displacement along a wave: amplitude, frequency, wavelength, and speed of the wave motion

Question

If the displacement along a wave is given by y=0.05 sin (20πt-0.1πx), where distances are in meters, and times in seconds, what are the amplitude, frequency, wavelength, and speed of the wave motion?

Answer 1

In general, a sine wave of the form y = A*sin(ωt -kx + φ) has amplitude A, frequency f = ω/2π, wavelength λ = 2π/k, and speed v = λf = ω/k. ω is known as the angular frequency, while f is the linear frequency. k is known as the wave number, and it can be thought of as a being a kind of spatial frequency i.e. it tells us how squished together or stretched out the wave crests are. φ is a phase shift, which is zero in this case.

For the given wave, the amplitude is 0.05 meters, the (linear) frequency is f = ω/2π = 20π/2π = 10 Hz, the wavelength is λ = 2π/k = 2π/0.1π = 20 m, and the wave's speed is v = fλ = (10 Hz)(20 m) = 200 m/s. You should check for yourself that we get the same result using v = ω/k. This is known as the "phase velocity" because it refers to how fast the wave crests themselves are moving. This is not necessarily the right speed to think about all of the time; sometimes it's more useful to look at the "group velocity".