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21 February, 09:58

The following four waves are sent along strings with the same

linear densities (x is in meters and t is in seconds). Rank the waves according to (a) their wave speed and (b) the tension in the strings along which they travel, greatest first:

1 $$y_1 = (3 mm) sin (x - 3t) $$, (1) $$y_2 = (6 mm) sin (2x - t) $$, (3) $$y_3 = (1 mm) sin (4x - t) $$, (4) $$y_4 = (2 mm) sin (x - 2t) $$,

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  1. 21 February, 10:27
    0
    y_1 = (3 mm) sin (x - 3t)

    comparing it with standard wave equation

    y = A sin (ωt-kx)

    we see

    ω = - 3, k = - 1

    velocity = ω / k

    = 3

    y_2 = (6 mm) sin (2x - t)

    we see

    ω = - 1, k = - 2

    velocity = ω / k

    =.5

    y_3 = (1 mm) sin (4x - t)

    we see

    ω = - 1, k = - 4

    velocity = ω / k

    =.25

    y_4 = (2 mm) sin (x - 2t)

    we see

    ω = - 2, k = - 1

    velocity = ω / k

    = 2

    So greatest velocity to lowest velocity

    y_1 = (3 mm) sin (x - 3t), y_4 = (2 mm) sin (x - 2t), y_2 = (6 mm) sin (2x - t), y_3 = (1 mm) sin (4x - t)

    b)

    Given the mass per unit length of wire the same, velocity is proportional to

    √ T, where T is tension

    so in respect of tension in the wire same order will exist for highest to lowest tension.
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