4

Give the Fourier coefficients $a_{0}, a_{n},$ and $b_{n}$ of the waveform in Fig. $16.47 .$ Plot the amplitude and phase spectra....

Question

Give the Fourier coefficients $a_{0}, a_{n},$ and $b_{n}$ of the waveform in Fig. $16.47 .$ Plot the amplitude and phase spectra.

Give the Fourier coefficients $a_{0}, a_{n},$ and $b_{n}$ of the waveform in Fig. $16.47 .$ Plot the amplitude and phase spectra.



Answers

Three sinusoidal waves of the same frequency travel along a string in the positive direction of an $x$ axis. Their
amplitudes are $y_{1}, y_{1} / 2,$ and $y_{1} / 3,$ and their phase constants
are $0, \pi / 2,$ and $\pi,$ respectively. What are the (a) amplitude and
(b) phase constant of the resultant wave? (c) Plot the wave
form of the resultant wave at $t=0,$ and discuss its behavior as $t$
increases.

So for this question, basically what we're asked to do is to see to modified frequency and amplitude and see how those two variables changes in those two variables can without on the graph. So here we have the original graph that we're gonna bother five based on and I'll grab. You can see that the original, um, time period, T IHS. I'm here to here, so and it's gonna be a point for second have been very t close to 14 seconds. And we can also see that aptitude this here. So we're gonna say a close to to. So since we have the original crab eaters here, now we can take a look at a MP and see what we can do to change on the graph. So in question A, um, we need to find the frequency. And since we know that t because 2.4 seconds now we can clogging the equation. Ash, you close to one over tea and hook 14 So convincing. One over point for will give us the frequency off 2.5 hertz. So, um, this is different. Quincy, 25 hertz. This the original frequency that's call it off. Not the original frequency on on the graph. And now, since the question is asking us to graph, uh um graph. Ah, Austin. Either That has twice the frequency eschewing autograph. We knew too. Multiply Have not. So that's kind f Hey, is, um it's a new F. It is twice the old frequency have not. So it is twice 2.5. We shall give us five hertz and the new period. T would also change plays down the FAA. So Greece A t a you close to a one over half a, you will give us one know very five hertz, and we'll get 0.2 seconds. So if we go back to the graph, um, are amplitude wouldn't change just by the frequency, But, um, we're gonna have a shorter T. So it's now ISS. It's, um, points two sevens comparing 2.4 this graph, its basic on a shrink horizontally. So if we tried in New York, where I found the original wonder, it's gonna be like this. Oops. It's got a vicious streak again. According to go down like this, that's gonna be new brush for part A. Okay, so now we're gonna be moving into part B, and then you raise off the calculations and here, so we have more room to do. Part B. Um, so part b of the problem is asking us about, um, how the changing amplitude would Rick slapped on a graph? So pretty much we have to sing. Um, t. But the A be changed from two two centimeters, too. Four centimeters. So we're gonna see all part B. They be would be force and meters because it's twice the original amplitude. And, um, this one is pretty easy. We don't even do any calculation. We're just gonna expand it, um, on the wire access direction. So if Majaida graph out like this, it's going to start from four instead of two on the original one. And but the horizontal Alsace is would still be the suit. So 0.1 12 and one Syrian point for, So it's gonna start from four. It was down to one, 23 and Bridges goes back to the original decision at 0.4. So you can see there. How is this stretched? Um vertically comparing to the original graph

We can see that it was too big. Was minus one in equal three and M equals two. They're full If off de ik waas to boss Treaty my ass cause duty Is the guns indictable dealer to buy If dee dee dee So this dickens dear to buy I do Goss three d my ass cause Jody did he So this becomes to upon tree sign treaty, my ass Sign duty a bon too. Did you buy? This is equal doom. Do it on three. Sign three by minus one upon dual sign Dubai minus two. But three Sign Joe minus one upon to Zion, you know. So this becomes zero minus zero minus zero minus. I do. So what? All dance. It is judo. This is the answer to the problem.

Okay, I'm just gonna sketch out what the given graph is gonna look like just so we can see. As you can see, it's oscillating and varying directions and to varying degrees. Therefore, we know now that if we let you equals three t, then we have d t equals 1/3 d U. And this is for you equals three teeth. So this means that we are two times 1/3 from 03 pie of sign you. Do you? Therefore we have 2/3 negative co sign you from 0 to 3 pie giving us. If you plug in the bounds of zero and three pie, you should end up with 1/4. Okay, Now we know that if we're letting you equals two t, so can you page will you equals two teeth and we know that d t to D cheek was to use a DT equals 1/2 d'you. We cannot write this part out as 1/2 the integral from 0 to 2 pi. So we're looking at to pry this time because previously we were doing free pie of sign you d you giving us 1/2 times negative coast on you from 0 to 2 pi, which gives us 1/2 times negative co sign to PI plus co sign zero, which should give us 1/2 francs is negative. One plus one, not a bump. It's 100 times. 1/2 is simply zero. So as you can see therefore we know that the answer to our statement is gonna be 4/3 because we know that two comprise euro sign of three t t t plus from zero pie sign of to t deed. Thio is equivalent to from zero pie Oh, tea DDT and this is equivalent to 4/3.


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