Wave Propagation

This tutorial explores wave propagation. Whether used in radio frequency systems, microwave systems, optical system or other, electromagnetic waves propagation obey Maxwell's equations. Here we try to examine one of the simplest solution to Maxwell's equation and understand its meaning. An electromagnetic field with only one component E_x and independent of x and y is solution to the differential equation

The above equation has many solutions. However one of the simplest and most useful is the one where time and z variations are sinusoidal and is given by.

The electric field component E_x is a function of two variables: t and z. To study this function we will change time t in steps and plot E_x as a function of z. This is done in the applet below.

Interactive Wave Propagation Simulator

Explore wave propagation using the equation: \[ E_x = E_0 \cos(\omega t - kz) \]

\[ E_x = E_0 \cos(\omega t - kz) \]

\[ \omega = 2\pi f \quad \text{and} \quad k = \frac{2\pi}{\lambda} \]

1.0 Hz

Frequency (f)

1.0 s

Period (T)

1.0 m

Wavelength (λ)

1.0 m/s

Wave Speed (v = fλ)

Amplitude E₀: 1.0

Frequency f: 1.0 Hz

Wavelength λ: 1.0 m

Time Speed: 0.5x

Wave Parameters and Relationships

Frequency (f) - Number of complete oscillations per second, measured in Hertz (Hz). Higher frequency means more oscillations per second.

Period (T) - Time for one complete oscillation: \[ T = \frac{1}{f} \]. This is the time it takes for the wave to repeat itself at a fixed position.

Angular Frequency (ω) - Related to frequency: \[ \omega = 2\pi f \]. This is the rate of change of phase in radians per second.

Wavelength (λ) - Distance between two consecutive identical points on the wave (e.g., peak to peak). This is the spatial period of the wave.

Wave Number (k) - Related to wavelength: \[ k = \frac{2\pi}{\lambda} \]. This is the spatial frequency in radians per meter.

Wave Speed (v) - Speed of wave propagation: \[ v = f\lambda = \frac{\omega}{k} \]. For electromagnetic waves in vacuum, this equals the speed of light (c ≈ 3×10⁸ m/s).

Understanding Wave Propagation

The red curve shows the electric field Eₓ as it varies with position z at the current time t.

Time Evolution (Period T): At a fixed position (choose any vertical line), watch how the wave oscillates up and down. The time between successive peaks at that position is the period T.

Spatial Variation (Wavelength λ): At a fixed time (pause the animation), measure the distance between two successive peaks. This distance is the wavelength λ.

Wave Propagation: The wave moves to the right because of the minus sign in \[ \omega t - kz \]. This means that as time increases, the wave pattern shifts to the right.

Key Relationships:

Higher frequency = shorter period (faster oscillations in time)
Longer wavelength = more spread out in space
Wave speed = frequency × wavelength: \[ v = f\lambda \]
Angular frequency = 2π × frequency: \[ \omega = 2\pi f \]
Wave number = 2π ÷ wavelength: \[ k = \frac{2\pi}{\lambda} \]

Try adjusting the frequency and wavelength sliders independently to see how they affect the wave's appearance and speed.