FFT for a non coherent wave
Oh yes.  This is a known issue.  What you have just discovered is called "spectral leakage". Entire books could be written on this topic. What are you going to want to do is to use a "window" to reduce the effects of spectral leakage. I have some tutorial material on my website that may help you.   

http://mechanicalvibration.com/windowed_DFT.html