Traveling Waves
Sources:
- Fawwaz T. Ulaby & Umberto Ravaioli. (2020). Chapter 1. Introduction: Waves and Phasors. Fundamentals of Applied Electromagnetics (8th ed., pp. 22-30). Pearson.
Traveling Waves
Notation
| Symbol | Type | Description |
|---|---|---|
| \(\lambda\) | \(\mathbb{R}^+\) | Wavelength, the spatial period of a sinusoidal wave (unit: \(\mathrm{m}\)) |
| \(T\) | \(\mathbb{R}^+\) | Temporal period, the time it takes for the wave pattern to repeat (unit: \(\mathrm{s}\)) |
| \(f\) | \(\mathbb{R}^+\) | Frequency, the reciprocal of the temporal period \(f = \frac{1}{T}\) (unit: \(\mathrm{Hz}\)) |
| \(c\) | \(\mathbb{R}^+\) | Speed of light in a vacuum \(c \approx 3 \times 10^8\) (unit: \(\mathrm{m/s}\)) |
| \(u_p\) | \(\mathbb{R}^+\) | Phase velocity, the speed of wave propagation \(u_p = f \lambda\) (unit: \(\mathrm{m/s}\)) |
| \(A\) | \(\mathbb{R}^+\) | Amplitude of the wave, the maximum displacement of the wave (unit: \(\mathrm{m}\)) |
| \(\phi(x, t)\) | \(\mathbb{R}\) | Phase of the wave, a function of space and time \(\phi(x, t) = \omega t - \beta x + \phi_0\) (unit: \(\mathrm{rad}\)) |
| \(\phi_0\) | \(\mathbb{R}\) | Reference phase, the initial phase shift (unit: \(\mathrm{rad}\)) |
| \(\alpha\) | \(\mathbb{R}^+\) | Attenuation constant of a lossy medium (unit: \(\mathrm{Np/m}\)) |
| \(\beta\) | \(\mathbb{R}^+\) | Phase constant (wavenumber), \(\beta = \frac{2\pi}{\lambda}\) (unit: \(\mathrm{rad/m}\)) |
| \(\omega\) | \(\mathbb{R}^+\) | Angular frequency, \(\omega = 2\pi f\) (unit: \(\mathrm{rad/s}\)) |
| \(y(x, t)\) | \(\mathbb{R}\) | Wave displacement as a function of space \(x\) and time \(t\) (unit: \(\mathrm{m}\)) |
| \(\mathcal{E}\) | \(\mathbb{R}^+\) | Energy carried by an electromagnetic wave (unit: \(\mathrm{J}\) or \(\mathrm{kg \cdot m^2/s^2}\)) |
| \(\mathrm{Np/m}\) | Unit | Neper per meter, unit for attenuation constant |
| \(\mathrm{rad/m}\) | Unit | Radians per meter, unit for phase constant |
| \(\mathbb{R}\) | Set | Set of real numbers |
| \(\mathbb{R}^+\) | Set | Set of positive real numbers |
Properties of waves
Waves are a natural consequence of many physical processes. Various types of waves, indluding sound waves, mechanical waves, and electromagnetic waves, exhibit a number of common properties, including:
- Moving waves carry energy.
- Waves have velocity; it takes time for a wave to travel from one point to another. Electromagnetic waves in a vacuum travel at a speed of \(3 \times 10^8 \mathrm{~m} / \mathrm{s}\), and sound waves in air travel at a speed approximately a million times slower, specifically \(330 \mathrm{~m} / \mathrm{s}\). Sound waves cannot travel in a vacuum.
- Many waves exhibit a property called linearity. The total of two linear waves is simply the sum of the two waves as they would exist separately.
- Electromagnetic waves are linear, as are sound waves.
An essential feature of a propagating wave is that it is a self-sustaining disturbance of the medium through which it travels. If this disturbance varies as a function of one space variable, such as the vertical displacement of the string shown in Fig. 1-10, we call the wave one-dimensional.
A two-dimensional wave propagates out across a surface, like the ripples on a pond, and its disturbance can be described by two space variables. By extension, a three-dimensional wave propagates through a volume, and its disturbance may be a function of all three space variables.
To keep the presentation simple, we limit our discussion to sinusoidally varying waves whose disturbances are functions of only one space variable.
Sinusoidal waves in a lossless medium
By way of an example, let us consider a wave traveling on a lake’s surface, and let us assume for the time being that frictional forces can be ignored, thereby allowing a wave generated on the water’s surface to travel indefinitely with no loss in energy.
