Complex Waves: Phase, Frequency, and Amplitude

Introduction

In our last lesson, we accomplished a significant milestone by deriving the electromagnetic wave equation directly from Maxwell's laws. We demonstrated that the interconnected dance of changing electric and magnetic fields gives rise to self-propagating waves that travel at the speed of light. The equation we arrived at, for instance for the electric field, was:

\nabla^{2} E = \frac{1}{c ^{2}} \frac{\partial ^{2} E}{\partial t ^{2}}

The solutions to this equation are sinusoidal waves, which can be described using sine and cosine functions. However, working directly with these functions, especially when dealing with phase shifts and amplitudes, can be mathematically cumbersome.

This lesson introduces a more elegant and powerful mathematical framework for describing these waves: complex exponential notation. Your background in data analytics and signal processing may have already exposed you to the power of complex numbers in Fourier analysis, and we will now apply that same power to the physics of waves. We will learn how to represent an electromagnetic wave's core properties—its amplitude, phase, and frequency—within this compact notation. This skill is foundational for nearly all advanced topics in wave physics and radar signal processing.

The Intuition Behind Complex Representation

Before diving into the equations, let's build a strong visual intuition for why complex numbers are so naturally suited to describing oscillations. A sinusoidal wave, like a cosine, is simply a projection of uniform circular motion. The following video provides an excellent visual explanation of this connection.

How do Complex Numbers relate to Real Signals? ("Best explanation EVER!")

This video, "How do Complex Numbers relate to Real Signals?", masterfully illustrates how sine and cosine waves are just two different views of a single point rotating on a circle. It then connects this rotation to the mathematical expression for complex numbers.

Please watch the first three segments: The first part builds the core visual. Pay close attention to how looking at a rotating point from two orthogonal directions generates sine and cosine waves. The second part introduces Euler's formula, e^{j\theta} = \cos\theta + j\sin\theta, showing that a single complex number can represent the point on the circle, with its real and imaginary parts corresponding to the cosine and sine waves, respectively. The third part makes the final connection to real signals by relating the angle \theta to angular frequency and time (\theta = \omega t).

The video uses 'j' for the imaginary unit, which is common in engineering. In physics and mathematics, we typically use 'i'. They are identical: $i^{2} = j^{2} = - 1$ .

The key insight is that a single complex exponential, $e^{iω t}$ , contains both a cosine and a sine component. We can think of the physical wave as the projection of this complex rotation onto the real axis.

This diagram illustrates the same concept as the video. A point moving in a circle over time traces a helix. The projection of this helix onto one plane gives a cosine wave (the real part), and the projection onto an orthogonal plane gives a sine wave (the imaginary part). Euler's formula, $e^{it} = \cos(t) + i\sin(t)$, mathematically describes the point on the circle at any given time.

From Sines and Cosines to Complex Exponentials

Now let's formalize this. A general one-dimensional plane wave solution to the wave equation can be written as:

E (z, t) = E_{0} cos (k z - ω t + ϕ)

Here, $E_{0}$ is the real amplitude, $ω$ is the angular frequency, $k$ is the wavenumber, and $ϕ$ is a phase offset. To represent this using complex notation, we leverage Euler's formula.

The following text provides a step-by-step guide to this representation.

2.3: Representation of Waves via Complex Functions

This article from Physics LibreTexts offers a clear mathematical bridge from the familiar real-valued wave function to its complex representation.

First, quickly review the section on complex numbers to refamiliarize yourself with the terms modulus and argument. Then, read the main section that begins with Euler's theorem. Focus on how a real wave, \psi(x,t), is represented as the real part of a complex function, and how the real amplitude A and phase angle \varphi are combined into a single complex amplitude, \psi_0.

As the text explains, we can write our wave as:

E (z, t) = Re {E_{0} e^{i (k z - ω t + ϕ)}}

Using the properties of exponentials, we can separate the constant phase term:

E (z, t) = Re {(E_{0} e^{i ϕ}) e^{i (k z - ω t)}}

This allows us to define a complex amplitude, $E ~_{0} = E_{0} e^{i ϕ}$ . This single complex number now elegantly packages both the real amplitude and the initial phase of the wave.

The amplitude of the wave is the modulus of the complex amplitude: $E_{0} = ∣ E ~_{0} ∣$ .
The initial phase of the wave is the argument of the complex amplitude: $ϕ = ar g (E ~_{0})$ .

Our wave representation simplifies to:

E (z, t) = Re {E ~_{0} e^{i (k z - ω t)}}

The Physicist's Shorthand: Phasor Notation

In practice, writing $Re {\dots}$ all the time is tedious. It's a common convention in physics and engineering to work directly with the complex form, with the understanding that the physical quantity is just its real part. This is known as phasor notation.

