Anatomy of a harmonic wave

The spatial profile of the harmonic wave from the previous section

$LaTeX: \psi (x,t)=A \cos (k x-\omega t +\phi )$ $\psi (x,t)=A \cos (k x-\omega t +\phi )$

is given by evaluating this at t=0. The following graph shows the relevant quantities associated with harmonic waves. The equation for the plot is:

$LaTeX: \psi (x,t)|_{t=0}=\psi(x)=A \cos (k x+\phi )=f(x)$ $\psi (x,t)|_{t=0}=\psi(x)=A \cos (k x+\phi )=f(x)$

Anatomy of a wave

where

$LaTeX: \boxed{k=\frac{2 \pi }{\lambda }}$ $\boxed{k=\frac{2 \pi }{\lambda }}$

is a positive semi-definite constant known as the propagation number or angular wave number (the wave number is the same constant without the 2π). It is important to note that the product of $LaTeX: k\times x$ $k\times x$ cannot have physical units so k must have units of radians per meters. This is due (at least geometrically) to the fact that sine is the ratio of two legs of a triangle so is unit-less.

The constant appearing in the denominator of the definition of k=2π/λ is the spatial period of the wave, also known as the wavelength. The customary measure of λ (at least for light) is the nanometer. That is 1 nanometer is 1 billionth of a meter or 1 nm = 10^-9m. We often specify the wavelength of the disturbance in micrometers as well. In this case 1 μm = 10^-6 m, or one millionth of a meter.

In a similar fashion, now we discuss the variable ω that we tacitly included in our trial solution. If we were to remain stationary at one spatial position say, x=0, the disturbance would take the form

$LaTeX: \psi (x,t)|_{x=0}=\psi(t) =A \cos (\omega t +\phi )=f(t)$ $\psi (x,t)|_{x=0}=\psi(t) =A \cos (\omega t +\phi )=f(t)$

This would be represented by the motion of the black dot in the animation

The black dot shows harmonic motion

In this case we see that ω must have units of radians per time and is known as the angular temporal frequency of the wave. The relationship between angular frequency and pure frequency is $LaTeX: \omega=2\pi f$ $\omega=2\pi f$ (often $LaTeX: \nu$ $\nu$ is used as the temporal frequency as well but is not used here to avoid confusion with speed v). The temporal period of the wave is then $LaTeX: T=1/f=2\pi / \omega$ $T=1/f=2\pi / \omega$ . If the wave is traveling at some speed in the medium v, then we should have the relationship that

$LaTeX: \lambda =v T=\frac{v}{f}$ $\lambda =v T=\frac{v}{f}$

or written another way

$LaTeX: v=\lambda f$ $v=\lambda f$

i.e., the speed of the wave is the wavelength multiplied by the frequency. Using this expression, we see that the angular temporal frequency is given by

$LaTeX: \boxed{\omega = \frac{2\pi v}{\lambda}}$ $\boxed{\omega = \frac{2\pi v}{\lambda}}$

The two variables that we have neglected explaining, until now are $LaTeX: \phi$ $\phi$ and A. The sine function ranges from -1 to 1 in height so the constant A must be the maximum value of $LaTeX: \psi (x,t)$ $\psi (x,t)$ . This maximum of the disturbance is known as the wave amplitude. The constant $LaTeX: \phi$ $\phi$ simply shifts the wave along the axis and is known as a constant phase shift or initial phase.