If \(y\) denotes the height of the water’s surface relative to the mean height (undisturbed condition) and \(x\) denotes the distance of wave travel, the functional dependence of \(y\) on time \(t\) and the spatial coordinate \(x\) has the general form \[ \begin{equation} \label{eq1.17} y(x, t)=A \cos \left(\frac{2 \pi t}{T}-\frac{2 \pi x}{\lambda}+\phi_0\right)(\mathrm{m}) \end{equation} \]
where \(A\) is the amplitude of the wave, \(T\) is its time period, \(\lambda\) is its spatial wavelength, and \(\phi_0\) is a reference phase. The quantity \(y(x, t)\) also can be expressed in the form
\[ y(x, t)=A \cos \phi(x, t) \]
where
\[ \phi(x, t)=\left(\frac{2 \pi t}{T}-\frac{2 \pi x}{\lambda}+\phi_0\right) \quad(\mathrm{rad}) \] The angle \(\phi(x, t)\) is called the phase of the wave, and it should not be confused with the reference phase \(\phi_0\), which is constant with respect to both time and space.
Explanation of wavelengh
At first, the concept of wavelength may seem confusing. Simply put, the wavelength \(\lambda\) is the spatial period over which the wave pattern repeats, analogous to the temporal period \(T\), which describes the interval of time over which the wave pattern repeats.
For example, consider the simple case with \(\phi_0=0\) :
\[ y(x, t)=A \cos \left(\frac{2 \pi t}{T}-\frac{2 \pi x}{\lambda}\right) \quad(\mathrm{m}) \]
The plots in Fig. 1-12 illustrate \(y(x, t)\) as a function of \(x\) at \(t=0\) and as a function of \(t\) at \(x=0\).
The wave pattern repeats spatially over a distance \(\lambda\) along \(x\), and temporally with a period \(T\) along \(t\). This is why \(\lambda\) is referred to as the spatial wavelength.
Phase velocity
If we take time snapshots of the water's surface, the height profile \(y(x, t)\) would exhibit the sinusoidal patterns shown in Fig. 1-13.
All three profiles correspond to three different values of \(t\), and the spacing between peaks is equal to the wavelength \(\lambda\), even though the patterns are shifted relative to one another because they correspond to different observation times.
Because the pattern advances along the \(+x\) direction at progressively increasing values of \(t\), \(y(x, t)\) is called a wave traveling in the \(+x\) direction. If we track a given point on the wave, such as the peak \(P\), and follow it in time, we can measure the phase velocity of the wave. At the peaks of the wave pattern, the phase \(\phi(x, t)\) is equal to zero or multiples of \(2 \pi\) radians. Thus, \[ \phi(x, t)=\frac{2 \pi t}{T}-\frac{2 \pi x}{\lambda}=2 n \pi, \quad n=0,1,2, \ldots \]
Had we chosen any other fixed height of the wave, say \(y_0\), and monitored its movement as a function of \(t\) and \(x\), this again would have been equivalent to setting the phase \(\phi(x, t)\) constant such that
\[ y(x, t)=y_0=A \cos \left(\frac{2 \pi t}{T}-\frac{2 \pi x}{\lambda}\right) \] or
\[ \begin{equation} \label{eq1.23} \frac{2 \pi t}{T}-\frac{2 \pi x}{\lambda}=\cos ^{-1}\left(\frac{y_0}{A}\right)=\text { constant. } \end{equation} \]
By taking the time derivative of \(\eqref{eq1.23}\), we obtain
\[ \frac{2 \pi}{T}-\frac{2 \pi}{\lambda} \frac{d x}{d t}=0 \]
which gives the phase velocity \(u_{\mathrm{p}}\) as
\[ u_{\mathrm{p}}=\frac{d x}{d t}=\frac{\lambda}{T} \quad(\mathrm{~m} / \mathrm{s}) \] The phase velocity, also called the propagation velocity, is the velocity of the wave pattern as it moves across the water’s surface.
