Superposition of solutions

Using the Gaussian plume model and the other relations presented, it is possible to compute ground level concentrations C, at any receptor point (Xq, in the region resulting from each of the isolated sources in the emission inventory. Since Equation (2) is linear for zero or linear decay terms, superposition of solutions applies. The concentration distribution is available by computing the values of C, at various receptors and summing over all sources. 

Since the partial differential equation (2.6) is linear, any linear superposition of solutions is also a solution. Therefore, the most general solution of equation (2.6) for a time-independent potential energy V(x) is... 

See, for example, Chap. 2 in G. Goertzel and N. Tralli, Some Mathematical Methods of Physics, McGraw-Hill, New York, 1960. Because Eq. 4.34 is a set of linear rate laws, although coupled, it is possible to express their solutions as the superposition of solutions of uncoupled (i.e., parallel-reaction) rate laws, as in Eq. 4.35. The number of terms in the superposition will be the same as the number of rate laws (two in the present case). The parameters in Eq. 4.35 are then chosen to make the solutions meet all mathematical conditions imposed by the problem to be solved. 

Predicting fast and slow rates of sorption and desorption in natural solids is a subject of much research and debate. Often times fast sorption and desorption are approximated by assuming equilibrium portioning between the solid and the pore water, and slow sorption and desorption are approximated with a diffusion equation. Such models are often referred to as dual-mode models and several different variants are possible [35-39]. Other times two diffusion equations were used to approximate fast and slow rates of sorption and desorption [31,36]. For example, foraVOCWerth and Reinhard [31] used the pore diffusion model to predict fast desorption, and a separate diffusion equation to fit slow desorption. Fast and slow rates of sorption and desorption have also been modeled using one or more distributions of diffusion rates (i.e., a superposition of solutions from many diffusion equations, each with a different diffusion coefficient) [40-42]. 

The time-dependent wave function ip(x,y,z t) for the internal motion is expanded as a superposition of solutions of the form (2.15) in the following way... 

Now, in view of the linearity of these equations, a superposition of solutions is also a solution, and we can thus represent any arbitrary disturbance in terms of a superposition of normal or Fourier modes. In terms of dimensional variables, a typical such mode can be represented in the form... 

The critical difficulty with this problem is that the solution depends on the orientations of U and f2 relative to axes fixed in the particle, as well as on the relative magnitudes of U and f2. Thus, for every possible orientation of U and/or f2, a new solution appears to be required to calculate u, p, F, or G. Fortunately, however, the possibility of constructing solutions of a problem as a sum (or superposition) of solutions to a set of simpler problems means that this is not actually necessary in the creeping-flow limit. Rather, to evaluate u, p, F, or G for any arbitrary choice of U and f2, we will show that it is sufficient to obtain detailed solutions for translation in three mutually orthogonal directions (relative to axes fixed in the particle) with unit velocity U = e, and il = 0, and for rotation about three mutually orthogonal axes with unit angular velocity il = e, and U = 0. 

Note that since the problem of Stokes flow is linear, one can find the velocity and pressure fields in translational-shear flows as the superposition of solutions describing the translational flow considered in Section 2.2 and shear flows considered in the present section. 

E. One has to be aware of the fact that the time-dependent equation is not only solved by ilj E,x,t), but also by arbitrary superpositions of solutions with different values of E. 

It has been shown that, for property variations for which superposition of solutions is permitted, a series of solutions corresponding to a step in surface temperature can be utilized to represent an arbitrary surface temperature [22]. This approach is identical with the Duhamel method used in heat conduction problems to satisfy time-dependent boundary conditions... 

Stepwise and Arbitrary Heat Flux Distribution. It is often necessary to evaluate the surface temperatures resulting from a prescribed heat flux distribution. The superposition of solutions yields the surface enthalpy distribution as... 

