Wave Equations
Wave Equations in Electromagnetic Theory
Wave equations form the mathematical foundation of electromagnetic wave propagation. They describe how electric and magnetic fields travel through space and time. In microwave engineering, transmission lines, antennas, waveguides, and RF systems are all analyzed using these equations.
The starting point for deriving electromagnetic wave equations is Maxwell's equations. These equations relate electric fields, magnetic fields, charge density, and current density within a medium.
Maxwell's Equations in Time Domain
Maxwell's equations describe the behavior of electric and magnetic fields as functions of time and space.
$ \nabla \times E = -\frac{\partial B}{\partial t} $
$ \nabla \times H = J + \frac{\partial D}{\partial t} $
$ \nabla \cdot D = \rho_v $
$ \nabla \cdot B = 0 $
These four equations form the basis of electromagnetic field theory and are used to derive the wave equations for both electric and magnetic fields.
The Vector Operator (Del Operator)
The vector differential operator, represented by the symbol ∇ (del), is used to calculate divergence, gradient, and curl operations.
Cartesian Coordinates
$ \nabla = \frac{\partial}{\partial x}u_x + \frac{\partial}{\partial y}u_y + \frac{\partial}{\partial z}u_z $
Cylindrical Coordinates
$ \nabla = \frac{\partial}{\partial r}u_r + \frac{1}{r}\frac{\partial}{\partial \phi}u_{\phi} + \frac{\partial}{\partial z}u_z $
Spherical Coordinates
$ \nabla = \frac{\partial}{\partial r}u_r + \frac{1}{r}\frac{\partial}{\partial \theta}u_{\theta} + \frac{1}{r\sin\theta}\frac{\partial}{\partial \phi}u_{\phi} $
Wave Equations in Electromagnetic Theory
Every electromagnetic wave, whether it travels through a transmission line, waveguide, antenna, or free space, follows a mathematical relationship known as a wave equation. Before deriving the wave equation, it is necessary to understand the fundamental laws that govern electric and magnetic fields. These laws are known as Maxwell's equations.
Maxwell's equations form the foundation of electromagnetic field theory and microwave engineering. They describe how electric fields, magnetic fields, charges, and currents interact with one another.
Maxwell's Equations in Time Domain
The derivation of electromagnetic wave equations begins with Maxwell's four equations expressed in the time domain.
Faraday's Law of Electromagnetic Induction
$ \nabla \times E = -\frac{\partial B}{\partial t} $
This equation states that a time-varying magnetic field produces an electric field. The negative sign indicates that the induced electric field opposes the change in magnetic flux according to Lenz's law.
In simple terms, whenever a magnetic field changes with time, an electric field is generated around it.
Ampere-Maxwell Law
$ \nabla \times H = J + \frac{\partial D}{\partial t} $
This equation shows that magnetic fields are produced by two sources:
- Conduction current density (J)
- Displacement current density (∂D/∂t)
The displacement current term was introduced by Maxwell and made electromagnetic wave propagation possible. Without this term, the theory of electromagnetic waves would be incomplete.
Gauss's Law for Electric Fields
$ \nabla \cdot D = \rho_v $
This equation states that electric charges act as sources or sinks of electric flux. The divergence of electric flux density is equal to the volume charge density present in the medium.
Gauss's Law for Magnetic Fields
$ \nabla \cdot B = 0 $
This equation indicates that magnetic monopoles do not exist. Magnetic field lines always form closed loops and never start or end at a single point.
The Vector Operator (Del Operator)
The symbol ∇, called the del operator or vector differential operator, is widely used in Maxwell's equations. It performs operations such as gradient, divergence, and curl.
The mathematical form of the operator depends on the coordinate system being used.
Del Operator in Cartesian Coordinates
For rectangular coordinate systems:
$ \nabla = \frac{\partial}{\partial x}u_x + \frac{\partial}{\partial y}u_y + \frac{\partial}{\partial z}u_z $
Cartesian coordinates are commonly used for rectangular waveguides and many transmission line problems.
Del Operator in Cylindrical Coordinates
For cylindrical structures such as coaxial cables:
$ \nabla = \frac{\partial}{\partial r}u_r + \frac{1}{r}\frac{\partial}{\partial \phi}u_\phi + \frac{\partial}{\partial z}u_z $
Del Operator in Spherical Coordinates
For spherical geometries and antenna analysis:
$ \nabla = \frac{\partial}{\partial r}u_r + \frac{1}{r}\frac{\partial}{\partial \theta}u_\theta + \frac{1}{r\sin\theta} \frac{\partial}{\partial \phi}u_\phi $
Constitutive Relations of Electromagnetic Fields
Maxwell's equations alone do not completely describe wave propagation. The properties of the medium must also be considered. These properties are represented through constitutive relations.
