1. Basic properties of waves
One-dimensional wave
t=0 t
phase
measures the position of a wave feature, such as a wave peak or trough
wavelength
distance between repeating units of a propagating wave
wave-number
circular wave-number
period T duration of one cycle in a repeating event
frequency
number of occurrences of a repeating event per unit time
angular frequency
phase speed

2. Euler formula
i: imaginary unit
Harmonic wave : one amplitude and one frequency
Right moving wave
Left moving wave
Waves superimpose together
Example 1: standing wave

3. Example 2: Non-dispersive and dispersive wave
Specific case 1
Non-dispersive wave-packet
Specific case 2
Dispersive wave-packet
Non-dispersive wave: wave phase speed dos not depend on wave number.
Dispersive wave: wave phase speed depends on wave number.

4. Because of the reinforcement or cancellation of wave amplitude, the
Specific case 3
Specific case 4
Because of the reinforcement or cancellation of wave amplitude, the
energy of wave group will be concentrated in regions where the wave
amplitude is large, and those regions with small wave amplitude
contain less energy. Thus, wave energy does not propagate at the
phase speed of individual waves, but at the speed of wave envelope,
called group velocity.

5. Group velocity
high frequency carrier wave
slow moving envelope wave
group velocity