Waves traveling, equation

A. The simplest solution to this equation is the uniform plane wave traveling in an arbitrary direction denoted by the vector k ... 

The two component plane waves in equation (1.10) travel with equal phase velocities co/k, but in opposite directions. Using equations (A.31) and (A.32), we can express equation (1.10) in the form... 

When a wave, such as that on the sea, travels in a particular direction, we can characterize it by a number of parameters - the velocity of travel, its amplitude (the height of the waves), its wavelength (distance between adjacent peaks), and its frequency (the number of waves per unit time). The simplest form of wave to consider is a sine wave, the equation for... 

It is important to note that the velocity of the wave in the direction of propagation is not the same as the speed of movement of the medium through which the wave is traveling, as is shown by the motion of a cork on water. Whilst the wave travels across the surface of the water, the cork merely moves up and down in the same place the movement of the medium is in the vertical plane, but the wave itself travels in the horizontal plane. Another important property of wave motion is that when two or more waves traverse the same space, the resulting wave motion can be completely described by the sum of the two wave equations - the principle of superposition. Thus, if we have two waves of the same frequency v, but with amplitudes A and A2 and phase angles

resulting wave can be written as ... 

For a building with a flat roof (pitch less than 10°) it is normally assumed that reflection does not occur when the blast wave travels horizontally. Consequently, the roof will experience the side-on overpressure combined with the dynamic wind pressure, the same as the side walls. The dynamic wind force on the roof acts in the opposite direction to the overpressure (upward). Also, consideration should be given to variation of the blast wave with distance and time as it travels across a roof element. The resulting roof loading, as shown in Figure 3.8, depends on the ratio of blast wave length to the span of the roof element and on its orientation relative to the direction of the blast wave. The effective peak overpressure for the roof elements are calculated using Equation 3.11 similar to the side wall. 

The velocity with which a pure thermal wave travels through an insulated packed bed may be obtained from equation 17.100 by putting Uq = 0 and (d/dT)(CsAH) = 0 to give ... 

Equation (3.2.24b) (or more generally for m < 0, the appropriate integral of (3.2.21b), satisfying the boundary condition (3.2.20b)), represents a monotonic wave, travelling from left to right with speed c. In order to specify c the boundary condition (3.2.20b) has to be modified to... 

It is clear that sound, meaning pressure waves, travels at finite speed. Thus some of the hyperbolic—wavelike-characteristics associated with pressure are in accord with everyday experience. As a fluid becomes more incompressible (e.g., water relative to air), the sound speed increases. In a truly incompressible fluid, pressure travels at infinite speed. When the wave speed is infinite, the pressure effects become parabolic or elliptic, rather than hyperbolic. The pressure terms in the Navier-Stokes equations do not change in the transition from hyperbolic to elliptic. Instead, the equation of state changes. That is, the relationship between pressure and density change and the time derivative is lost from the continuity equation. Therefore the situation does not permit a simple characterization by inspection of first and second derivatives. 

This solution is the equation for a wave traveling to the right (+x direction, the first term) and to the left (—x direction, second term). We can impose a boundary condition, namely, we can specify the particle is traveling in the +x direction. Then we have... 

A preliminary step to dielectric measurement by wave-transmission techniques is to relate the basic wave parameter, called the propagation factor, 7 of the material, to permittivity. In terms of the propagation factor the equations for the electric and magnetic fields of a plane wave travelling in the x-direction in a uniform, infinite material are ... 

The subscripts 1 and 2 refer to the material the wave travels in and the material that is reflected by or transmitted into, respectively. These equations show that the maximum transmission of ultrasound occurs when the impedances and Z2 of the two materials are identical. The materials are then said to be acoustically matched. If the materials have very different impedances, then most of the US is reflected. The reflection and transmission of ultrasound at boundaries has important implications on the design of ultrasonic experiments and the interpretation of their results. In addition, measurements of the reflection coefficient are often used to calculate the impedance of a material. 

The simplest type of wave in three dimensions is the plane wave in which, by definition, the disturbance at any instant of time has the same value at all points in any given plane that is perpendicular to the direction of propagation. Such a wave traveling along the x direction is described by the equation... 

Let us consider an isolated molecule perturbed by an electromagnetic field. According to the semiclassical approach, the external radiation is described as a plane monochromatic wave traveling with velocity c and obeying the Maxwell equations [21] (i.e., the fields are not quantized). 

Let us analyze the space and time structure of the elastic displacement field in detail. We will demonstrate that equation (13.26) describes the propagation of two types of body waves in an elastic medium, i.e., compressional and shear waves travelling at different velocities and featuring different physical properties. To this end, let us recall the well-known Helmholtz theorem according to which an arbitrary vector field, in particular an elastic displacement field U(r), may be represented as a sum of a potential, Up(r), and a solenoidal, Us(r), field (Zhdanov, 1988) ... 

As we can see here, both the potential and solenoidal components of the elastic displacement field satisfy wave equations and therefore represent waves traveling in space at velocities Cp and Cg respectively. Let us examine them in detail. 

We can see that the first term in equation (13.67) characterizes a spherical wave traveling from the origin of the coordinates to infinity, while the second one is a spherical wave traveling from infinity to the origin of the coordinates. Since the physical formulation of the problem contains only one source at the origin of coordinates, the function g2 in equation (13.67) must be taken as equal to zero. Thus, the Green s function G (r, t), as one could anticipate, represents a divergent spherical wave ... 

One can assume that the Kirchhoff integral over the surface Or characterizes the superposition of waves traveling across that surface from the surrounding outer space inside domain K-- Therefore, equations (13.199) and (13.200) show that in the given model, only outward waves travel across the surface Or, providing the radius r is large enough. 

Equation (22) describes an ellipsoid (see Fig. 2) called the index ellipsoid. The latter is very useful in deriving the refractive index of optical waves with different polarization and propagation direction. A wave traveling in a uniaxial polymer at an angle d with respect to the optic axis experiences two different index depending on its polarization if the wave is s-polarized (perpendicular to the plane of incidence) the refractive index is n and is independent of 0 for a p-polarized wave (polarization in the plane of incidence) the refractive index is given by... 

This equation describes propagation of sound waves, with wave speed Cs = jEjp, and is therefore usually referred to as the wave equation. In fact, the general solution may be expressed as u(x, t) =fR x — Cgt) +/l(x + Cgt), where the first term represents a wave traveling to the right and the second, a wave traveling to the left. Here, /r and are arbitrary functions of the indicated arguments. Although this particular derivation pertains to sound waves, all wave motions are in fact described by equations of the same form. 

One possible solution of the first equation in (5), corresponding to the plane waves, traveling along the z axis and having the same amplitude and phase everywhere [24], has the form [14,24,25]... 

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