Electric Flux Density Relation
$ D=\varepsilon E $
Electric flux density is directly proportional to the electric field intensity. The proportionality constant is the permittivity of the medium.
Magnetic Flux Density Relation
$ B=\mu H $
Magnetic flux density is directly proportional to magnetic field intensity. The proportionality constant is the permeability of the medium.
Current Density Relation
$ J=\sigma E $
This equation relates current density to electric field intensity through the conductivity of the material.
Permittivity of a Medium
$ \varepsilon=\varepsilon_r\varepsilon_0 $
Here:
- ε = Absolute permittivity
- εr = Relative permittivity
- ε0 = Permittivity of free space
Permeability of a Medium
$ \mu=\mu_r\mu_0 $
Where:
- μ = Absolute permeability
- μr = Relative permeability
- μ0 = Permeability of free space
Source-Free Medium Assumptions
In most waveguide, transmission line, and free-space propagation problems, the region through which the wave travels contains no free charges and no external current sources.
To simplify the derivation of wave equations, two important assumptions are made.
Assumption 1: No Free Charge Density
$ \rho_v=0 $
Substituting this into Gauss's Law gives:
$ \nabla \cdot D = 0 $
Assumption 2: No Conduction Current
$ J=0 $
This assumption is valid for ideal dielectric media where electrical conduction is negligible.
Maxwell's Equations After Applying Source-Free Assumptions
Using the constitutive relations and assuming a source-free medium, Maxwell's equations become:
$ \nabla \times E = -\mu\frac{dH}{dt} $
$ \nabla \times H = \varepsilon\frac{dE}{dt} $
$ \nabla \cdot D = 0 $
$ \nabla \cdot B = 0 $
These simplified equations provide the starting point for deriving the electric and magnetic wave equations. The next step is to assume a sinusoidal time variation and convert the equations into their phasor form.
Time Harmonic Fields and Phasor Representation
After simplifying Maxwell's equations for a source-free medium, the next challenge is solving the time derivatives that still appear in the equations. Directly solving these differential equations in the time domain is often difficult. To make the analysis easier, electromagnetic fields are assumed to vary sinusoidally with time.
This assumption is extremely important in microwave engineering because most RF and microwave systems operate using sinusoidal signals at a fixed frequency.
Why Time Harmonic Fields Are Assumed
Many practical electromagnetic systems operate with signals that continuously oscillate at a single frequency.
Examples include:
- Radio transmitters
- Microwave communication systems
- Radar systems
- Satellite communication links
- Waveguide-based microwave circuits
Because these signals are sinusoidal, it is convenient to represent electric and magnetic fields using a sinusoidal time dependence.
Mathematical Representation of a Time Harmonic Electric Field
Assume the electric field varies sinusoidally with angular frequency ω.
The electric field can be represented as:
$ E = E_0 e^{j\omega t} $
where:
- E = Electric field intensity
- E₀ = Field amplitude
- ω = Angular frequency
- t = Time
- j = Imaginary unit
This representation is known as the complex exponential form of the electromagnetic field.
Using Euler's Formula
The complex exponential can be expanded using Euler's identity:
$ e^{j\omega t} = \cos(\omega t) + j\sin(\omega t) $
Substituting into the electric field equation:
$ E = E_0 \left( \cos\omega t + j\sin\omega t \right) $
In physical systems, only the real part of the expression represents the actual measurable electric field.
Therefore:
$ E = \Re \left\{ E_0 e^{j\omega t} \right\} $
The imaginary component is retained only because it makes mathematical manipulation much easier.
First Time Derivative of the Electric Field
To simplify Maxwell's equations, we need to determine how differentiation affects a time harmonic field.
