it was noted by Bohr that the existence of any sort of wave packet implies uncertainty in the wave frequency and wavelength, since a spread of frequencies is needed to create the packet itself.

In quantum mechanics, where all particle momenta are associated with waves, it is the formation of such a wave packet which localizes the wave, and thus the particle, in space.

In states where a quantum mechanical particle is bound, it must be localized as a wave packet, and the existence of the packet and its minimum size implies a spread and minimal value in particle wavelength, and thus also momentum and energy.

In quantum mechanics, as a particle is localized to a smaller region in space, the associated compressed wave packet requires a larger and larger range of momenta, and thus larger kinetic energy.

In chemistry, Schrödinger, Pauling, Mulliken and others noted that the consequence of Heisenberg's relation was that the electron, as a wave packet, could not be considered to have an exact location in its orbital.

Max Born suggested that the electron's position needed to be described by a probability distribution which was connected with finding the electron at some point in the wave-function which described its associated wave packet.

In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number n for each orbital became known as an n-sphere in a three dimensional atom and was pictured as the mean energy of the probability cloud of the electron's wave packet which surrounded the atom.

Once the electronic and nuclear variables are separated ( within the Born – Oppenheimer representation ), in the time-dependent approach, the wave packet corresponding to the nuclear degrees of freedom is propagated via the time evolution operator ( physics ) associated to the time-dependent Schrödinger equation ( for the full molecular Hamiltonian ).

The most popular methods for propagating the wave packet associated to the molecular geometry are:

Again, though, other physicists believe that tunneling experiments in which particles appear to spend anomalously short times inside the barrier are in fact fully compatible with relativity, although there is disagreement about whether the explanation involves reshaping of the wave packet or other effects.

Solid line: A wave packet.

Dashed line: The envelope of the wave packet.

Consider a wave packet as a function of position x and time t: α ( x, t ).

where ω is implicitly a function of k. We assume that the wave packet α is almost monochromatic, so that A ( k ) is nonzero only in the vicinity of a central wavenumber k < sub > 0 </ sub >.

As a result, the envelope of the wave packet not only moves, but also distorts.

Eventually, the wave packet gets stretched out.

For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x < sub > 0 </ sub > ( neither an eigenstate of position nor of momentum ).

The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity ( like a classical particle with no forces acting on it ).

However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain with time.

This also has the effect of turning a position eigenstate ( which can be thought of as an infinitely sharp wave packet ) into a broadened wave packet that no longer represents a ( definite, certain ) position eigenstate.

These areas are eroded at a faster pace creating a hole or crevasse that, through time, by means of wave action and erosion, becomes a cave.

The noise travels around the loop and is amplified and filtered until very quickly it becomes a sine wave at a single frequency.

In this Feynman diagram, an electron and a positron annihilate, producing a photon ( represented by the blue sine wave ) that becomes a quark-antiquark pair.

When pushed all the way in, the specified component wave form becomes absent from the mix.

In 1678, Huygens proposed that every point to which a luminous disturbance reaches becomes a source of a spherical wave ; the sum of these secondary waves determines the form of the wave at any subsequent time.

Because the group velocity varies with k, the shape of the wave packet broadens with time, and the particle becomes less localized.

For an asymmetric wave ( periodic pulses in one direction, for example ), the peak amplitude becomes ambiguous.

It is straightforward to show that if a plane wave is incident at any arbitrary angle θ < sub > i </ sub >, the grating equation becomes:

At the heart of Fresnel's wave theory is the Huygens-Fresnel principle, which states that every unobstructed point of a wavefront becomes the source of a secondary spherical wavelet and that the amplitude of the optical field E at a point on the screen is given by the superposition of all those secondary wavelets taking into account their relative phases.

Waves must be sampled at more than two points per wavelength, or the wave arrival direction becomes ambiguous.

Bowl: A section of the wave that becomes steep approaching a peak where the bodyboarder can hit and become airborne.

However, the situation becomes more complicated when the finite speed of electromagnetic wave propagation is introduced ( see retarded potential ).

However, as the speed increases and the shock wave increasingly approaches the sides of the craft, there comes a point where the two start to interact and the flowfield becomes very complex.

* Under the effect of the pressure of the detonation wave, the tube deforms and becomes a cone which contacts the helically wrapped coil, diminishing the number of turns not short-circuited, compressing the magnetic field and creating an inductive current ;

Now, even if the local speed of the air on the upper surface of the wing becomes supersonic, a shock wave cannot form there because it would have to be a sweptback shock-swept at the same angle as the wing-ie., it would be an oblique shock.

# Those in which the energy associated with the wave is used to excite some other phenomenon within the region of space where the original traveling wave becomes evanescent ( for example, as in the total internal reflection fluorescence microscope )

In the limit of large field the state becomes a good approximation of a noiseless stable classical wave.

In the Eastern United States a heat wave can occur when a high pressure system originating in the Gulf of Mexico becomes stationary just off the Atlantic Seaboard ( typically known as a Bermuda High.

However, because their deaths have been hastened by the heat wave, in the months that follow the number of deaths becomes lower than average.

It follows that the energy and momentum flux in this wave field only becomes significant at extremely short wavelengths where directional coupler technology is currently lacking.

This plane wave spectrum representation of the electromagnetic field is the basic foundation of Fourier Optics ( this point cannot be emphasized strongly enough ), because when z = 0, the equation above simply becomes a Fourier transform ( FT ) relationship between the field and its plane wave content ( hence the name, " Fourier optics ").