Two identical traveling waves, moving in the same direction, are out of phase by π/5.0 rad. What is the amplitude of the resultant wave in t

Question

Two identical traveling waves, moving in the same direction, are out of phase by π/5.0 rad. What is the amplitude of the resultant wave in terms of the common amplitude ym of the two combining waves? (Give the answer as the ratio of the total amplitude to the common amplitude.)

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Diễm Thu 5 years 2021-07-20T09:38:18+00:00 1 Answers 83 views 0

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  1. niczorrrr
    0
    2021-07-20T09:39:49+00:00
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    Answer:

    Therefore the amplitude of the resultant wave is =0.95 y_m

    Explanation:

    The equation of wave:

    y=A sin (kx-ωt)

    For wave 1:

    y₁=A sin (kx-ωt) = y_{m}sin (kx-ωt)

    For wave 2:

    y₂=A sin (kx-ωt+Φ) = y_{m}sin (kx-ωt+Φ)

    Where A= amplitude=y_m

    The angular frequency \omega=\frac{2\pi}{T}

    k=\frac{2\pi}{\lambda} , \lambda= wave length.

    t= time

    T= Time period

    \phi = phase difference =  \frac{\pi}{5}

    The resultant wave will be

    y = y₁ + y₂

     =y_m sin (kx-ωt) + y_m sin (kx-ωt+Φ)

     =y_m {sin (kx-ωt) + sin (kx-ωt+Φ)}

     =y_m\ sin(\frac{kx-\omega t +\phi + kx-\omega t }2)\ cos(\frac{kx-\omega t +\phi -kx+\omega t}2)

     =y_m\ sin({kx-\omega t +\frac\phi 2)\ cos(\frac{\phi }2)

    =y_m\ cos(\frac{\phi }2) sin({kx-\omega t +\frac\phi 2)

    Therefore the amplitude of the resultant wave is

    =y_m\ cos(\frac{\phi }2)

    =y_m\ cos(\frac{\pi }{10})

    =0.95 y_m

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