Start with:
$ E = E_0 e^{j\omega t} $
Differentiate with respect to time:
$ \frac{dE}{dt} = \frac{d}{dt} \left( E_0 e^{j\omega t} \right) $
Since E₀ is a constant:
$ \frac{dE}{dt} = E_0 \frac{d}{dt} \left( e^{j\omega t} \right) $
The derivative of an exponential function is:
$ \frac{d}{dt} \left( e^{j\omega t} \right) = j\omega e^{j\omega t} $
Therefore:
$ \frac{dE}{dt} = j\omega E_0 e^{j\omega t} $
Recognizing that:
$ E = E_0 e^{j\omega t} $
gives:
$ \frac{dE}{dt} = j\omega E $
Important Observation
The time derivative operation has now been replaced by multiplication with jω.
Instead of performing calculus, we can simply multiply by jω.
This is one of the biggest advantages of phasor analysis.
Therefore:
$ \frac{\partial}{\partial t} \Longleftrightarrow j\omega $
Second Time Derivative of the Electric Field
Differentiate once more:
$ \frac{d^2E}{dt^2} = \frac{d}{dt} \left( j\omega E \right) $
Since jω is constant:
$ \frac{d^2E}{dt^2} = j\omega \frac{dE}{dt} $
Substituting:
$ \frac{dE}{dt} = j\omega E $
gives:
$ \frac{d^2E}{dt^2} = j\omega(j\omega E) $
Therefore:
$ \frac{d^2E}{dt^2} = (j\omega)^2E $
Since:
$ j^2=-1 $
we obtain:
$ \frac{d^2E}{dt^2} = -\omega^2E $
Second Important Transformation
The second derivative operator becomes:
$ \frac{\partial^2}{\partial t^2} \Longleftrightarrow (j\omega)^2 $
or equivalently:
$ \frac{\partial^2}{\partial t^2} \Longleftrightarrow -\omega^2 $
Applying Phasor Transformation to Maxwell's Equations
Recall the source-free Maxwell equations obtained previously:
$ \nabla\times E = -\mu \frac{dH}{dt} $
$ \nabla\times H = \varepsilon \frac{dE}{dt} $
Replace each time derivative by jω.
Modified Faraday's Law
$ \nabla\times E = -\mu (j\omega H) $
Therefore:
$ \nabla\times E = -j\omega\mu H $
Modified Ampere-Maxwell Law
$ \nabla\times H = \varepsilon (j\omega E) $
Therefore:
$ \nabla\times H = j\omega\varepsilon E $
Including Conductive Media
If the medium possesses conductivity σ, the conduction current term must also be included.
Using:
$ J=\sigma E $
Ampere-Maxwell Law becomes:
$ \nabla\times H = \sigma E + j\omega\varepsilon E $
Factoring E:
$ \nabla\times H = (\sigma+j\omega\varepsilon)E $
Final Maxwell Equations in Phasor Form
The frequency-domain Maxwell equations are:
$ \nabla\times E = -j\omega\mu H $
$ \nabla\times H = (\sigma+j\omega\varepsilon)E $
$ \nabla\cdot D = \rho_v $
$ \nabla\cdot B = 0 $
These equations are much easier to manipulate than their time-domain counterparts and form the starting point for deriving the electric and magnetic wave equations.
Derivation of the Electric Field Wave Equation
Now that Maxwell's equations have been converted into phasor form, the next objective is to derive a single equation containing only the electric field. This equation is known as the electric field wave equation.
The derivation begins with Faraday's Law in phasor form and uses vector calculus identities together with source-free medium assumptions.
Step 1: Start with Faraday's Law
From the phasor-domain Maxwell equations:
$ \nabla \times E = -j\omega\mu H $
This equation relates the electric field to the magnetic field. Since our goal is to obtain an equation involving only the electric field, the magnetic field must eventually be eliminated.
Step 2: Take the Curl of Both Sides
Applying the curl operator to both sides gives:
$ \nabla \times \left( \nabla \times E \right) = \nabla \times \left( -j\omega\mu H \right) $
Because j, ω, and μ are constants, they can be moved outside the curl operator.
Therefore:
$ \nabla \times \nabla \times E = -j\omega\mu \left( \nabla \times H \right) $
At this stage, the equation still contains the magnetic field term. The next step is to remove it using another Maxwell equation.
Step 3: Substitute Ampere-Maxwell Law
For a lossless source-free dielectric medium:
$ \nabla \times H = j\omega\varepsilon E $
Substituting this into the previous equation:
$ \nabla \times \nabla \times E = -j\omega\mu \left( j\omega\varepsilon E \right) $
Multiplying the constants:
$ \nabla \times \nabla \times E = (-j)(j) \omega^2 \mu \varepsilon E $
Since:
$ j^2=-1 $
it follows that:
$ (-j)(j)=1 $
Therefore:
$ \nabla \times \nabla \times E = \omega^2 \mu \varepsilon E $
This is an important intermediate result, but the left-hand side still contains a double curl operation that must be simplified.
