Why do standing waves filter out all other frequencies but its natural frequencies I was wondering about how standing waves filter out all other frequencies. I read online that it's because the other frequencies destructively interfere with one another. However as hard as I try, I am unable to picture how they destructively interfere. 
Furthermore, if the frequency of the wave was NOT one of the resonant frequencies, how does it affect the frequency that you hear. i.e. if the string had multiple frequencies on it, what would happen to the frequency that was amplified (is it softer? different?)
Thank you.
 A: The following experiment will shine light on this.

What is happening here? The string has fixed boundary conditions but the left side of the string is driven by a small amplitude oscillation. The driving frequency of each string increases as you go down. The first and last string both are driven by a resonant frequency. Recall that $f=c/\lambda$ with $c$ the propagation speed and $\lambda$ the wavelength of an oscillation with frequency $f$. For resonance to happen we need that the wavelength is a multiple of twice the strings length: $\lambda=2nL$. Indeed the frequency of the upper string is $\frac{c}{2L}$ and for the lowest string it is $\frac c{L}$.
So how does this happen? Imagine a wave starts travelling at the left. Due to the fixed boundary conditions it reflects of the right wall. Generally it will reflect multiple times. When the conditions I mentioned above are met, each of these reflected waves has exactly the same form. They add up constructively and make the amplitude larger. In the animation you see that the amplitude grows after the waves have had the time to reflect a few times.
When the conditions are not met each reflected wave is slightly offset from the other waves. After enough reflections the phase is basically random compared to the newest wave. When you add lots of sine waves that are offset from each other they sum to about zero. If you add some damping you get that the frequency that don't match resonance quickly die out while the resonance frequency survives because each reflection add to the overall amplitude. This is why the sound a guitar produces is mostly overtones. The overtones survive in the string and cause the air to virbrate at those frequencies.
If you are only slightly offset from the resonance frequency you get that each wave still add to the overall amplitude, just slightly less. So you still get a large amplitude just not as large as when you hit perfect resonance.
