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Math Help - Plz Help , stuck on final exercise !

  1. #1
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    Arrow Plz Help , stuck on final exercise !

    Plz Help with the following math problem ! I really appreciate it ! :

    Two pure notes of the same amplitude and of frequencies 15 Hz. and 17 Hz. sounded together to produce beats. Each note is modeled as a sine wave.
    Quests :

    a - What values should the constants a and b take in the function Y1 = sin(at) + sin(bt) if it is to model the sound of the two notes playing together ?

    b - What values should the constants c and d take in the function Y2 = 2 sin(ct) cos(dt) if it is to model the sound of the two notes playing together ?

    c - Plot both functions Y1 and Y2 on your calculator,choosing a suitable window to show the beats over a period of one second , and then sketch what is on the screen , labeling the axes.

    Thanksss in advance !
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  2. #2
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    Quote Originally Posted by speed01 View Post
    Plz Help with the following math problem ! I really appreciate it ! :

    Two pure notes of the same amplitude and of frequencies 15 Hz. and 17 Hz. sounded together to produce beats. Each note is modeled as a sine wave.
    Quests :

    a - What values should the constants a and b take in the function Y1 = sin(at) + sin(bt) if it is to model the sound of the two notes playing together ?
    a and b represent angular frequencies, so:
    f = 15 Hz = a/(2*(pi)) ==> a = 2*(pi)*15 = 30(pi) rad/s.
    f = 17 Hz = b/(2*(pi)) ==> b = 2*(pi)*17 = 34(pi) rad/s.

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by speed01 View Post
    Plz Help with the following math problem ! I really appreciate it ! :

    Two pure notes of the same amplitude and of frequencies 15 Hz. and 17 Hz. sounded together to produce beats. Each note is modeled as a sine wave.
    Quests :

    b - What values should the constants c and d take in the function Y2 = 2 sin(ct) cos(dt) if it is to model the sound of the two notes playing together ?
    We propose that
    sin(at) + sin(bt) = 2sin(ct)cos(dt)
    for some given values of a, b, c, and d and for all t.

    Now,
    sin(ct)cos(dt) = (1/2)*[sin((c + d)t) + sin((c - d)t)]

    Thus we require:
    sin(at) + sin(bt) = sin((c + d)t) + sin((c - d)t)

    So let
    a = c + d
    b = c - d

    Thus
    c = (1/2)(a + b) = (1/2)(30(pi) + 34(pi)) = 32(pi) rad/s
    d = (1/2)(a - b) = (1/2)(30(pi) - 34(pi)) = -2(pi) rad/s

    -Dan
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