Angular frequency and wave number

The sinusoidal wave in equation (6.2) has a complex amplitude $\tilde{A}=A e^{i \psi}$. Here $A$, the magnitude of $\tilde{A}$ determines the magnitude of the wave. We refer to $\phi(x,t)=\omega t - k x +\psi$ as the phase of the wave, and the wave can be also expressed as

$$\tilde{a}(x,t)=A e^{i \phi(x,t)}$$ (6.3)

If we study the behaviour of the wave at a fixed position $x_1$, we have

$$\tilde{a}(t)=[\tilde{A}e^{-i kx_1}] e^{i \omega t}=\tilde{A}' e^{i \omega t}\,.$$ (6.4)

We see that this is the familiar oscillation (SHO) discussed in detail in Chapter 1. The oscillation has amplitude $\tilde{A}'=[\tilde{A}e^{- i k x_1}]$ which includes an extra constant phase factor. The value of has sinusoidal variations. Starting at $t=0$, the behaviour repeats after a time period $T$ when $\omega T=2 \pi$. We identify $\omega$ as the angular frequency of the wave related to the frequency $\nu$ as

$$\omega=\frac{2 \pi}{T}= 2 \pi \nu \,.$$ (6.5)

We next study the wave as a function of position $x$ at a fixed instant of time $t_1$. We have

$$\tilde{a}(x)=\tilde{A}e^{i \omega t_1} e^{- i k x} = \tilde{A}^{''} e^{- i k x}$$ (6.6)

where we have absorbed the extra phase $e^{i \omega t}$ in the complex amplitude $\tilde{A}^{''}$. This tells us that the spatial variation is also sinusoidal as shown in Figure 6.2. The wavelength $\lambda$ is the distance after which repeats itself. Starting from $x=0$, we see that $a(x)$ repeats when $k x=2 \pi$ which tells us that $k \lambda = 2 \pi$ or

$$k=\frac{2 \pi}{\lambda}$$ (6.7)

where we refer to $k$ as the wave number. We note that the wave number and the angular frequency tell us the rate of change of the phase $\phi(x,t)$ with position and time respectively

$$k=-\frac{\partial \phi}{\partial x} \, \, {\rm and} \, \, \omega =\frac{\partial \phi}{\partial t}$$ (6.8)

Figure 6.2: