Lect. 06 phase & group velocities Prepared By Dr. Eng. Sherif Hekal

Adding up waves of different frequencies

Constructing a wave packet by adding up several waves ………… ___________________________________ If several waves of different wavelengths (frequencies) and phases are superposed together, one would get a resultant which is a localized wave packet

A wave packet describes a particle ____________________________ A wave packet is a group of waves with slightly different wavelengths interfering with one another in a way that the amplitude of the group (envelope) is non- zero only in the neighbourhood of the particle A wave packet is localized – a good representation for a particle!

Wave packet, phase velocity and group velocity The velocities of the individual waves, which superpose to produce the wave packet representing the particle, are different - the wave packet as a whole has a different velocity from the waves that comprise it Phase velocity: The rate at which the phase of the wave propagates in space Group velocity: The rate at which the envelope of the wave packet propagates.

Phase velocity Phase velocity is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate. You could pick one particular phase of the wave and it would appear to travel at the phase velocity. The phase velocity is given in terms of the wave's angular frequency ω and wave vector k by

Group velocity where: vg is the group velocity; Group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space. The group velocity is defined by the equation where: vg is the group velocity; ω is the wave's angular frequency; k is the wave number. The function ω(k), which gives ω as a function of k, is known as the dispersion relation.

The Phase Velocity v = l f v = w / k How fast is the wave traveling? Velocity is a reference distance divided by a reference time. The phase velocity is the wavelength / period: v = l / t Since f = 1/t : In terms of k, k = 2p / l, and the angular frequency, w = 2p / t, this is: v = l f v = w / k

The Group Velocity This is the velocity at which the overall shape of the wave’s amplitudes, or the wave ‘envelope’, propagates. (= signal velocity) Here, phase velocity = group velocity (the medium is non-dispersive)

Dispersion: phase/group velocity depends on frequency Black dot moves at phase velocity. Red dot moves at group velocity. This is normal dispersion (refractive index decreases with increasing λ)

Dispersion: phase/group velocity depends on frequency Black dot moves at group velocity. Red dot moves at phase velocity. This is anomalous dispersion (refractive index increases with increasing λ)

Normal dispersion of visible light Shorter (blue) wavelengths refracted more than long (red) wavelengths. Refractive index of blue light > red light.

Dispersive and Non-dispersive Media In a non-dispersive medium all of the different frequency components travel at the same speed so the wave function doesn't change at all as it travels. In a dispersive media, the wave speed depends on the frequency of the wave. The blue wave pulse in the animation is the same Gaussian function as the black pulse, consisting of a large number of frequency components added together. However, the wave speed depends on frequency, with higher frequencies traveling faster than lower frequencies. As a result, the wave pulse spreads out and changes shape as it travels.