The complex wave is written as:

E ~ (z, t) = E ~_{0} e^{i (k z - ω t)}

The time-dependent part is often separated, leaving just the complex spatial field, or phasor:

E ~ (z) = E ~_{0} e^{ik z}

The full time-dependent field can be recovered by multiplying by $e^{- iω t}$ and taking the real part. (Note: some engineering fields use $e^{+ iω t}$ , but the $e^{- iω t}$ convention is more common in physics and electromagnetism).

This notation isn't just for brevity; it drastically simplifies calculus. For example, taking the time derivative of the wave becomes simple multiplication:

\frac{\partial E ~ ( z , t )}{\partial t} = \frac{\partial}{\partial t} (E ~_{0} e^{ik z} e^{- iω t}) = - iω (E ~_{0} e^{ik z} e^{- iω t}) = - iω E ~ (z, t)

This turns differential equations into algebraic equations, a significant simplification that you will appreciate as we move forward.

The following resource provides a formal treatment of this approach, applying it directly to the solutions of the wave equation we derived.

Chapter 2 The Wave Equation

This document from ETH Zurich rigorously develops the time-harmonic solution to the wave equation using complex notation.

Please read the section "Homogeneous Solution in Free Space" (Section 2.1). This introduces the time-harmonic electric field E(\mathbf{r}, t) = \text{Re}\{E(\mathbf{r}) e^{-i\omega t}\} and derives the Helmholtz equation. Then, continue with the subsection "Plane Waves" (Section 2.1.1). This introduces the full plane wave solution E(\mathbf{r}, t) = \text{Re}\{E_0 e^{\pm ik\cdot r - i\omega t}\} and defines the relationship between the fields and the direction of propagation.

Defining the Properties of an Electromagnetic Wave

Let's consolidate everything we've learned into the full 3D description of a plane electromagnetic wave and clearly define its properties.

A plane electromagnetic wave propagating in the +x direction. The mathematical expression we've developed describes this physical phenomenon.

The complex representation of the electric field vector for a plane wave is:

E (r, t) = E_{0} e^{i (k \cdot r - ω t)}

Here, we've dropped the $Re {}$ as per convention. Let's break down each component:

Complex Amplitude ( $E_{0}$ ): This is a complex vector.
- Amplitude: The magnitude of this vector, $∣ E_{0} ∣$ , determines the peak strength of the electric field.
- Polarization: The direction of this vector determines how the electric field is oriented in the plane perpendicular to propagation. We will explore this important property in a later lesson.
- Initial Phase: The argument of the complex components of $E_{0}$ sets the wave's phase at $r = 0, t = 0$ .
Wave Vector ( $k$ ): This real vector points in the direction the wave is propagating. Its magnitude, $k = ∣ k ∣$ , is the wavenumber, which is related to the wavelength $λ$ by $k = 2 π / λ$ .
Angular Frequency ( $ω$ ): This scalar value determines how rapidly the wave oscillates in time. It is related to the temporal frequency $f$ (in Hertz) by $ω = 2 π f$ . For EM waves in a vacuum, $ω = c k$ .
Phase ( $Φ (r, t) = k \cdot r - ω t$ ): This is the argument of the complex exponential. It describes the state of the oscillation at any point in space ( $r$ ) and time ( $t$ ). All points on a plane defined by $k \cdot r = constant$ have the same phase, which is why these are called "plane waves."

Conclusion

In this lesson, we have established a powerful and efficient method for describing electromagnetic waves. By moving from sines and cosines to complex exponentials, we have found a notation that is not only more compact but also vastly simplifies the mathematical operations needed to analyze wave behavior.

The key takeaways are:

Sinusoidal waves can be represented as the real part of a complex exponential function, $E (t) = Re {E ~_{0} e^{- iω t}}$ .
Euler's formula, $e^{i θ} = cos θ + i sin θ$ , is the mathematical link between circular motion in the complex plane and real-world oscillations.
A complex amplitude ( $E ~_{0}$ ) neatly combines the wave's real amplitude ( $∣ E ~_{0} ∣$ ) and its initial phase ( $ar g (E ~_{0})$ ) into a single number.
The complete description of a plane wave is given by $E (r, t) = E_{0} e^{i (k \cdot r - ω t)}$ , where we can identify its amplitude, frequency, phase, and direction of travel.

You have now completed the foundational module on electromagnetism and waves. You understand what EM waves are, where they come from, and how to describe them mathematically. In our next lesson, we will begin the second module, "Radar Principles and Wave-Matter Interaction," by using this knowledge to derive the foundational radar range equation, which governs how much power a radar system receives from a target.

Recommended resources