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