Step 4: Apply the Curl-of-Curl Vector Identity
Vector calculus provides a useful identity:
$ \nabla \times (\nabla \times E) = \nabla(\nabla\cdot E) - \nabla^2E $
This identity converts the complicated double curl into divergence and Laplacian terms.
Substituting into the previous equation:
$ \nabla(\nabla\cdot E) - \nabla^2E = \omega^2 \mu \varepsilon E $
At this point, the divergence term must be simplified.
Step 5: Apply the Charge-Free Condition
For most microwave propagation problems, the region contains no free charge.
Therefore:
$ \rho_v=0 $
Using Gauss's Law:
$ \nabla\cdot D = \rho_v $
Substituting ρv = 0:
$ \nabla\cdot D = 0 $
Since:
$ D=\varepsilon E $
and ε is constant throughout the medium:
$ \nabla\cdot(\varepsilon E) = 0 $
Therefore:
$ \varepsilon (\nabla\cdot E) = 0 $
Since ε ≠ 0:
$ \nabla\cdot E = 0 $
This result is extremely important because it eliminates the divergence term from the wave equation derivation.
Step 6: Eliminate the Divergence Term
Recall:
$ \nabla(\nabla\cdot E) - \nabla^2E = \omega^2 \mu \varepsilon E $
Since:
$ \nabla\cdot E = 0 $
it follows that:
$ \nabla(\nabla\cdot E) = 0 $
Therefore:
$ -\nabla^2E = \omega^2 \mu \varepsilon E $
Step 7: Rearrange the Equation
Multiplying both sides by −1:
$ \nabla^2E = -\omega^2 \mu \varepsilon E $
This is the electric field wave equation in phasor form.
Final Electric Field Wave Equation
The final result is:
$ \nabla^2E = -\omega^2 \mu \varepsilon E $
or equivalently:
$ \nabla^2E + \omega^2 \mu \varepsilon E = 0 $
This form is known as the Helmholtz Equation for the electric field.
Why This Equation Is Important
The electric field wave equation describes how electromagnetic energy propagates through a medium.
It is used extensively in:
- Waveguide analysis
- Transmission line theory
- Antenna design
- Radar systems
- Microwave circuits
- Optical wave propagation
Once the electric field distribution is known, the magnetic field can be calculated directly using Maxwell's equations.
Physical Interpretation of Each Term
The Laplacian term:
$ \nabla^2E $
describes how the electric field varies throughout space.
The term:
$ \omega^2\mu\varepsilon $
contains information about frequency and material properties.
Together, these terms determine how electromagnetic waves travel, reflect, refract, and attenuate within a medium.
Step 5: Applying the Vector Identity to Simplify the Wave Equation
At this stage, we have obtained:
$ \nabla \times \nabla \times E = -j\omega\mu(\sigma+j\omega\varepsilon)E $
The left-hand side contains the curl of a curl, which is difficult to work with directly. To simplify it, we use a standard vector calculus identity:
$ \nabla \times (\nabla \times E) = \nabla(\nabla \cdot E)-\nabla^2E $
Substituting this identity into the previous equation gives:
$ \nabla(\nabla \cdot E)-\nabla^2E = -j\omega\mu(\sigma+j\omega\varepsilon)E $
This expression is still general and valid for any electromagnetic medium.
Why We Need Another Assumption
The term ∇(∇·E) makes the equation complicated. Fortunately, most waveguide and transmission problems are analyzed in regions where no free charge exists.
In a charge-free region:
$ \rho_v=0 $
From Gauss's Law:
$ \nabla \cdot D=\rho_v $
Substituting \(\rho_v=0\):
$ \nabla \cdot D=0 $
Since:
$ D=\varepsilon E $
and \(\varepsilon\) is constant throughout the medium, we obtain:
$ \nabla \cdot E=0 $
Therefore:
$ \nabla(\nabla \cdot E)=0 $
The wave equation now becomes much simpler:
$ -\nabla^2E = -j\omega\mu(\sigma+j\omega\varepsilon)E $
Multiplying both sides by \((-1)\):
$ \nabla^2E = j\omega\mu(\sigma+j\omega\varepsilon)E $
Special Case: Lossless Dielectric Medium
Most microwave waveguide derivations assume a lossless dielectric medium.
