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3 September, 01:46

Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is yR (x, t) = 0.70 m sin⎛ ⎝3.00 m-1 x - 6.28 s-1 t + π/16 rad⎞ ⎠. What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?

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  1. 3 September, 04:57
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    two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is yR (x, t) = 0.70 m sin⎛ ⎝3.00 m-1 x - 6.28 s-1 t + π/16 rad⎞ ⎠. What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?

    ω = 6.28 s - 1,

    k = 3.00 m - 1,

    φ = π rad,

    A R = 2 A cos (φ 2),

    A = 0.37 m

    Explanation:

    y1 (x, t) = A sin (k x - ω t + φ),

    y 2 (x, t) = A sin (k x - ω t).

    from the principle of superposition which states that when two or more waves combine, there resultant wave is the algebriac sum of the individual waves

    y1 (x, t) = A sin (k x - ω t + φ), is generaL form of thw wave eqaution

    A=amplitude

    k=angular wave number

    ω=angular frequency

    φ = phase constant

    k=2π/lambda

    ω=2π/T

    yR (x, t) = 0.70 m sin{3.00 m-1 x - 6.28 s-1 t + π/16 rad} ... *

    two waves superposed to give the above, assuming they are moving in the + x direction

    y1 (x, t) = A sin (k x - ω t + φ), ... 1

    y 2 (x, t) = A sin (k x - ω t) ... 2

    adding the two equation will give

    A sin (k x - ω t + φ) + A sin (k x - ω t) ... 3

    A (sin (k x - ω t + φ) + sin (k x - ω t)), ... 4

    similar to the following trigonometry identity

    sina+sinb=2cos (a-b) / 2sin (a+b) / 2

    let a = (k x - ω t

    b=k x - ω t + φ)

    y (x, t) = 2Acos (φ/2) sin (k x - ω t + φ/2)

    k=3m^-1

    lambda=2π/k=2.09m

    ω=6.28 = T=2π/6.28

    T=1s

    φ/2=π/16

    φ=π/8rad

    amplitude

    2Acos (φ/2) = 0.70 m

    A=0.7/2cos (π/8)

    A=0.37 m
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