Traveling wave coordinate

A useful technique which can be applied in cases where constant velocity solutions arise is that of changing from a fixed coordinate system (the present x coordinate has an origin whose position is fixed in space) to travelling-wave coordinates. With the latter, the origin moves from left to right with the front at the same constant velocity c we are constantly adjusting our frame of reference so that within the frame the reaction front appears stationary. This new coordinate z is defined in terms of x and t by... 

If both reactions (11.1) and (11.30) occur, the dimensionless reaction-diffusion equation for the concentration of B in terms of the travelling-wave coordinate z can be written as... 

The boundary condition after the front (at z = — oo) is simply p = dp/ dz = 0. Ahead of the wave we have 0 = 0+ and d0/dz = 0 at some (unknown) z. In order to fix these pre-front conditions at + oo, we must rescale our distance coordinate. If we introduce a new travelling wave coordinate defined by... 

The a-u phase plane is shown in Figure 3. The evolution of the system, now considered as a function of the traveling wave coordinate z, corresponds to a trajectory across this plane. The initial starting point for this trajectory will be the state a,u) = (0,0), the boundary condition at z = -00. An acceptable solution trajectory must approach the final state (a, u) = (1,0) as z -1-00. (Note that the initial point in terms of z corresponds to complete reaction, i.e., the post-front composition in time.)... 

In Fig. 5, we indicate the coordinate system and the notation used to model the waveguide configuration. The electric and magnetic fields are decomposed into two traveling waves a positive (Ei, Hi) and a negative (E2, H2) propagating wave along the z-direction. Thus, for each medium j we write ... 

W(ri,T,p) Chemical reaction rate for gas-solid systems X Spatial coordinate in traveling wave frame see Eq. (10)... 

Since the system (3.22), (3.23) is already written in a moving coordinate system, the traveling wave solution is a stationary solution of this problem,... 

We assume that the basic solution loses stability when y, exceeds a critical value, and we seek solutions appearing as a result of this instability. Let us hrst describe the conditions for the loss of stability. Consider the problem (3.87) linearized about the basic solution and written in the moving coordinate system attached to the traveling wave... 

We observe that the solutions we seek are not traveling wave solutions in the strict sense of the word. For a traveling wave solution u, we must have u = u x), where x = x + ct for some constant speed c. It can be also written d x = x — ip t) with (p t) = —ct. Here, ip t) is the coordinate of the front (or any other characteristic point of the solution). We seek solutions with (fi = ip y,t). That is, we look for solutions u of the problem (3.87) having the form... 

In this section, we determine stationary solutions of the above problem, which correspond to uniformly propagating one-dimensional traveling waves in the laboratory coordinate system. We solve the following reactionless system ahead of x < 0) and behind x > 0) the front... 

Figure 8.1.2 Propagation of a traveling wave with density, normalized particle displacement, particle velocity and sound pressure as a function of normalized time or coordinate.

It was noted in Section III. A that, in addition to quasista-tionaiy Rossby waves, travehng waves are also present in the stratosphere. For a wave of the form (31), a traveling wave is simply one forwhichthe phase speed c is nonzero the perturbation associated with such a wave moves with respect to the space coordinates. 

In this equation, a one-dimensional coordinate system z is laid alongthe direction of traveling waves and the electron beam. It is assumed that the interaction of electrons and traveling electromagnetic waves started at z = 0 and the time t = 0. In Eq. (6.2), Urn is the magnitude of velocity modulation at the initial location, the attenuation constant of the velocity modulation, Pe the phase constant of the electronic velocity modulation waves in the beam, and co the operating microwave frequency. 

Forward waves In a traveling wave tube, microwaves propagate in the same direction with electrons in the electron beam. Electromagnetic waves travel in the positive direction of the coordinate system. 

Travelling waves with a constant velocity u on an unbounded interval can be studied upon coordinate transformation — mt which brings the partial differential equations (1)... 

The periodicity of the crystal permits description of the displacement of atoms in the form of traveling waves with wavevector k. Solution for the oscillatory displacement of coordinate i takes the form ... 

Let X and z be the horizontal and vertical coordinates, respectively. The bottom of the layer is taken at z = —1, the free surface at z = rf x, f), and the top of the air layer at z = /i, where t is time and r] x t) describes the surface deformation. Thus as already mentioned we restrict consideration to (1 + 1 )D flow motions. To search for only long traveling wave motions in a shallow layer we redefine the horizontal variable, = e x - Ct), where (7 is a phase velocity to be determined. In addition we scale horizontal velocity, pressure and deformation of the surface T] with e, vertical velocity with and introduce the slow time scale r = The scale for temperature is determined by the leading convective contribution to the temperature field which is of order e. Accordingly, the equations governing long wave disturbances are... 

To describe traveling wave solutions, it is customary to introduce a wave coordinate f = X - uf, with soliton velocity v. Integrating the equation of motion once gives... 

We will see that a solution to the time-independent Schrodinger equation provides both a coordinate wave function if and an energy value E. We can immediately write a solution to the time-dependent equation by multiplying a coordinate wave function by the time factor. This type of solution, with the coordinate and time dependence in separate factors, corresponds to a standing wave, because any nodes are stationary. There are also solutions of the time-dependent Schrodinger equation that are not products of a coordinate factor and a time factor. These solutions can correspond to traveling waves. 

The traveling wave equation develops thereby a double periodicity on the coordinate x and on time t. One can, for instance, fix a particle coordinate (jc = const.) and consider its displacement as a function of time. Alternatively, one can fix a moment of time (t = const.) and consider particle displacement as a function of coordinates. So, standing on a pier one can take a picture of the surface of the sea at time instant t, or having thrown an object into the sea (i.e., having fixed a coordinate jc), one can check its oscillation in time. Both these cases are given as graphs in Figure 2.18. 

The traveling wave equation can be obtained as the solution of a differential wave equation referred to as the wave equation. Knowing the solution in the forms (2.8.5) and/or (2.8.7) we can find the wave equation itself. Differentiation of the equation of a plane wave (x, 0 twice upon the time and upon the coordinate gives... 

Figure 7.2 Coordinates for a plane wave. Having defined a reference point O (origin), an arbitrary spatial point of the wave travelling with wavenumber vector k into the K-direction is indicated by r. Cases (a) and (b) differ in the direction of the selected reference axis z (the quantization axis) which does or does not coincide with the direction of the wave, respectively. In case (c) the spin of an electron wave is also indicated it is shown as the double arrow pointing into the direction e against which the spin projection is assumed to...

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