For a lossless medium:
$ \sigma=0 $
Substituting \(\sigma=0\):
$ \nabla^2E = j\omega\mu(j\omega\varepsilon)E $
Multiplying the two imaginary terms:
$ j \times j=-1 $
Therefore:
$ \nabla^2E = -\omega^2\mu\varepsilon E $
This is the standard electric-field wave equation in phasor form.
It is commonly written as:
$ \nabla^2E+\omega^2\mu\varepsilon E=0 $
Physical Meaning of the Electric Wave Equation
This equation describes how an electromagnetic wave propagates through space.
The Laplacian operator \(\nabla^2\) represents spatial variation of the electric field.
The term \(\omega^2\mu\varepsilon\) determines how rapidly the field changes with distance and frequency.
A larger frequency produces faster field variations. A medium with higher permeability or permittivity also changes the propagation characteristics of the wave.
This equation forms the mathematical foundation of:
- Rectangular waveguides
- Circular waveguides
- Transmission lines
- Microwave resonators
- Antennas
- Optical waveguides
Key Result Obtained
After applying the vector identity and assuming a charge-free, lossless medium, Maxwell's equations reduce to:
$ \nabla^2E = -\omega^2\mu\varepsilon E $
This is the fundamental electric-field wave equation used throughout microwave engineering and electromagnetic field theory.
Step 6: Derivation of the Magnetic Field Wave Equation
The electric-field wave equation was obtained by taking the curl of Faraday's Law. The magnetic-field wave equation is derived in exactly the same manner, starting from Ampere-Maxwell's equation in phasor form.
For a lossless medium:
$ \nabla \times H = j\omega \varepsilon E $
This equation shows that a time-varying magnetic field produces an electric field.
Taking the Curl of Both Sides
Apply the curl operator to both sides:
$ \nabla \times (\nabla \times H) = \nabla \times (j\omega\varepsilon E) $
Since \(j\omega\varepsilon\) is a constant, it can be moved outside the curl operator:
$ \nabla \times \nabla \times H = j\omega\varepsilon(\nabla \times E) $
Substituting Faraday's Law
From Maxwell's phasor equation:
$ \nabla \times E = -j\omega\mu H $
Substituting into the previous expression:
$ \nabla \times \nabla \times H = j\omega\varepsilon(-j\omega\mu H) $
Rearranging:
$ \nabla \times \nabla \times H = -j^2\omega^2\mu\varepsilon H $
Since:
$ j^2=-1 $
Therefore:
$ \nabla \times \nabla \times H = \omega^2\mu\varepsilon H $
Applying the Curl-Curl Identity
Using the vector identity:
$ \nabla \times (\nabla \times H) = \nabla(\nabla \cdot H)-\nabla^2H $
Substituting into the equation:
$ \nabla(\nabla \cdot H)-\nabla^2H = \omega^2\mu\varepsilon H $
Using Gauss's Law for Magnetism
One of Maxwell's equations states:
$ \nabla \cdot B=0 $
Since:
$ B=\mu H $
and \(\mu\) is constant:
$ \nabla \cdot H=0 $
Therefore:
$ \nabla(\nabla \cdot H)=0 $
The wave equation becomes:
$ -\nabla^2H = \omega^2\mu\varepsilon H $
Multiplying both sides by \((-1)\):
$ \nabla^2H = -\omega^2\mu\varepsilon H $
Or equivalently:
$ \nabla^2H+\omega^2\mu\varepsilon H=0 $
Final Magnetic Field Wave Equation
The magnetic field satisfies the same wave equation form as the electric field:
$ \nabla^2H = -\omega^2\mu\varepsilon H $
This proves that both electric and magnetic fields propagate as electromagnetic waves through space.
Comparison of Electric and Magnetic Wave Equations
Electric field wave equation:
$ \nabla^2E = -\omega^2\mu\varepsilon E $
Magnetic field wave equation:
$ \nabla^2H = -\omega^2\mu\varepsilon H $
Notice that both equations have exactly the same mathematical structure. The only difference is whether the unknown quantity is the electric field or the magnetic field.
What Does This Mean Physically?
Neither field can exist independently in a propagating electromagnetic wave.
A changing electric field creates a magnetic field.
A changing magnetic field creates an electric field.
