A long string carries a wave; a 7.00-m segment of the string contains five complete wavelengths and has a mass of 180 g. The string vibrates sinusoidally with a frequency of 55.0 Hz and a peak-to-valley displacement of 20.0 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 =

y = 0.1sin (4.5x - 345.6t)

Explanation:

Parameters given:

Length of 5 wavelengths = 7 m

Length of one wavelength, λ = 7/5 = 1.4 m

Frequency of wave, f = 55 Hz

Peak to Valley displacement = 20 cm = 0.2 m

Amplitude is half of the Peak to Valley displacement = 0.1 m

The wave function of a wave traveling in the positive x direction is given generally as:

y = Asin (kx - wt)

Where A = amplitude

k = Wave factor = 2π/λ

x = displacement on the x axis

w = angular frequency = 2πf

t = time taken

=> Wave factor, k = 2π/λ = 2π / (1.4) = 4.5 m^ (-1)

=> Angular frequency, w = 2πf = 2π * 55 = 345.6 Hz

Therefore, inserting all the necessary parameters, we get that the wave function is:

y = 0.1sin (4.5x - 345.6t)