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Standing Wave's shape?

  1. Oct 6, 2011 #1
    What is the shape of the string at time=0?

    This is my wave:

    y(x,t)=(0.03)sin( (3π/4)x)cos(200πT)

    I assume that x=vt and replace in sin. Then sin(0)=0 and then the shape of the string is a line.

    Is this correct?
     
  2. jcsd
  3. Oct 7, 2011 #2

    ehild

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    This is a standing wave, a two-variable function of the independent variables x and t. y(x,t) is the displacement from horizontal of a point at x distance from one end. This displacement changes with time as a cosine function, cos(200πt). Substitute t=0 and see how the string looks like.

    ehild
     
  4. Oct 7, 2011 #3
    Cosine of 0 is 1. Then, y(x,t)=Asin(Kx)

    There's no information of x. I thought that the amplitude or displacement was Asin(Kx)

    I don't know, what's the shape of the Standing wave.
     
  5. Oct 7, 2011 #4

    ehild

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    Last edited: Oct 7, 2011
  6. Oct 7, 2011 #5
    So, I guess that at time 0, there's no movement and therefore a line?
     
  7. Oct 7, 2011 #6

    ehild

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    At t=0, the shape of the wave is y(x,0)=0.03sin((3pi/4)x) You know the sine function, do you not? Try to plot the function in terms of x. At what value of x is it zero? Where is maximum and minimum?


    ehild
     
  8. Oct 7, 2011 #7
    I know very little about the sine function, but I've seen it. I tried to plot sine with a changing x from 0 to 5.
    I noticed that Y gives me 0 for every case. Then, I assume that it's always zero and therefore a line.

    Is there a way to know this behavior without having to plot it?
    Thank you.
     
  9. Oct 8, 2011 #8

    ehild

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    How did you get zero for all x? Note that the argument of the sine is in radians. Set your calculator to "rad" .

    If x=1 3pi/4 =2.356. Set your calculator to rad and input sin(2.356). It is 0.707.
    You can also transform the argument to angles. 3pi/4 x =135x degrees. If x=1, you have sin(135).

    The sine function varies between -1 and 1. The amplitude of your wave is 0.03, so it changes between -0.03 and 0.03.

    See this: http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html#sincostan and use the Applets

    ehild
     
    Last edited: Oct 8, 2011
  10. Oct 8, 2011 #9
    I got ( 0.707), but I rounded it up and in some cases I got ( -0. something else), but did the same.

    Why do you infer that x is 1?
    If the wave changes from -0.03 to 0.03, can I say that it is a line or looks like a line?

    Thank you ehild.
     
  11. Oct 8, 2011 #10

    ehild

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    X=1 is just an example to show that y is not zero.
    Values between -0.03 and 0.03 are not too small - with respect to what?? The string is not a straight line, and it does not look a straight line. You do not know the unit of the amplitude. If it is in meters, 0.03m is 3 cm and it is definitely not zero.
    Such standing waves appear in musical instruments. There are standing waves on the string of a violin, for example. You hear the sound even when the amplitude is a few mm.

    ehild
     
  12. Oct 8, 2011 #11
    This is the whole problem:
    •• A 2.00-m-long string fixed at one end and free at the other (the free
    end is fastened to the end of a long, light thread) is vibrating in its third harmonic
    with a maximum amplitude of 3.00 cm and a frequency 100 Hz. (a) Write the
    wave function for this vibration. (b) Write a function for the kinetic energy of a
    segment of the string of length dx, at a point a distance x from the fixed end, as a
    function of time t. At what times is this kinetic energy maximum? What is the
    shape of the string at these times? (c) Find the maximum kinetic energy of the
    string by integrating your expression for Part (b) over the total length of the
    string.


    I forgot to mentioned that the maximum kinetic energy happens at time equal 0. Does this make the wave a line? Why?
     
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