This continuous interaction allows electromagnetic energy to travel through free space, waveguides, transmission lines, optical fibers, and antennas.
The wave equations obtained above are the mathematical proof of electromagnetic wave propagation predicted by Maxwell.
Key Results Obtained So Far
Electric field wave equation:
$ \nabla^2E+\omega^2\mu\varepsilon E=0 $
Magnetic field wave equation:
$ \nabla^2H+\omega^2\mu\varepsilon H=0 $
These two equations form the starting point for deriving:
- Rectangular waveguide equations
- Circular waveguide equations
- TE mode field components
- TM mode field components
- Cutoff frequency equations
- Phase velocity and group velocity equations
- Propagation constants
Step 7: Converting the Phasor Wave Equation into the Time-Domain Wave Equation
So far, the wave equations have been derived in the phasor domain:
$ \nabla^2E+\omega^2\mu\varepsilon E=0 $
$ \nabla^2H+\omega^2\mu\varepsilon H=0 $
These equations are extremely useful for sinusoidal steady-state analysis. However, they only describe fields that vary sinusoidally with time.
To obtain the general wave equation that applies to any time-varying electromagnetic field, we must convert the equations back into the time domain.
Recall the Time-Harmonic Assumption
Earlier, the electric field was assumed to vary as:
$ E=E_0e^{j\omega t} $
Taking the first derivative with respect to time:
$ \frac{\partial E}{\partial t} = j\omega E $
Taking the second derivative:
$ \frac{\partial^2E}{\partial t^2} = (j\omega)^2E $
Since:
$ j^2=-1 $
we obtain:
$ \frac{\partial^2E}{\partial t^2} = -\omega^2E $
Rearranging:
$ \omega^2E = -\frac{\partial^2E}{\partial t^2} $
Substituting into the Electric Field Wave Equation
Starting with:
$ \nabla^2E+\omega^2\mu\varepsilon E=0 $
Replace \(\omega^2E\) with the equivalent time-domain expression:
$ \nabla^2E + \mu\varepsilon \left( -\frac{\partial^2E}{\partial t^2} \right) =0 $
Simplifying:
$ \nabla^2E - \mu\varepsilon \frac{\partial^2E}{\partial t^2} =0 $
Moving the second term to the right-hand side:
$ \nabla^2E = \mu\varepsilon \frac{\partial^2E}{\partial t^2} $
Final Electric Field Wave Equation
Therefore, the electric field wave equation in time domain becomes:
$ \nabla^2E = \mu\varepsilon \frac{\partial^2E}{\partial t^2} $
This equation describes how an electric field propagates through a medium as a wave.
Deriving the Time-Domain Magnetic Field Wave Equation
The same procedure is applied to the magnetic-field wave equation.
Starting with:
$ \nabla^2H+\omega^2\mu\varepsilon H=0 $
Using:
$ \omega^2H = -\frac{\partial^2H}{\partial t^2} $
Substituting:
$ \nabla^2H + \mu\varepsilon \left( -\frac{\partial^2H}{\partial t^2} \right) =0 $
Therefore:
$ \nabla^2H = \mu\varepsilon \frac{\partial^2H}{\partial t^2} $
Final Magnetic Field Wave Equation
$ \nabla^2H = \mu\varepsilon \frac{\partial^2H}{\partial t^2} $
This equation describes the propagation of the magnetic field through the medium.
The Standard Electromagnetic Wave Equations
The final time-domain wave equations are:
Electric Field Wave Equation
$ \nabla^2E = \mu\varepsilon \frac{\partial^2E}{\partial t^2} $
Magnetic Field Wave Equation
$ \nabla^2H = \mu\varepsilon \frac{\partial^2H}{\partial t^2} $
These are among the most important equations in electromagnetic theory.
They describe how electromagnetic waves travel through free space, dielectric materials, transmission lines, waveguides, antennas, and optical fibers.
Important Observation
Notice that both equations have exactly the same mathematical structure.
The only difference is the field variable:
- \(E\) for electric fields
- \(H\) for magnetic fields
This confirms that electric and magnetic fields travel together as a coupled electromagnetic wave.
Key Results Derived
$ \nabla^2E = \mu\varepsilon \frac{\partial^2E}{\partial t^2} $
$ \nabla^2H = \mu\varepsilon \frac{\partial^2H}{\partial t^2} $
These equations are the final wave equations obtained directly from Maxwell's equations for a charge-free lossless medium.