Nonuniform Surface Temperature. Nonuniform surface temperatures affect the convective heat transfer in a turbulent boundary layer similarly as in a laminar case except that the turbulent boundary layer responds in shorter downstream distances The heat transfer to surfaces with arbitrary temperature variations is obtained by superposition of solutions for convective heating to a uniform-temperature surface preceded by a surface at the recovery temperature of the fluid (Eq. 6.65). For the superposition to be valid, it is necessary that the energy equation be linear in T or i, which imposes restrictions on the types of fluid property variations that are permitted. In the turbulent boundary layer, it is generally required that the fluid properties remain constant however, under the assumption that boundary layer velocity distributions are expressible in terms of the local stream function rather than y for ideal gases, the energy equation is also linear in T [%]. 

When the strengths of two neighbouring disclinations are equal and opposite, the brushes connecting them are circular. By superposition of solutions of the type (3.5.3)... 

In such problems, the pressure and velocity fields are described by Stokes equations (8.1) and (8.2). In the most general case particles can perform both translational and rotational motion. Since the Stokes equations and the appropriate boundary conditions are linear, it is possible to consider both motions separately and then form a superposition of solutions. For small volume concentrations of particles, the analysis may be limited to pair interactions between them. For translational motion of particles, the boundary conditions at the particle s surface are ... 

Step 5 Use superposition of solutions to define the inversion back to the H domain. 

Here, r is the distance between the particles, is the radius of the circular contact line at which the membrane detaches fi om the colloid, a is the angle with respect to the horizontal at which it does so, and the Kj, are modified Bessel functions of the second kind. This solution is analytical, simple, and wrong. Or more accurately, it only holds when r 2 ro, a restriction that excludes the interesting tensionless limit in which 2 oo. The mathematical reason is that superposition in the way celebrated here is not allowed yes, superpositions of solutions to linear equations are still solutions, but superpositions of solutions, each of which only satisfies some part of aU pertinent boundary conditions, generally do not satisfy any boundary condition and are thus not the solutions we are looking for. The physical reason why the superposition ansatz in this case fails is because the presence of one colloid on the membrane, which creates a local dimple, will abet a nearby coUoid to tilt, thereby changing the way in which that second colloid interacts with the membrane and, in turn, the first one. 

Here Gxy and Gxy are the shear moduli of the damaged and intact laminate, respectively. This expression, Eq. (14), is similar to the semi-empirical expression suggested by Tsai and Daniel (1992) on the basis of the superposition of solutions for a single set of cracks... 

Equation (2.45) represents a speoial solution to the equation of motion (2.6). According to (2.33,35), there are N independent values of q lying in the first Brillouin zone -rr/a < q 4 ir/a. If q > 0, A(S) is the amplitude of a wave travelling to the right while A( S) is a wave travelling to the left. Since the two waves are independent, there are no necessary conditions between A(S) and A( S). The general solution will therefore be a superposition of solutions of the type (2.45), where the summation extends ever all modes... 

In anticipation of the periodic nature of solar illumination we use a periodic function for the time factor. In general, solar radiation is not a simple sinusoidal term, but must be expressed by a Fourier series. To simplify notation we treat only the lowest diurnal frequency of the series higher order terms can be considered later by superposition of solutions. Substituting Eq. (8.5.5) into Eq. (8.5.4) leads to... 

When a mechanical part is made from a polymer, and when it is to be used as a loadcarrying component, obviously it is not necessarily always going to be subject to a constant stress as in the creep test. It generally has to be designed to withstand some history of stress variation. How will the polymer respond to the stress history Can its response be predicted Fortunately, for hnear viscoelastic behavior, predicting the response is possible, because of the principle of superposition of solutions to linear differential equations. The student, of course, remembers that if y,(Ji ) and y2(x) are both solutions of an ordinary differential equation for y x), then the sum y (x) + yj (x) is also a solution. This is the basis of the Boltzmann Superposition Principle for linear polymer behavior. 

We must pay attention to one more important property of the Schrodinger equation (7.4.1), apparent from its uniformity any superposition of solutions is also a solution of this equation. Such a superposition property often leads to what in chemistry is called hybridization. 

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