Wave motion is characterized by two velocities: the velocity with which the wave moves in the medium (e.g., air or a string) and the velocit

Question

Wave motion is characterized by two velocities: the velocity with which the wave moves in the medium (e.g., air or a string) and the velocity of the medium (the air or the string itself). Consider a transverse wave traveling in a string. The mathematical form of the wave is y(x,t)=Asin(kx−ωt).

a)Find the velocity of propagation v_p of this wave.

Express the velocity of propagation in terms of some or all of the variables A, k, and omega.

b)Find the y velocity v_y(x,t) of a point on the string as a function of x and t.

Express the y velocity in terms of omega, A, k, x, and t.

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Philomena 1 month 2021-08-06T05:55:35+00:00 1 Answers 0 views 0

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    2021-08-06T05:57:01+00:00

    Answer: a) v = ω /k, b) v = – ωAcos( kx −ωt)

    Explanation:

    y(x,t)=Asin(kx−ωt) defines the wave equation.

    a)

    We are asked to find wave speed (v)

    Recall that v = fλ

    From the wave equation above,

    k = 2π/ λ where k is the wave number and λ is the wavelength, λ = 2π /k

    ω = 2πf where f is the frequency and ω is the angular frequency.

    f = ω/ 2π.

    By substituting for λ and ω into the wave speed formulae, we have that

    v =( ω/ 2π) × (2π /k)

    v = ω/k

    b)

    y(x,t)=Asin(kx−ωt)

    The first derivative of y with respect to x give the velocity (vy)

    By using chain rule, we have that

    v = dy/dt = A cos( kx −ωt) × (−ω)

    v = – ωAcos( kx −ωt)

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