form $\frac{2L}{N}$ where N is a natural number.
If the mass density is increased without changing the frequency of the driving source then it is obvious that the wavelength won't be equal to $\frac{2L}{N}$ as a lower velocity will lead to a lower wavelength for the same frequency. So, you won't observe a standing wave unless the change in velocity is such that the new wavelength is also of the form $\frac{2L}{N}$. In this case, you will see a different mode of the standing wave.