Another representation of phase velocity
The frequency of a sinusoidal wave, \(f\), is the reciprocal of its time period \(T\) :
\[ f=\frac{1}{T} \quad(\mathrm{~Hz}) \]
Combining the preceding two equations yields
\[ u_{\mathrm{p}}=f \lambda \quad(\mathrm{~m} / \mathrm{s}) \] Afterwards, \(\eqref{eq1.17}\) can be rewritten in a more compact form as
\[ \begin{equation} \label{eq1.28} \begin{aligned} y(x, t) & =A \cos \left(2 \pi f t-\frac{2 \pi}{\lambda} x +\phi_0 \right) \\ & =A \cos (\omega t-\beta x +\phi_0) \end{aligned} \end{equation} \]
(wave moving along \(+x\) direction) where \(\omega\) is the angular velocityof the wave and \(\beta\) is its phase constant (or wavenumber), defined as \[ \begin{array}{ll} \omega=2 \pi f & (\mathrm{rad} / \mathrm{s}) \\ \beta=\frac{2 \pi}{\lambda} & (\mathrm{rad} / \mathrm{m}) \end{array} \]
In terms of these two quantities,
\[ \begin{equation} \label{eq1.30} u_{\mathrm{p}}=f \lambda=\frac{\omega}{\beta} \end{equation} \]
So far, we have examined the behavior of a wave traveling in the \(+x\) direction. To describe a wave traveling in the \(-x\) direction, we reverse the sign of \(x\) in \(\eqref{eq1.28}\):
\[ y(x, t)=A \cos (\omega t+\beta x +\phi_0) \]
(wave moving along \(-x\) direction)
Explanation of reference phase
We now examine the role of the phase reference \(\phi_0\) given previously in \(\eqref{eq1.17}\).
\[ y(x, t)=A \cos \left(\omega t-\beta x+\phi_0\right) \]
A plot of \(y(x, t)\) as a function of \(x\) at a specified \(t\) or as a function of \(t\) at a specified \(x\) is shifted in space or time, respectively, relative to a plot with \(\phi_0=0\) by an amount proportional to \(\phi_0\). This is illustrated by the plots shown in Fig. 1-14.
We observe that when \(\phi_0\) is positive, \(y(t)\) reaches its peak value, or any other specified value, sooner than when \(\phi_0=0\). Thus, the wave with \(\phi_0=\pi / 4\) is said to lead the wave with \(\phi_0=0\) by a phase lead of \(\pi / 4\); and similarly, the wave with \(\phi_0=-\pi / 4\) is said to lag the wave with \(\phi_0=0\) by a phase \(\operatorname{lag}\) of \(\pi / 4\).
Sinusoidal waves in a lolssy medium
If a wave is traveling in the \(x\) direction in a lossy medium, its amplitude decreases as \(e^{-\alpha x}\). This factor is called the attenuation factor, and \(\alpha\) is called the attenuation constant of the medium and its unit is neper per meter ( \(\mathrm{Np} / \mathrm{m}\) )1. Thus, in general, \[ y(x, t)=A e^{-\alpha x} \cos \left(\omega t-\beta x+\phi_0\right) . \]
The wave amplitude is now \(A e^{-\alpha x}\), not just \(A\). Figure 1 -15 shows a plot of \(y(x, t)\) as a function of \(x\) at \(t=0\) for \(A=10 \mathrm{~m}\), \(\lambda=2 \mathrm{~m}, \alpha=0.2 \mathrm{~Np} / \mathrm{m}\), and \(\phi_0=0\). Note that the envelope of the wave pattern decreases as \(e^{-\alpha x}\).
The electromagnetic spectrum
Visible light belongs to a family of waves arranged according to frequency and wavelength along a continuum called the electromagnetic spectrum (Fig. 1-16). Other members of this family include gamma rays, X-rays, infrared waves, and radio waves. Generically, they all are called EM waves because they share the following fundamental properties: - A monochromatic (single frequency) EM wave consists of electric and magnetic fields that oscillate at the same frequency \(f\).
The phase velocity of an EM wave propagating in a vacuum is a universal constant given by the velocity of light \(c\).
In a vacuum, the wavelength \(\lambda\) of an EM wave is related to its oscillation frequency \(f\) by \[ \lambda=\frac{c}{f} . \]
Whereas all monochromatic EM waves share these properties, each is distinguished by its own wavelength \(\lambda\), or equivalently by its own oscillation frequency \(f\).
Naming of electromagnetic waves
Because of \(\eqref{eq1.30}\), the connection between wavelength \((\lambda)\) and frequency \((f)\) is given by: \[ \lambda=\frac{c}{f} \]
Due to this relationship, spectral ranges can be specified either by their wavelength range or frequency range.
One well-known spectral band is the millimeter-wave band, named after its wavelength range, which extends from 1 mm (corresponding to 300 GHz ) to 1 cm (corresponding to 30 GHz ).
The real unit of \(\alpha\) is \((1 / \mathrm{m})\); the neper \((\mathrm{Np})\) part is a dimensionless, artificial adjective traditionally used as a reminder that the unit \((\mathrm{Np} / \mathrm{m})\) refers to the attenuation constant of the medium, \(\alpha\). A similar practice is applied to the phase constant \(\beta\) by assigning it the unit ( \(\mathrm{rad} / \mathrm{m}\) ) instead of just \((1 / \mathrm{m})\).↩︎