Sound of a limited wave after removing main frequency? From my old studies in signals I can remember that "a signal limited in frequency domain is unlimited in time domain" and viceversa (a signal limited in time domain is unlimited in frequency domain).
So, if I take a window over a sinusoidal signal:

The result is that we change the "pure frequency" domain to a domain with other frequencies:

Well, how do "those extra frequencies" sound? So in practice, the limited signal will be some set of frequencies centered around the main frequency. I want to know how does that sound when removing the frequencies nearest to the main frequency but leaving all others.

I'm asking for (approximate) sound of:

You are allowed to shift those frequencies and to normalize the amplitude in order to bring them into an "audible range".
EDIT: I'm not sampling a sinus wave in a limited time, but sampling it in a long time with a limited signal.. in example:

If accidentally this question helps to find a way to syntethize some kind of noise (highly doubt, but), it will be already covered by stackexchange contents license.
 A: This is treated in numerous places on the web. For example, you can find it in here. 
The formula for the Fourier transform of a cut-off sine wave is $$f(\omega) = \frac{2a}{\omega-\omega_0} sin
\frac{(\omega-\omega_0)\,\tau}{2}, $$
where $\omega$ is the frequency of the original sine wave and 
$\tau$ is the widgth of the window.
It basically has a sharp central peak, and the tails are a sine wave decaying inversely with distance from the peak.
So when you remove the central peak, you get a sine wave in the frequency domain that decays with inverse distance from the (removed) peak. I really don't know what this sounds like. 
A: Lets take a step back and look at this logically without using much math or frequency domain abstractions because what you start with is a time domain signal, and you also want your end result to be a time domain signal which you can listen to.
You know what silence sounds like.  It is a pure DC signal of any amplitude in your speaker which may also include the special case of 0 signal amplitude.  It can also be an AC signal(including all harmonics) that resides completely above, or completely below an individual's auditory range.
You know what a sine wave sounds like.  It is a pure AC signal in your speaker.
What you termed the sound of "silence" is the click/pop heard at the beginning or end of a digital audio program without an edge filter in place. It is a well known phenomenon but most work aims to eliminate it, not isolate it. 
You want to isolate only the transitions, the few samples that are clearly not DC, but still too early in the sine period or too late to be recognized as a sine wave when viewed(heard) on its own.  That is what you want to listen to.
It is certainly possible with direct digital synthesis to generate the signal you want to hear.  I would personally use these tools because I am familiar with them and I would like to use knobs to vary frequency, window placement, and phase.  But I am sure there are pure software solutions you can run on your computer.
Where the initial transition ends, and final transition begins is not completely clear.  So my implementation would allow varying those points and listening to several iterations to see if I could notice any difference.
To simplify things we'll look at the single case that involves the inflection points.
Beginning with 
0v DC signal left of t.0, 
then low frequency audible sine from t.0 to t.pi1/2, 
inaudible, 
then low frequency audible sine from t.pi3/4 to t.pi2,
finally 0v DC signal right after t.pi2.  
Or, from ground up to the top of the hill, inaudible, then from bottom of valley up to ground.
The problem is the inaudible portion.  It needs to join the top of the hill to the bottom of the valley.  You cannot just splice the 2 ends together because that would be a discontinuous signal.  You cannot insert a DC level because that would not join the discontinuities, you cannot join the points with a straight line because that would produce an audible signal.
The solution is to substitute the removed central portion of the original sine wave with the similar portion of a subaudible sine wave.
In order to minimize noise from change of slope at the join points, the 2 frequencies should be as close as possible while still being clearly audible/inaudible.  Such as audible 160Hz, inaudible 20Hz.  
Adjust as necessary to suit your personal hearing abilities.  Amplitude of inaudible section and length of original program need to be adjusted to match the join points.  To simplify, the audible frequency should be an even multiple of the inaudible.  For our example, use 8 periods of 160Hz.  remove all but the beginning and ending transition and replace with 1 period of 20Hz.
piecewise function
if t <  0,                                    v = 0
if t >= 0         && t < 0.0015625,           v(t) = sin(2*pi*160)
if t >= 0.0015625 && t < 0.0015625+0.0375,    v(t) = sin(2*pi*20 + pi/16)
if t >= 0.0390625 && t < 0.0390625+0.0015625, v(t) = sin(2*pi*160)
if t > 0.040625,                              v - 0
(I'll work on formating in an edit)
A: Fourier transform is a linear operation. This means that the infinite sinusoidal signal can be written as the sum of the sinus in the window plus the sinus outside the window. If $f(t)$ is your window function this means
$$
\underbrace{\sin(t)}_{g_0(t)} = \underbrace{\sin(t) f(t)}_{g_+(t)} + \underbrace{(1-f(t)) \sin(t)}_{g_-(t)}
$$
or in the fourier domain
$$
g_0(\omega)= g_+(\omega) + g_-(\omega)
$$
What you are asking for is in frequency domain $g_+(\omega)-g_0(\omega)$, which is in the time domain just
$$
g_+(t)-g_0(t) = (f(t)-1)\sin(t) = -g_-(t)
$$
So the "silence" sounds like a simple sine with a pause.
The spectra also look somewhat different than what you have drawn. The finite sinus has a finite power spectrum and no delta peak. The "silence" contains the delta peak.

The other possible way to understand your questions is that we look at the fourier transform of the windowed sinus and apply a filter in the frequency domain to cut out the main finite peak. The resulting spectrum would look like this.
Here I took a broader window function than in the first figure.
The signal in the time domain then looks like this
and sounds like this.
