For that reason, it is called the heat equation in mathematics, even though it applies to many other physical quantities besides temperatures. This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in a homogeneous isotropic non-conducting solid. This differential equation is called "the" wave equation in mathematics, even though it describes only one very special kind of waves.

Consider a traveling transverse wave which may be a pulse on a string the medium. Consider the string to have a single spatial dimension. Consider this wave as traveling. A generalized representation of this wave can be obtained [6] as the partial differential equation. General solutions are based upon Duhamel's principle. Constant values of this argument correspond to constant values of F , and these constant values occur if x increases at the same rate that vt increases.

That is, the wave shaped like the function F will move in the positive x -direction at velocity v and G will propagate at the same speed in the negative x -direction. The amplitude of a wave may be constant in which case the wave is a c. The outline of the variation in amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written in the form: [10] [11] [12]. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation. There are two velocities that are associated with waves, the phase velocity and the group velocity.

Phase velocity is the rate at which the phase of the wave propagates in space : any given phase of the wave for example, the crest will appear to travel at the phase velocity.

The units of the amplitude depend on the type of wave. These are related by:.

### Do the Wave, Everyone Else Is!

It is related to the frequency or period by. Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, the incoming wave undulates with a varying local wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth. Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems , in the presence of dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude, making it easy to analyze.

The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves that exist for a limited span in space and duration in time.

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## Mechanical and Electromagnetic Vibrations and Waves | Ebook | Ellibs Ebookstore

An arbitrary wave shape can be decomposed into an infinite set of sinusoidal waves by the use of Fourier analysis. As a result, the simple case of a single sinusoidal wave can be applied to more general cases. A standing wave, also known as a stationary wave , is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. The sum of two counter-propagating waves of equal amplitude and frequency creates a standing wave.

Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example, when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut , where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node.

Halfway between two nodes there is an antinode , where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time. One-dimensional standing waves; the fundamental mode and the first 5 overtones. A two-dimensional standing wave on a disk ; this is the fundamental mode.

## Electromagnetic Waves

A standing wave on a disk with two nodal lines crossing at the center; this is an overtone. Waves normally move in a straight line that is, rectilinearly through a transmission medium. Such media can be classified into one or more of the following categories:. Absorption of waves means, if a kind of wave strikes a matter, it will be absorbed by the matter. When a wave with that same natural frequency impinges upon an atom, then the electrons of that atom will be set into vibrational motion.

If a wave of a given frequency strikes a material with electrons having the same vibrational frequencies, then those electrons will absorb the energy of the wave and transform it into vibrational motion. When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line normal to the surface equals the angle made by the reflected wave and the same normal line.

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Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the phase velocity changes. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law. A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening.

Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave. Waves that encounter each other combine through superposition to create a new wave called an interference pattern. Important interference patterns occur for waves that are in phase.

The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions. Transverse waves can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of linear polarization. A transverse wave is linearly polarized if it oscillates in only one direction or plane. In the case of linear polarization, it is often useful to add the relative orientation of that plane, perpendicular to the direction of travel, in which the oscillation occurs, such as "horizontal" for instance, if the plane of polarization is parallel to the ground.

Electromagnetic waves propagating in free space, for instance, are transverse; they can be polarized by the use of a polarizing filter. Longitudinal waves, such as sound waves, do not exhibit polarization.

For these waves there is only one direction of oscillation, that is, along the direction of travel. A wave undergoes dispersion when either the phase velocity or the group velocity depends on the wave frequency. Dispersion is most easily seen by letting white light pass through a prism , the result of which is to produce the spectrum of colours of the rainbow.

Isaac Newton performed experiments with light and prisms, presenting his findings in the Opticks that white light consists of several colours and that these colours cannot be decomposed any further. Seismic waves are waves of energy that travel through the Earth's layers, and are a result of earthquakes, volcanic eruptions, magma movement, large landslides and large man-made explosions that give out low-frequency acoustic energy.

A shock wave is a type of propagating disturbance. When a wave moves faster than the local speed of sound in a fluid , it is a shock wave. Like an ordinary wave, a shock wave carries energy and can propagate through a medium; however, it is characterized by an abrupt, nearly discontinuous change in pressure , temperature and density of the medium.

An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. In the 19th century, James Clerk Maxwell showed that, in vacuum , the electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed of light.

From this emerged the idea that light is an electromagnetic wave. Electromagnetic waves can have different frequencies and thus wavelengths , giving rise to various types of radiation such as radio waves , microwaves , infrared , visible light , ultraviolet , X-rays , and Gamma rays. Solutions of this equation are wave functions which can be used to describe the probability density of a particle.

The Dirac equation is a relativistic wave equation detailing electromagnetic interactions. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. Louis de Broglie postulated that all particles with momentum have a wavelength.

This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. A wave representing such a particle traveling in the k -direction is expressed by the wave function as follows:. However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet , [28] a waveform often used in quantum mechanics to describe the wave function of a particle.

Waves take up different forms, and hence it is vital to learn about the types of waves. Below mentioned are the three categories of waves which are discussed with their sub-segments:. Mechanical Waves behave as a propagation of a particular disturbance traveling through a material medium as a result of the constant periodic motion of particles.

Under this, the disturbance is transferred from one particle to the next. To be more specific, energy and momentum transmit by the motion of particles in the medium. One must note that mechanical waves cannot propagate through the vacuum. Learn about Transverse and Longitudinal Wave in more detail here. In these types of waves, the periodic changes occur in electric and magnetic fields ; therefore, it is termed as Electromagnetic Wave.