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Model a wave equation forced oscillation, A(t)?

Posted 1 month ago
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Tenenbaum, 28D, p339. Forced undamped oscillation.

y[t]''+w0^2 y[t]==F Sin[w1 t+b]
yc==c1 Sin[w0 t+d]
yp==F/(w0^2-w1^2) Sin[w1+b]
(*constants w0,w1,b,d,c1*)

I am told |c|+|F...| is Amplitude(max) (and note is a constant), so that if w0 is near w1, I am told A will be infinite within (one) period. (serway 3rd phy), showing same, says: "due to limitations it will not actually grow infinite"

I am then told (tenenbaum): "if w0 and w1 are equal" (the undet. coeff. solves differently, which i confirmed by doing)

yp==-t F/(2 w0) Cos[w0+b]

Which grows infinite over unbounded t (and no doubt has phy limitations) and A==A(t).

My question is how can minute differences in w0-w1 yield large Amplitude than equality of w0==w1 can (and at that, in a far shorter time). I suspect my "reasoning" here is simply mislead?

My 2nd question is if there are youtube videos or URL showing actual pertinent experiment (or data, but not calculated data) proving the case that small differences are infinitely more powerful than equality - assuming I'm wrong.

I understand F Sin[w0 t+b] (with a w0) may constitute an impossible machine but if is I don't understand it's use in the books. I don't understand the result of a this "machine" causing a wave to be infinite in one period while it has the same frequency unless the problem is ... malformed in practice (a silly machine). If i'm wrong here and machines just such as this are (with limitation) common in electronics, please say so. The use in the books introduce it as a simple equation physically bound, not as any kind of mathematical (unlikely) machine. Which is confusing.

I think I've just read something wrong or am inexperience with electronics scope results. Which is why I ask.

2 Replies

You can explore the issue with a Manipulate:

Clear[y, t, w0, w1];
y[t_, w0_, w1_] /; w0 != w1 = 
  DSolveValue[{y''[t] + w0^2*y[t] == Sin[w1*t], y[0] == 1, 
    y'[0] == 0}, y[t], t];
y[t_, w0_, w1_] /; w0 === w1 = 
  DSolveValue[{y''[t] + w0^2*y[t] == Sin[w0*t], y[0] == 1, 
    y'[0] == 0}, y[t], t];
Manipulate[
 Plot[{y[t, 1, w1], y[t, 1, 1]}, {t, 0, tmax}, 
  PlotLabel -> Row[{"w0 = 1,   w1 = ", w1}]],
 {{w1, 2}, 1, 3}, {{tmax, 10 Pi}, 10 Pi, 100 Pi}]

Bring the slider for w1 to the left and look at what happens to the solution of the nonresonant equation (in blue) compared to the resonant case (in gold). They will overlap better and better. Then enlarge the tmax to see what happens in the longer run.

I think you've been mislead visually.

2 Sin[(2 Pi/4) t +0] + (10/((2 Pi/4)^2 - ((-.0001 + 2 Pi/4)^2))) Sin[(2 Pi/4) t + 0]

2 Sin[(2 Pi/4) t + 0] - (10/(2 Pi/4)) t Cos[(2 Pi/4) t + 0]

try plotting those two, (A==2, w0==2Pi/4, w1 very near w0)

the reason you won't see the same plot? has to do automatic plotting range (that w0 and w1 never approach each other in your plot, and mm adjusts the plot to run them together). as a general rule you should graph them separately and insure nothing strange happens when they combine, because all of strange adjustments can occur.

i'm open to the issue i may be wrong, please correct me

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