Elastic wave

Elastic wave . They are a tension disturbance that propagates along an elastic medium. For example, seismic waves cause tremors that can be treated as elastic waves that propagate through the ground, which can cause damage in areas where there are urban settlements.

Summary

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  • 1 Transverse and longitudinal waves
  • 2 Speed
  • 3 Energy and intensity
  • 4 Damping
  • 5 Source

Transverse and longitudinal waves

Oscillations that occur at any one point in an elastic medium are transmitted to neighboring points, which also begin to oscillate. The process of transmitting the oscillations from one point to another is characteristic not only of the elastic media, but also of the electromagnetic field .

Waves are the disturbances of the state of a substance or a field that propagate in space. The oscillations of the substance cause an elastic wave; those of the electromagnetic field, an electromagnetic wave .

In a long tube, filled with a gas or a liquid, a piston is inserted that performs harmonic oscillations. The oscillations of this piston, by virtue of the action of the forces of elasticity, are transmitted to the gas, which means that Along the tube an elastic wave propagates. This wave constitutes a system of regions of compression and thinning of the medium that periodically change their state: if in a certain instajite a compression is observed at any point in the medium and in the neighbor a thinning, after half a period in the first region it will produce thinning and in the second, compression and so on.

It is noted that in this case the oscillations of the particles of the elastic medium take place in the same direction in which the oscillations are transmitted from layer to layer, that is, along the direction in which the wave propagates. When the oscillations of the particles occur along the direction in which the wave propagates, the wave is said to be longitudinal.

The waves produced on the surface of a liquid are not due to the elasticity of the medium, but due to the forces of surface tension or gravity. The peculiarity of these waves is that the liquid particles oscillate in a vertical direction, while the wave propagates in a horizontal plane. When the oscillations of the particles in the medium are perpendicular to the direction in which the wave propagates, it receives the name of transversal.

In solids, both longitudinal and transverse waves are possible. The longitudinal wave is produced as a result of a deformation by compression or rarefying, the same as in gases and liquids. The transverse wave is due to shear deformation. Gases and liquids lack shear elasticity and no transverse waves are produced in them.

The locus of the points that always oscillate the same, that is, in the same phase, is called the surface or wavefront. If the wave surfaces are planes perpendicular to the direction in which the wave propagates, the waves are called flat.

An example of spherical waves can be the waves that are produced in the air around a small isotropic source of sound, eg, around a bell.

Lightning is the line whose tangent at each of its points coincides with the direction of propagation of the wave, that is, with the direction of energy transport . In a homogeneous medium the ray is a straight line perpendicular to the wavefront.

For example, if the wave is excited by a point source, the wavefront is sphere-shaped and the rays are radial lines.

Speed

High-amplitude elastic waves are called shock waves; those of small amplitude (or waves of small disturbances), sound or acoustic waves. In § 30J the expressions Formulas were obtained for the speed of acoustic waves in gases

In the air: Formulas

For T = 273 K we obtain a = 330 m / s, and for T = 293 K we have a = 343 m / s, which agrees well with the experimental results.

The speed of sound waves in solids and liquids depends on their compressibility (elasticity) and their density.

To calculate the velocity of the transverse wave (S wave) in solids, the compressibility modulus K must be replaced by the stiffness modulus G:

The modulus of stiffness is approximately 2-4 times less than that of compressibility, so the velocity of the transverse waves is approximately 1.5 times less than the velocity of the longitudinal waves. Experience confirms this result. Thus, in granite, the velocity of the longitudinal wave ap = 5400 m / s and that of the transverse wave as 3300 m / s; in the basalt ap = 6300 m / s and as = 3700 m / s.

The seismic prospecting methods for minerals are based on this speed difference . In a hole drilled underground, a charge is exploded; the initial moment of the explosion is registered by means of a sensor. The waves reflected in different regions of the terrain are recorded in a set of seismographs, from which the oscillations are transmitted to a seismic station . In this they are amplified and, together with the time signals , they are recorded on a tape. The analysis of the seismograms gives an idea of ​​how the minerals are distributed. This method is widely used in prospecting for oil , gas , minerals, etc.

If the longitudinal wave is propagated by a rod, and not in a medium of unlimited extension, the compressibility modulus K must be replaced by Young E:

Energy and intensity

By mentally isolating a certain region of an ethical medium, of volume V, in which a wave of amplitude A and frequency o propagates. The energy in this volume W = ‘/ 2 mo2A’. Dividing it by the volume, the expression of the average density of the wave energy is obtained: Formulas. where p is the density of the medium.

The intensity of a wave is called the magnitude equal to the energy that on average transports the wave through the surface unit in the time unit .

Let P be the power of the wave. Suppose At »T where T is the period of the oscillations. During the time At the energy contained in the volume AV = SuAt, in which u is the wave velocity, will pass through the surface: the allergy AW = wAV = wSuAt.

The magnitude equal to the product of the density of the medium by the speed of sound in it z = pu, is called acoustic impedance and characterizes the wave properties of it.

Cushioning

Elastic waves are always absorbed by the substance, with the particularity that the degree of absorption depends on many factors. Deduce the law of absorption of flat waves (of parallel rays). For light this law was discovered and founded by P. Bouguer in 1729 .

Suppose the plane wave passes through a layer of substance whose thickness is x. The intensity of the wave varies from magnitude 1 to 1 <Jo. Transparency D of the given layer of substance for this wave will be called the ratio of the intensity of the wave transmitted to the initial intensity: Formulas

Admit it that the transparency of the given layer of substance depends only on its thickness, and not on the intensity of the wave: Formulas

It is not difficult to verify that the functional equation can be satisfied by the exponential function f (x) – a °. Choose the number a = 2 as the base. Taking into account that 1 (x) is a decreasing function, it is observed that the coefficient a contained in the exponent must be a negative number:

The law of damping of plane waves (Bourguer’s law) is written as follows: Formula

Or, otherwise, Formulas and Image

The magnitude L is called the semi-absorption layer. Indeed, if the wave passes through a layer of thickness x = L, Formula, that is, the intensity of the wave is halved. Linear absorption coefficient Formula

The assumption that the transparency of the medium layer is independent of wave intensity plays an important role in deducing the law of absorption. In the case where the transparency depends on the intensity, the absorption law is no longer expressed by Formula 1. This occurs mainly with the shock waves.

 

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