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What is the resultant amplitude at X?

  1. Jan 18, 2010 #1
    1. The problem statement, all variables and given/known data
    Two wave generators S1 and S2 produce water waves of wavelength 0.5 m. A detector X is placed on the water surface 3 m from S1 and 2 m from S2 as shown in the diagram. Each generator produced a wave of amplitude A at X when operated alone. The generator are operating together and producing waves which have a constant phase difference of [tex]\pi[/tex] radians. What is the resultant amplitude at X?

    a. 0
    b. A/2
    c. A/3
    d. A
    e. 2 A

    untitled-1.png

    2. Relevant equations



    3. The attempt at a solution
    Phase difference [tex]\pi[/tex] means that the two waves are completely out of phase.

    The path difference (p) between the two sources = 1 m = 2 [tex]\lambda[/tex]. If p = n[tex]\lambda[/tex], it should produce constructive interference, but I think we have to consider the phase difference...

    Now I'm confused how to relate these two informations to obtain the answer. Thanks
     
    Last edited: Jan 18, 2010
  2. jcsd
  3. Jan 18, 2010 #2

    ehild

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    Re: wave

    D path difference corresponds to 2pi*D/lambda phase difference, and the total phase difference counts in interference.

    ehild
     
  4. Jan 18, 2010 #3
    Re: wave

    Hi ehild

    Sorry but I don't get it...:frown:

    You mean : phase difference = 2pi*D/lambda ?
     
  5. Jan 18, 2010 #4

    ehild

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    Re: wave

    Lambda path difference is equivalent to 2pi phase difference, 2lambda path difference + the original phase difference between the sources is 4pi + pi phase difference at the meeting point of the waves.

    ehild
     
  6. Jan 19, 2010 #5
    Re: wave

    I think I get it. It will be destructive interference.

    Thanks ehild !! :smile:
     
  7. Jan 19, 2010 #6

    ehild

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    Re: wave

    OK, you got it!!:smile:
     
  8. Jan 19, 2010 #7
    Re: wave

    Hi ehild

    I want to ask another condition. What if the generator are operating together and producing waves which have a constant phase difference of pi/2 radians?

    The phase difference will be 4.5 pi and I think a complete destructive or constructive interference won't occur. How to determine the amplitude at X?

    Thanks
     
    Last edited: Jan 20, 2010
  9. Jan 20, 2010 #8

    ehild

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    Re: wave

    When two waves meet in a point, and one produces a displacement (Displacement can be a lot of things. Change of pressure if it is sound, electric field strength, if it is an electromagnetic wave) y1=A1sin(wt) and the other y2=A2sin(wt+phi) the resultant will be a sin function again with a new amplitude and phase constant.

    y1+y2=Asin(wt+alpha)-->
    A1 sin (wt)+
    A2[sin(wt)cos(phi)+cos(wt)sin(phi)]=A[sin(wt)cos(alpha)+cos(wt)sin(alpha)]

    collect the terms containing sin(wt) and do the same with term containing cos(wt).

    sin(wt)[A1+A2cos(phi)-Acos(alpha)]=cos(wt)[Asin(alpha)-A2sin(phi)]

    The equation should hold for any t. If sin(wt)=0, the right side is also 0, but cos(wt) is different from 0, so

    Asin(alpha)-A2sin(phi)=0

    The same holds for the left side: A1+A2cos(phi)-Acos(alpha)=0

    From this two equations, you get both A (the new amplitude) and alpha( the new phase)

    A sin (alpha)=A2sin(phi)
    Acos(alpha)=A1+A2cos(phi)

    Take the square of both equations and add them up, you get the square of the new amplitude:

    A2=A12+2A1*A2*cos(phi)+A22

    Assume that the amplitudes are equal A1=A2 and the phase difference is phi = pi/4 + 2k*pi, cos(phi)=sin(phi)=1/sqrt(2).

    A=A1*sqrt[2+sqrt(2)]

    ehild
     
  10. Jan 20, 2010 #9
    Re: wave

    Hi ehild
    We take cos(wt) = 0 to get A1+A2cos(phi)-Acos(alpha)=0 ?

    Thanks
     
  11. Jan 20, 2010 #10

    ehild

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    Re: wave

    We can take cos(wt) anything between -1 and +1, the equation must hold. It must hold for cos(wt)=0 (that is, for wt=pi/2+2kpi) One side of the equation is 0, then the other side must be also 0. The right side is the product of sin(wt) and A1+A2cos(phi)-Acos(alpha). One of them must be 0. But it can not be sin(wt), because the sine is either +1 or -1. Therefore A1+A2cos(phi)-Acos(alpha)=0.


    ehild

    ehild
     
  12. Jan 23, 2010 #11
    Re: wave

    Thanks a lot ehild !!
     
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