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6 March, 06:15

Two sinusoidal waves are identical except for their phase. When these two waves travel along the same string, for which phase difference will the amplitude of the resultant wave be a maximum?

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  1. 6 March, 06:29
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    zero or 2π is maximum

    Explanation:

    Sine waves can be written

    x₁ = A sin (kx - wt + φ₁)

    x₂ = A sin (kx - wt + φ₂)

    When the wave travels in the same direction

    Xt = x₁ + x₂

    Xt = A [sin (kx-wt + φ₁) + sin (kx-wt + φ₂) ]

    We are going to develop trigonometric functions, let's call

    a = kx + wt

    Xt = A [sin (a + φ₁) + sin (a + φ₂)

    We develop breasts of double angles

    sin (a + φ₁) = sin a cos φ₁ + sin φ₁ cos a

    sin (a + φ₂) = sin a cos φ₂ + sin φ₂ cos a

    Let's make the sum

    sin (a + φ₁) + sin (a + φ₂) = sin a (cos φ₁ + cos φ₂) + cos a (sin φ₁ + sinφ₂)

    to have a maximum of the sine function, the cosine of fi must be maximum

    cos φ₁ + cos φ₂ = 1 + 1 = 2

    the possible values of each phase are

    φ1 = 0, π, 2π

    φ2 = 0, π, 2π,

    so that the phase difference of being zero or 2π is maximum
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