📂Physics

Sine Waves and Complex Wave Functions

Definitions

A wave expressed as a sine function is called a sine wave.

Description

The general form of a sine wave is as follows. The reason why it’s referred to as a sine wave even though the equation is $\cos$ is explained below, as the real part of the complex wave function is $\cos$. $\sin$ is the imaginary part.

$$f(x,t) = A \cos \big( k(x-vt)+\delta \big)$$

Here, $A$ is the wave’s amplitude, the variable $k(z-vt)+\delta$ of the cosine function is called the phase, and $\delta$ is the phase constant. Adding $2\pi$ to the phase constant does not change $f(x,t)$. Therefore, in most cases, values within the range of $0\le \delta \lt 2\pi$ are used as the phase constant. $k$ is the wave number and has the following relationship with the wavelength $\lambda$.

$$k=\dfrac{2\pi}{\lambda}$$

The time it takes for a wave to complete one full cycle is called the period. Since time=distance/speed, the period of the wave $T$ is

$$T=\dfrac{\lambda}{v} = \dfrac{2 \pi}{kv}$$

Since the period is the time it takes for one vibration, the frequency $\nu$, which is the number of vibrations per unit time, is naturally the inverse of the period.

$$\nu=\dfrac{1}{T}=\dfrac{v}{\lambda}$$

Angular frequency is commonly denoted as $\omega$ and represents the vibration in terms of uniform circular motion. It’s the angle turned per unit time, measured in radians.

$$\omega=2\pi \nu=2\pi\dfrac{1}{T}=kv$$

If you represent $(1)$ in terms of angular frequency, you get

$$f(x,t)=A \cos \big( kx-\omega t +\delta \big)$$

This is the wave function moving to the right with wave number $k$ and angular frequency $\omega$.

As shown in the figure above, $\dfrac{\delta}{k}$ is defined as the distance the wave function lags behind the origin. Therefore, if the direction of the wave propagation changes, the sign of the phase constant also changes. If the wave moves to the left, it means it lags behind as it moved to the right. Thus, the wave function moving to the left with wave number $k$ and angular frequency $\omega$ is as follows.

$$f(x,t)=A \cos \big( kx+\omega t -\delta \big)$$

However, since the cosine function is an even function, the above equation is the same as the equation below. $$f(x,t)=A \cos \big( -kx-\omega t +\delta \big)$$

This indicates that compared to the right-moving wave function $(2)$ with wave number $k$ and angular frequency $\omega$, only the sign of the wave number $k$ is different. In other words, by changing the sign of the wave number $k$, you get a wave that has the same amplitude, phase constant, frequency, wavelength, etc., but moves in the opposite direction.

Complex Wave Function

Since the wave function can be expressed as a cosine, it can also be represented in the form of a complex exponential function using Euler’s formula. The reason for specifically dealing with the complex wave function, including its imaginary part, is because in many respects, complex functions are more convenient for calculations. Using $e^{ix}=\cos x +i\sin x$ to express $(2)$ results in

$$f(x,t)=\text{Re}(Ae^{i(kx-\omega t +\delta)})$$

Here, $\text{Re}(a+ib)=a$ represents the real part. Since $f$ is the function representing only the real part of $Ae^{i(kx-\omega t +\delta)}$, let’s call it $\tilde{f}=Ae^{i(kx-\omega t +\delta)}$. In other words, $\text{Re}(\tilde{f})=f$. Then, it can be simplified as follows.

$$\tilde{f}(x,t)=Ae^{i(kx-\omega t+\delta)}=Ae^{i\delta}e^{i(kx-\omega t)}=\tilde{A}e^{i(kx-\omega t)}$$

The wave function $f(x,t)$ discussed in this article is the real part of the complex wave function.

$$f(x,t)=\text{Re}\big( \tilde{f}(x,t) \big)$$