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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- Differential equations can be used to model a variety of physical systems.
- The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application.
- As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond.
- All of them may be described by the same second-order partial-differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water.
- Conduction of heat is governed by another second-order partial differential equation, the heat equation .
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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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- 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: