Examples of wave equation in the following topics:
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- is a solution of the wave equation for $c= \frac{\omega}{k}$.
- is a solution to the wave equation.
- Converting back to the original variables of x and t, we conclude that the solution of the original wave equation is
- Any function that contains "x+ct" or "x-ct" can be a solution of the wave equation.
- A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge.
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- There are two main types of waves.
- More generally, waveforms are scalar functions $u$ which satisfy the wave equation, $\frac{\partial^2u}{\partial t^2}=c^2\nabla^2u$.
- By taking derivatives, it is evident that the wave equation given above holds for $c = \frac{\omega}{k}$, which is also called the phase speed of the wave.
- One important aspect of the wave equation is its linearity: the wave equation is linear in u and it is left unaltered by translations in space and time.
- Since a wave with an arbitrary shape can be represented by a sum of many sinusoidal waves (this is called Fourier analysis), we can generate a great variety of solutions of the wave equation by translating and summing sine waves that we just looked closely into.
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- A sea wave is an example of a wave in which water molecules are moving up and down as waves propagate towards the shore.
- Waves transfer energy not mass.
- While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse.
- The description of waves is closely related to their physical origin for each specific instance of a wave process.
- A brief introduction to the wave equation, discussing wave velocity, frequency, wavelength, and period.
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- The most common symbols for a wave function are ψ(x) or Ψ(x) (lowercase or uppercase psi, respectively), when the wave function is given as a function of position x.
- The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time.
- The wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation.
- This explains the name "wave function" and gives rise to wave-particle duality.
- Relate the wave function with the probability density of finding a particle, commenting on the constraints the wave function must satisfy for this to make sense
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- Waves are defined by its frequency, wavelength, and amplitude among others.
- Waves have certain characteristic properties which are observable at first notice.
- where v is called the wave speed, or more commonly,the phase velocity, the rate at which the phase of the wave propagates in space.
- Finally, the group velocity of a wave is the velocity with which the overall shape of the waves' amplitudes — known as the modulation or envelope of the wave — propagates through space.
- Conversely we say that the purple wave has a high frequency.
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- An antenna is a device that converts electric power into radio waves, and vice versa.
- Maxwell's equations predicted that all light waves have the same structure, regardless of wavelength and frequency.
- As a consequence, visible light and radio waves should share common characteristics.
- Maxwell's 1865 prediction passed an important test in 1888, when Heinrich Hertz published the results of experiments in which he showed that radio waves could be manipulated in the same ways as visible light waves.
- These serve to direct the radio waves into a beam or other desired radiation pattern.
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- Wave–particle duality postulates that all physical entities exhibit both wave and particle properties.
- Wave–particle duality postulates that all physical entities exhibit both wave and particle properties.
- In 1861, James Clerk Maxwell explained light as the propagation of electromagnetic waves according to the Maxwell's equations.
- De Broglie's wave (matter wave): In 1924, Louis-Victor de Broglie formulated the de Broglie hypothesis, claiming that all matter, not just light, has a wave-like nature.
- So, why do we not notice a baseball acting like a wave?
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- The force you feel from a wave hitting you at the beach is an example of work being done and, thus, energy being transfered by a wave in the direction of the wave's propagation.
- Energy transportion is essential to waves.
- It is a common misconception that waves move mass.
- Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields .
- This is a direct result of the equations above.
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- Electromagnetic waves are the combination of electric and magnetic field waves produced by moving charges.
- These waves oscillate perpendicularly to and in phase with one another.
- When it accelerates as part of an oscillatory motion, the charged particle creates ripples, or oscillations, in its electric field, and also produces a magnetic field (as predicted by Maxwell's equations).
- Electromagnetic waves are a self-propagating transverse wave of oscillating electric and magnetic fields.
- Notice that the electric and magnetic field waves are in phase.
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- There are two different kinds of sound waves: compression waves and shear waves.
- Compression waves can travel through any media, but shear waves can only travel through solids.
- The speed of sound is usually denoted by $c$, and a general equation can be used to calculate it.
- From this equation, it is easy to see that the speed of sound will increase with stiffness and decrease with density.
- This is a very general equation, there are more specific derivations, for example: