The Traveling Electromagnetic Wave, Quantitatively — Question 10

Question 10

A harmonic waveform can translate rigidly without being sinusoidal

A vacuum field is $E_y=E_0[\sin\phi+\tfrac12\sin2\phi]$, where $\phi=kx-\omega t$ and omega=ck. Seek a single +x traveling waveform with no static field offset.

1. Find Bz and show how the second harmonic frequency and wave number are related.

2. At phase zero find the time and spatial derivatives of Ey and use them to verify the transport equation.

Question 10 - Solution

A harmonic waveform can translate rigidly without being sinusoidal

(a) Method and calculation

$$\boxed{B_z=\frac{E_0}{c}[\sin\phi+\tfrac12\sin2\phi]}.$$$$k_2=2k,\quad\omega_2=2\omega\Rightarrow\boxed{\omega_2/k_2=c}.$$

(b) Interpretation and further calculation

$$\left.\partial_xE_y\right|_{\phi=0}=kE_0(1+1)=2kE_0,\quad\left.\partial_tE_y\right|_{\phi=0}=-2\omega E_0.$$$$\boxed{\partial_tE_y+c\partial_xE_y=0}\quad\text{at all phases}.$$

Physical check

In nondispersive vacuum all harmonics share the same phase speed, so their sum preserves its shape.