Question

Given the wave function: Y(x,t) = 5sin27(0.2x - 3t); (x = meters, t = sec.): What are the amplitude, frequency, wavelength, angular frequency and phase velocity of the wave? In which direction is it traveling? (Note: this wave is not a light wave traveling in a vacuum so va = 3 x 108 is not valid)

Answer 1

`A=5 m, \omega=81s^(-1), f=12.89s^(-1), \lambda=1.16m, v=4.95(m)/(s)` and the wave is traveling in the positive x-direction, since we have a wave function like this,
`y(x,t) = A sin(kx - \omega t)`

Step-by-step explanation:

A transverse harmonic wave traveling in the x axis is defined by:

`y(x,t) = A sin(kx \pm \omega t \pm \phi)(1)`

Where A is the amplitude, k the wave number.
`\omega` the angular frequency,
`\phi` the phase constant and the
`\pm` of the term
`\omega t` gives us the direction of propagation

Recall that
`\lambda=(2\pi)/(k), f=(\omega)/(2\pi)` and
`v=\lambda f`

Where
`\lambda` is the wavelength,
`f` the frequency and
`v` the phase velocity.

We have:

`Y(x,t) = 5sin27(0.2x - 3t)\\Y(x,t) = 5sin(5.4x - 81t)`

So, you can relate our wave function with the general wave function(1):

`\phi=0\\A=5 m\\k=5.4m^(-1)\\ \omega=81s^(-1)\\ f=(81s^(-1))/(2\pi)=12.89s^(-1)\\ \lambda=(2\pi)/(5.4m^(-1))=1.16m\\v=(1.16m)(12.89s^(-1))=14.95(m)/(s)`

The wave is traveling in the positive x-direction, since we have a wave function like this,
`y(x,t) = A sin(kx - \omega t)`