Chapter 9 - Electromagnetic Waves

Reference "Introduction to Electrodynamics" (5e) by David Griffiths.

Griffiths defines a wave as "a disturbance of a continuous medium that propagates with a fixed shape at a constant velocity".

If we represent such a wave mathematically, then we can say that the displacement $z$ (from the origin) at some later time $t$ is $$f(z,t)=f(z-vt,0)=g(z-vt)$$ Or just $$f(z,t)=g(z-vt)$$ Solutions to this equation are determined via the wave equation. Let $u=z-vt$, such that $g(z-vt)=g(u)$. $$\frac{d^{2}g}{d{u}^2}=\frac{{\partial }^{2}f}{\partial z^2}=\frac{1}{v^2}\frac{{\partial }^{2}f}{\partial t^2}$$ here the speed of propagation $v$ is given by $$v=\sqrt{\frac{T}{\mu }}$$ Functions of the form $g(z-vt)$ are not the only solutions - the wave equation has the square of $v$, so another class of solutions looks like $h(z+vt)$ - so the most general solution to the wave equation is $$f(z,t)=g(z-vt)+h(z+vt)$$

Note that this is a linear equation thus the sum of any solutions is also solution.

Waves that travel across some medium (like waves on an ocean or guitar string, moving with velocity $v$) are called transverse. Those that act more like a slinky are called longitudinal waves.

Sinusoidal waves

Sinusoidal waves take the form $$f(z,t)=A\cos (k(z-vt)+\delta )$$

• $A$ is the amplitude.
• $\delta$ is the phase constant.
• $k$ is the wave number, and is related to wavelength by $$\lambda =\frac{2\pi }{k}$$

Cosine completes one cycle every $2\pi /k$.

The period $T$ is $$T=\frac{2\pi }{kv}=\frac{\lambda }{v}$$ while the frequency (oscillations per unit $t$) $$f=\frac{1}{T}=\frac{v}{\lambda }$$ Alternatively, the angular frequency can be more useful: $$w=2\pi f=kv$$

The direction of propagation is determined by the sign of $k$. A leftward wave has $+k$, a rightward wave $-k$.

Complex wave notation

Waves can also be written in complex notation, by $$g(z,t)=A{e}^{i(kz-\omega t+\delta )}$$ Or alternatively, with $B=A{e}^{i\delta }$, $$g(z,t)=B{e}^{i(kz-\omega t)}$$ The actual wavefunction is still given through the real component of this. $$f(z,t)=\text{Re}[g(z,t)]$$

Polarization

Transverse waves move along a single axis (the direction of propagation):

Thus, there will always be two dimensions orthogonal to the direction of propagation given by the polarization vector $\hat{n}$, with the polarization angle $\theta$. $$\hat{n}=\cos \theta \hat{x}+\sin \theta \hat{y}$$ Most waves can therefore be considered a superposition of two waves: one horizontally polarized, one vertically, depending on your axis orientation. $$g(z,t)=g_1(x,t)\cos (\theta )\hat{x}+g_2(z,t)\sin (\theta )\hat{y}$$

Electromagnetic Waves

Derived from Maxwell's equations, $${\nabla }^2\vec{E}=\mu ϵ\frac{{\partial }^2\vec{E}}{\partial t^2}{\nabla }^2\vec{B}=\mu ϵ\frac{{\partial }^2\vec{B}}{\partial t^2}$$

In a vacuum (outside of any material), replace $\mu \to {\mu }_0$ and $ϵ\to {ϵ}_0$.

Both satisfy the wave equation in three dimensions: $${\nabla }^{2}f=\frac{1}{v^2}\frac{{\partial }^{2}f}{\partial t^2}$$ The velocity is given by $$v=\frac{1}{\sqrt{\mu ϵ}}$$

In vacuum, $v=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}=c$, the speed of light.

Monochromatic Plane Waves

"Monochromatic" implies the frequency $\omega$ is constant.

If these waves are traveling in a single direction $z$ and have no dependence on $x$ or $y$, then they're called plane waves (fields uniform over every plane perpendicular to direction of propagation).