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  1. #1
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    Amplitude of SINE curve

    This is not really a puzzle per se, but a puzzle of mine needing some clarification.

    I know that given y = SIN(x), the amplitude of the SINE curve is 1. I also know that y=a SIN(x) has an amplitude of a [so, y = 3 SIN(x) has an amplitude of 3].

    What is the amplitude of: y = (x^2) SIN(x) ???

    Graphing this, the curve appears to be all over the place because of the x^2. So, I'm puzzled as to how one would actually determine where the amplitude (on the y-axis) is FIRST reached on the x-axis.

    Can someone clarify this? I can't find my old dusty trig book.

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  3. #2
    Silver Lounger mrjimphelps's Avatar
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    What's Trig?

    It's been a LONG time since I've done anything like the above. Back in the day, I could have solve this in my sleep.

  4. #3
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    I used to be able to do this easily also...groan! Maybe a young Maud will step in or someone else that's young or youthful and awake.

  5. #4
    Silver Lounger mrjimphelps's Avatar
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    I'm surprised Maud hasn't already chimed in!

  6. #5
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    I suspect Maud is trying to find a trig book.

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  8. #6
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    It is easier to explain if x-axis is time, t.
    So, SIN(t) is just a specific time dependent function, which repeats itself every 2pi interval. (2pi=2x3.14159=6.2832). SIN(t) peaks at the time tick of 0.5pi (0.5pi=1.5708).
    The magnitude of the generic SIN(t) is normalized to 1. Or A=1 in A*SIN(t).

    The Amplitude A can also be time dependent, a function of time, f(t), such as your
    y = (x^2) SIN(x), here rewritten as time dependent, to
    y = t^2 SIN(t).
    Think of it as a sine wave moving with time. Then a person also adjusts the amplitude knob based on time on his watch.

    To know the y wave/function, set t=0, find y. Set x=0.2, find y. And so on. Plot the points and you'll see the y waveform.

    By just looking at y=t^2 SIN(t), you can estimate that y is very small if t is smaller than 1. (Squaring a less-than-1 number makes it even smaller). When t is above 1, squaring a greater-than-1 number makes it bigger, and bigger even faster when t is getting much bigger than 1.
    That is, initially y is nearly inert near zero. When t approaches t=1 and t>1, it starts to grow, because of the squaring of a number great-than-1.
    Because the amplitude is squared, the amplitude is always positive (e.g. -1*-1=|1|). Time t can be negative but the amplitude is always positive.
    Just glancing at the equation, seems y is symmetrical around the Y-axis at x=zero. That is, if folded along Y-axis at x=0 it shows Y is symmetrical. Or, Y is same on both positive and negative x-axis.

    An equation simply describes a system's behavior. It does not have starts and ends (unless specified). So, time, t, can be negative in the past, or positive into the future. It is interesting to note that physics equations also have this behavior. That is, unless specified, the equation is valid in the past, and future, as well! That is why time travel could be possible.

    To find the system response, in your example, simply choose a specified time. The equation will give you the correct value for y. Finding many points and you can plot them as graph, and visualize the behavior of y.

  9. #7
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    What I think is that since there is an x^2 in front of the sin(x) function, the result has a maximum that's WAAAY up there on the y-axis. The max is dependent on the x^2 I think.

    I don't think this is symmetrical about the y-axis (see attached).

    graph.jpg
    Last edited by kweaver; 2013-09-17 at 22:34.

  10. #8
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    My bad.
    I forgot that sine fct itself can go negative itself as well. If it were [sin(t)]^2 it would be always positive. So, it should not be symmetrical. It goes to show plotting graph is far better than just observing/guessing.
    By looking at your graph, it seems a flipped image (flip one side vertically).

  11. #9
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    The equation is a blow-up function. It'll grow and grow as x increases. Yes, the amplitude value (being squared) dominates after x>1.
    There is no one single 'maxima' in this function. As x grows, higher and higher peak values are generated, one after the next.

    To find all the maximas, take the derivatives of the function and set the derivative to zero.

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