When a system is driven by a wave at its resonant frequency, how does
the shape of the wave come into play? For instance, will a square
wave at a resonant frequency have the same effect as a sine wave at
that frequency?
Thank you.
What happens to a signal of which a certain frequency is amplified(say an
ideal resonant "curve")?
If its just a pure sine way then that sine way gets amplified but if its a
square wave, which contains all frequencies, then only some of the
frequencies get amplified.
What happen is that the resonance sorta picks out the frequency of the
signal. This is a sine wave that we get. Its not perfect but as the
resonance becomes sharper we get more of a pure sine wave and less of the
original signal. So think of the signal as "morphing" to a sine wave at the
resonant frequency as the resonance gets stronger.
Mathematically we can think of S(w) as the signal in the frequency domain
and then we are multiplying by a function like exp(-(w-w0)^2/q).
so we get
S(w)*exp(-(w-w0)^2/q)
No matter what S(w) is, as q->0 we get get a dirac(w-w0). When converted
back into the time domain this is just A*sin(w0*t).
What you can think of is that the resonance "shrinks" as a band pass filter.
What happens to a signal when you do this? You remove the lower frequencies
and the higher frequencies. What happens when you do this to a square wave?
Removing the higher frequencies makes it more into a sine like wave...
Better to look at what actually happens:
The fourier series of a square wave is
sum(sin(k*w*t)/k)
(doesn't matter about constants or harmonicity because we just want the
general behavor. i.e., it doesn't matter if k is even or not in the sum
above(using a sawtooth results in the same logic_).
Now what is this in the frequency domain? Its just a sum of impulses with
frequencies kw.
If we bandpass that it means we are removing the lower and higher
frequencies from the sum.
Essentially resulting in something like
sum(sin(kwt)/k,k=K1..K2)
(actually its a convolution with our filter but in the limit it works out to
be something like this)
where K1 and K2 get closer together and center around our resonant
frequency.
Since the signal here contains all frequencies it will pick out one and be
ok(although the signal could be quite complex).
But if you just have a pure sine wave you then might "miss" the sine wave
and not amplify it at.
Basically the point is that all you have to do is think of a band pass for a
resonance curve in this case.
If your interested in seeing how it works then just take a function,
transform it in the frequency domain, filter it and then take it back into
the time domain.
You can see that this type of analysis applies to all functions with
continuous or discrete transforms. Just take your signal's fourier
representation and convolve it with your filter. If you are looking at an
"idea" resonance curve then it simplifies a great deal and essentially picks
out the frequency at resonance. If its not a perfect filter then it picks
out more frequencies around the resonance and so it can get quite
complicated(looking nothing like the original single).
Theoretically if you have a pure sine wave and an ideal resonance curve you
will not get any signal unless the resonance frequency is the frequency of
the sine wave. This also happens in the square save if your resonance
frequency is not an odd multiple the fundamental frequency. In real life
things are rarely ideal and you cannot have an ideal resonance curve.
Anyways...
Jon