# Wave number, frequency, and velocity in dispersive waves

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• Ahmed1029
In summary, dispersive waves can have well-defined constant wave number and frequency, but this is not always the case. The velocity of any point on the wave will stay constant in time, but this is only true for waves that are not localized. The interaction of force with waves depends on the specific type of wave, and forces can cause the wave to change its shape or velocity over time. It is important to understand both non-dispersive and dispersive harmonic waves before studying more complex systems. Landau and Lifshitz vol. 8 provide a comprehensive description of dispersive waves and their properties.
Ahmed1029
1) If I generate a dispersive wave, will it have well-defined constant wave number and frequency? Ones that don't change in time?

2) does the velocity of any point on the wave stay constant in time?

3) How does force interact with waves? Does a free wave act in analogy with free particles? with every point having the same velocity over time unless disturbed?

Ahmed1029 said:
1) If I generate a dispersive wave, will it have well-defined constant wave number and frequency? Ones that don't change in time?

2) does the velocity of any point on the wave stay constant in time?

3) How does force interact with waves? Does a free wave act in analogy with free particles? with every point having the same velocity over time unless disturbed?
1)One can produce such an harmonic wave but it will of necessity not be localized
2)Because it is not localized, the velocity is the phase velocity. It is, by its definition, a constant in time
3)You need to be much more specific here. Typically forces will not be constant in time and/or will reshape the wave.

There are many kinds of waves and the specifics depend upon the wave. For example, there are nonlinear systems which admit solitary wave solutions whose shape is fixed but whose amplitude determines the velocity. This world is complicated.

You should first study dispersive harmonic waves (after understanding the nondispersive ones!)

vanhees71, topsquark and Ahmed1029
For electromagnetic waves, have a look at Landau and Lifshitz vol. 8. There you get a comprehensive description from first principles for both the usual "temporal dispersion" (valid, e.g., for usual dielectrics and visible light) as well as including "spatial dispersion" (needed, e.g., for em. waves in a plasma).

hutchphd and Ahmed1029

## 1. What is the relationship between wave number, frequency, and velocity in dispersive waves?

The wave number, frequency, and velocity in dispersive waves are all related through the dispersion relation, which is a mathematical equation that describes the relationship between these three properties. In general, the higher the wave number, the higher the frequency and the lower the velocity. However, in dispersive waves, this relationship is not constant and can vary depending on the medium and the type of wave.

## 2. How does the dispersion of waves affect their behavior?

The dispersion of waves can affect their behavior in several ways. In dispersive waves, different frequencies travel at different speeds, causing the wave to spread out over time. This can result in the wave breaking up or changing shape as it propagates. Additionally, dispersion can cause waves to travel in different directions, which can lead to interference and the formation of complex wave patterns.

## 3. Can dispersive waves be found in all types of mediums?

Yes, dispersive waves can be found in all types of mediums, including solids, liquids, and gases. The degree of dispersion, however, can vary depending on the properties of the medium. For example, waves in water tend to exhibit more dispersion than waves in air.

## 4. How do dispersive waves differ from non-dispersive waves?

Non-dispersive waves, also known as monochromatic waves, have a constant frequency and wavelength and do not spread out or change shape as they propagate. In contrast, dispersive waves have a range of frequencies and wavelengths and can exhibit spreading and changing behavior. Non-dispersive waves are commonly found in simple systems, while dispersive waves are more common in complex systems.

## 5. What are some practical applications of understanding wave number, frequency, and velocity in dispersive waves?

Understanding wave number, frequency, and velocity in dispersive waves is crucial in various fields, including oceanography, meteorology, and seismology. In oceanography, the dispersion of waves can help predict the formation and behavior of ocean swells, which are essential for shipping and offshore activities. In meteorology, dispersion is important for understanding the propagation of weather systems, such as hurricanes and typhoons. In seismology, dispersion can help identify different types of seismic waves and provide information about the structure of the Earth's interior.

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