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That part of the North Atlantic bounded on the north by West Africa and on the east by Equatorial Africa. The small African nation São Tomé and Príncipe and the zero-point or origin of the Geographic coordinate system both lie within it.

Group velocity
The 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. For example, imagine what happens if you throw a small pebble into the middle of a very still pond. When the pebble hits the surface of the water, a circular pattern of waves appears. It soon turns into a circular ring of waves with a quiescent center. The ever expanding ring of waves is the group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The longer waves travel faster than the group as a whole, but they die out as the approach the leading edge. The shorter waves travel slower and they die out as they emerge from the trailing boundary of the group.

Formulas for group velocity
For more precision, a mathematical expression for group velocity is needed. The terminology follows that used in the Wave article.

A group of two wavelets
Consider a group consisting of the sum of two sinusoidal wavelets



d = \sin ((k+\Delta k)x - (\omega + \Delta\omega)t) + \sin ((k-\Delta k)x - (\omega - \Delta\omega)t) \, $$

where both $$ \Delta k $$ and $$ \Delta\omega $$ are small so that they have nearly the same wavenumber and angular frequency. Using the formula for the sine of the sum of two angles yields four terms, of which two cancel and the other two combine to give



d = 2 \sin (kx - \omega t) \cos (\Delta k\,x - \Delta\omega\,t). $$

which is the equation of a modulated wave where the sine factor is the carrier and the cosine factor is the envelope of the waveform that gives it its shape. The velocity of the carrier is $$ v_p = \omega / k \, $$, but the velocity of the envelope is $$ v_e = \Delta\omega / \Delta k $$. The velocity of the shape is that of the shape so $$ v_g = v_e = \Delta\omega / \Delta k $$.

The group velocity is defined by the equation


 * $$v_g \ \equiv\ \frac{\partial \omega}{\partial k}\,$$

where:
 * vg is the group velocity;
 * ω is the wave's angular frequency;
 * k is the wave number.

The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform.

The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.

Matter wave group velocity
Albert Einstein first explained the wave-particle duality of light in 1905. Louis de Broglie hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded then (but may be questioned today, see above), should always equal the group velocity of the corresponding wave. De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold.

Loligo pealei