Question

A sinusoidal wave traveling on a string in the negative x direction has amplitude 1.00 cm, wavelength 3.00 cm, and frequency 245 Hz. At t = 0, the particle of string at x = 0 is displaced a distance D = 0.80 cm above the origin and is moving upward. sketch the shape of the wave at t = 0 and determine the function of x and t that describe the wave.

Answer 1

Answer:

         y = 0.010 cos (  1539.4 t + 0.645)        

Explanation:

The general equation for a wave on a string is

           y = A sin (kx - wt)

Where k is the wave number

           k = 2π /λ

           k = 2π / 3 10⁻²

           k = 209.4 m⁻¹

The angular velocity is related to the frequency

          w = 2π f

          w = 2π 245

          w = 1539.4 rad / s

The maximum amplitude is

           .A = 0.0100 m

The wave equation is

            y = 0.01 sin (-209.4 x - 1539.4 t)

The particle that is at each point of this wave performs an oscillatory movement perpendicular to the wave discarded by

         y = yo cos (wt + fi)

The angular velocity of the particle is equal to the velocity of the wave

        w = 1539.4 rad / s

The breadth of displacement is

         y₀ = 0.01 m

With the initial data let's look for the phase

         0.008 = 0.01 cos (0+ Ф)

        Ф = cos⁻¹ (0.008 / 0.01)

        Ф = 0.645 rad

Therefore the equation is

            y = 0.010 cos (  1539.4 t + 0.645)