A sinusoidal wave on a string

Question

A sinusoidal wave on a string travelling in the -x direction has an amplitude of 1cm, wavelength 3cm and frequency 245 Hz. At t=0, the particle of string at x=0 is displaced by the distance D(0,0)=0.8cm above the origin and is moving upward. How can I determine the function of x and t that describes this wave?

Answer 1

The general wave function is f(x,t) = A sin (kx - ωt + Φ), where A is the amplitude, k is wave vector, ω is the angular frequency, and Φ is the phase angle. Let's go through each term in order.

The amplitude is the maximum effect of the wave, which in this case is the displacement of 1cm

The wave vector is defined as 2π / wavelength, so k = 2π / 3 cm = 2π / 0.03 m = 209 radians / m

The angular frequency is defined as 2π ⋅ ⋅ frequency, ω = 2π ⋅ 245 Hz = 1539 radians / s

For finding the phase angle (essentially the starting point of the wave), let's use the values that were given in the last phase of the problem:

f(x,t) = A sin (kx - ωt + Φ)

f(0,0) = 0.8 cm = 1 cm ⋅ sin(0 - 0 + Φ)

0.8 = sin Φ, so Φ = 0.927 radians

Lastly, since it is given that it moves upward as it travels in the -x-direction, we need to include a negative value in the front of our wave vector. Doing a quick graph of the function in terms of x may help you visualize this

Putting it all together:

f(x,t) = -0.01 m ⋅ sin( -209 radians/m ⋅ x - 1539 radians/s ⋅ t + 0.927 radians)