Along string carries a wave; a 5.40-m segment of the string contains three complete wavelengths and has a mass of 180 g. the string vibrates sinusoidally with a frequency of 45.0 hz and a peak-to-valley displacement of 16.0 cm. (the "peak-to-valley" distance is the vertical distance from the farthest positive position to the farthest negative position.)
(a) write the function that describes this wave traveling in the positive x direction. (use the following as necessary: x and t. x is in meters and t is in seconds.)

y (x, t) = 8.00 sin (3.49 x - 282.7 t)

Explanation:

The general expression for a sine wave is

     y = A sin (kx -wt)

Where

     k = 2π /λ

     w = 2π f

Let's build this equation with the given data

The amplitude (A) is the distance from the point of zero height to the maximum height (peak), in the problem they give the height between the peak and the valley

    A = 16.0 / 2 =

    A = 8.00 cm

The wave number (k) is related to the wavelength, they give us a segment of 5.40 m has three (3) wavelengths, so

    3 λ = d

    λ = d / 3

    λ = 5.40 m / 3

    λ = 1.80 m

The wave number is

   k = 2π / 1.80

   k = 3.49 m⁻¹

The angular velocity is related to the frequency

  w = 2π f

  w = 2π 45.0

  w = 282.7 rad / s

With this information we can write the equation of the traveling wave

    y (x, t) = 8.00 sin (3.49 x - 282.7 t)

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