signal analysis - Bandwidth of a trimmed sinus wave

I generate a sinus which duration is 15ms and frequeuncy of 13KHz. Is its bandwidth is (1/15ms) ? Or in other words, BW = 66.6Hz? I ask that because I have two freqs in my system: 13K & 13.1K. I know that they are seperated by 100Hz. And as the tone I generate have a lower duration I think it spreads into its friend area.

Answer

The spectrum of your windowed sinusoid is just a shifted sinc function (assuming a rectangular window). If $T=0.015s$ is the length of the signal then the magnitude of the spectrum at positive frequencies is given by

$$S(f)=T|\text{sinc}((f-f_0)T)|,\quad f>0$$

with $f_0$ the frequency of the sinusoid, and where I use the definition $\text{sinc}(x)=\sin(\pi x)/(\pi x)$. So with $\Delta f=100Hz$ (the distance to the other sinusoid), you get a relative attenuation of

$$|\text{sinc}(\Delta f\cdot T)|=0.21221\doteq 13.5\,\text{dB}$$

So at the frequency of the neighboring sinusoid, the spectrum of the other (finite length) sinusoid is about $13.5\,\text{dB}$ lower.