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Constructive Interference Problem in the Time Domain

  1. Apr 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Two waves on a string are given by the following functions:
    Y1 (x,t) = 4cos(20t-x)
    Y2 (x,t) = -4cos(20t+x)
    where x is in centimeters. The waves are said to interfere constructively when their superposition |Ys| = |Y1 + Y2| is a maximum and they interfere destructively when |Ys|
    is a minimum.

    if t = ∏/50 seconds, at what location x is the interference constructive?

    2. Relevant equations
    No particular equation relevant as far as I know.


    3. The attempt at a solution
    So to get a constructive interference, the summation of the two waves(|Y1 + Y2|) must be the largest possible. I plugged in the value for time, and got this simplified equation for Ys:

    Ys = |4[cos(2∏/5 - x) - cos(2∏/5 + x)]|

    Now i know |Ys| must be the largest it can be, and the only way i can think of of approaching this is constructing a X-Y table and seeing if there is a trend in the values, though I feel there must be an easier way to do this.
     
  2. jcsd
  3. Apr 5, 2012 #2

    berkeman

    User Avatar

    Staff: Mentor

    Yes, there is an easier way. If you add two sinusoidal functions that are in phase, what is the maximum amplitude that you can get?

    y = Asin(∅) + Bsin(∅)

    What is the amplitude of y?

    So it's the same situation when you have constructive interference...
     
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