Harmonic Traveling Waves

A harmonic traveling wave is a traveling wave where spatial disturbance is sinusoidal in time. Think of not just one pulse but a continuous stream of waves that create a sinusoidal appearance.

Wavelength and Wavenumber

Wavelength $\lambda$ is the length of one wave (peak to peak). Wavenumber $k$ is the radians per unit length. The following are some relations.

\[k = \frac{2\pi}{\lambda}\] \[v = \frac{\omega}{k} = \frac{\lambda}{T} = \lambda f\]

Equation

\[y(x, t) = A\cos(k(x\mp vt)+\phi)\] \[y(x, t) = A\cos(kx\mp\omega t+\phi)\]

If we fix position and graph the y motion of a particle over time, we will get a sinusoidal graph with the same amplitude $A$ and angular frequency $\omega$.

If we take a snapshot in time, we get the same amplitude $A$ and wavelength $\lambda$.