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Harmonic wave equation

  1. Jan 2, 2012 #1
    the harmonic wave equation is given by y(x,t)=Rsin{2[itex]\pi[/itex]/[itex]\lambda[/itex](vx-t)+[itex]\phi[/itex]}
    where R is amplitude
    [itex]\lambda[/itex]is wavelength
    v is velo of wave
    [itex]\phi[/itex] is initial phase.
    Could you please tell as well as explain me what are the parameters x and t where there in units of length and time respectively.
     
  2. jcsd
  3. Jan 2, 2012 #2

    Philip Wood

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    Gold Member

    x is distance from source, and t is time. (vx-t) is therefore dimensionally inhomogeneous and meaningless. It should be (x - vt). Otherwise your equation is correct, assuming you intend (x - vt) to be on the top line.

    If you need to know more, just ask; plenty of people on this forum can help.
     
    Last edited: Jan 2, 2012
  4. Jan 2, 2012 #3
    Hello anigeo, it is a little difficult to puzzle out your exact question, but here is my answer to what I think you mean.

    Strictly your equation is not 'the wave equation' (harmonic or otherwise) but one solution to it.

    The wave equation is a differential equation which connects the variation of distance and time for some property of interest so that the solution functions to this differential equation repeat themselves regularly in space and time. Your equation is such a solution function.
    Don't worry about this if you are not familiar with differential equations.


    Now what you haven't asked is 'what is y?' in your equation.

    y is the displacement of an oscillating particle of the wave. We usually take this in a direction perpendicular to the distance from some reference point and measure distance as the x variable.

    Remembering that x and t are independent variables, each of which affect the value of y we obtain an equation such as yours with

    y(x,t)= a function of x and t.

    In order to handle this your equation tells us that if we keep either x constant and vary t or; t constant and vary x we obtain a sine curve.

    So we can either stand in one place (x constant) and watch the wave particles going up and down in the y direction in a sine wave.

    or

    We can survey the wave at any instant (t constant) and see a sine wave of particle displacements as we look along the x direction.

    does this help?

    Oh and Philip is quite correct to spot your typographical error - it should indeed be (x-vt)
     
    Last edited: Jan 2, 2012
  5. Jan 2, 2012 #4
    ya sorry it was vt-x
     
  6. Jan 2, 2012 #5
    ya thanx.
    but could you please tell me what does x physically mean?the dist of what from the source
     
  7. Jan 2, 2012 #6
    I said nothing about a source, just a reference point.

    Take a rope.
    Tie one end and snake the other end up and down to generate sine waves along the rope.

    What is happening?

    At any instant you can see a series of peaks and troughs along the rope. x is distance along the rope, measured say from one end (either will do). The important point is that you are measuring all the way down the rope at the same instant.
    Of course the wavelength is the distance between two successive peaks.

    Similarly as the peaks travel down the rope, if we watch one particular point (say the middle) we can watch the peak approaching, passing and changing to a trough. That is at the point x=xhalfway the rope particles go up and down as a sine wave as time passes.

    Edit
    In this case the period is the distance along the time axis between successive peaks. That is the time interval between successive peaks passing out given point. Of course the frequency is the reciprocal of the period.

    If you like a 'wave' is actually two sine curves, one on a distance axis and one on a time axis.
     
    Last edited: Jan 2, 2012
  8. Jan 2, 2012 #7

    Philip Wood

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    Studiot has considered the important cases of the 'snapshot': the variation of y with x at one instant (one particular time), and the single particle oscillation: the variation of y with t for a particular value of x.

    One more instructive case is the value of y for particles at different distances, x, from the reference point, such that x = vt. So we consider particles with successively different x-values. Since (x-vt) is always zero for particles selected in this way, the argument of the sine is constant at points whose x value is given by x=vt. So points of constant phase move according to x=vt. So v is the (phase) velocity of the wave.
     
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