Principle of Superposition

3 Systems with multiple inputs 4.3.1 Principle of superposition 

A dynamic system is linear if the Principle of Superposition can be applied. This states that The response y t) of a linear system due to several inputs x t), 

It should be noticed that the denominators for equations (4.11) and (4.12) are identical. Using the Principle of Superposition, the complete response is given by 

The diffusion equation with constant diffusivity (Equation 3-8) is said to be linear, which means that if fand g are solutions to the equation, then any linear combination of f and g, i.e., u = af+ bg, where a and b are constants, is also a solution. To show this, we can write 

Since it is assumed that f and g are solutions to the diffusion equation, then 

This result is known as the principle of superposition. The principle is useful in solving diffusion equations with the same boundary conditions, but different initial conditions, or with the same initial conditions but different boundary conditions, or other more general cases. Suppose we want to find the solution to the diffusion equation for the following initial condition  

If the diffusion problem with the same boundary conditions but the initial condition of C t o = fhas been solved to be Ci, and the problem with the same boundary conditions but the initial condition of C t=o=g has been solved to be Cx, then the solution to the diffusion problem for the initial condition of 

1 Diffusion in an infinite medium with an extended source 

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A water body is considered to be a one-diiuensional estuary when it is subjected to tidal reversals (i.e., reversals in direction of tlie water quality parameter are dominant). Since the describing (differential) equations for the distribution of eitlier reactive or conserv ative (nomciictive) pollutants are linear, second-order equations, tlie principle of superposition discussed previously also applies to estuaries. The principal additional parameter introduced in the describing equation is a tid il dispersion coefficient E. Methods for estimating this tidiil coefficient are provided by Thomaim and Mueller... 

We shall begin with the principle of superposition, illustrated by Young s experiment (Fig. 1). If two pinholes separated by a in an opaque screen are illuminated by a small source S of wavelength A, under certain conditions it is observed that the intensity in front of the screen varies cosinusoidaUy with the angle 6 according to... 

Thus far we have been considering—in Young s experiment, Michelson interference, and even measurements of mutual intensity—what is essentially two source interference. We now turn to multiple source interference. For example, a diffraction grating divides up a beam into an array of sources, which then interfere. By the principle of superposition, the disturbance in the far field is the sum over the sources... 

In order to determine the force with which an arbitrary body acts on a particle located around the point p, we mentally divide the volume of the body into many elementary volumes, so their dimensions are much smaller than the corresponding distance from the particle p. It is clear that the magnitude and direction of each force depends on the position of the point q inside a body. Now, applying the principle of superposition, we can find the total force acting on the particle p. Summation of elementary forces gives ... 

Next, applying the principle of superposition, we arrive at an expression for the potential caused by a volume distribution of masses ... 

At the beginning we assumed that masses are absent inside the volume V and, correspondingly. Equation (1.107) describes the potential of the attraction field, caused by masses located outside V. Now suppose that there are also masses in this volume and their distribution is characterized by the density Then, applying the principle of superposition, we obtain... 

Further we assume that the masses are distributed uniformly that is, a — constant. Applying the principle of superposition we obtain for the normal component of the field due to all surface masses ... 

Now we will show that, making use of this equation, it is possible to calculate the gravitational field caused by masses in a two-dimensional body with an arbitrary cross section. With this purpose in mind, let us mentally divide the body cross section into a sufficient number of relatively thin layers with the thickness hi. Then, applying the principle of superposition and Equation (4.21) for elementary layer, we have... 

Based onEq. (3-51), the time response y(t) should be strictly overdamped. However, this is not necessarily the case if the zero is positive (or xz < 0). We can show with algebra how various ranges of K and X may lead to different zeros (—l/xz) and time responses. However, we will not do that. (We ll use MATLAB to take a closer look in the Review Problems, though.) The key, once again, is to appreciate the principle of superposition with linear models. Thus we should get a rough idea of the time response simply based on the form in (3-50). 

If ipi, ip2, p3 are acceptable wave functions for the system, then according to the principle of superposition the true wave function T could be expanded as the linear combination of them ... 

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 ... 

An important corollary of the principle of superposition is that a wave of any shape can be described mathematically as a sum of a series of simple sine and cosine terms, which is the basis of the mathematical procedure called the Fourier transform (see Section 4.2). Thus the square wave, frequently used in electronic circuits, can be described as the sum of an infinite superposition of sine waves, using the general equation ... 

Equation (8.30) describes how Am affects the ith output Xj using the step-response coeflicient h(+i. Note that the sum of the indices in Am and b is always i -I- 1. The summation gives the effects of all the SC terms using the principle of superposition. 

The Gij s aie, in general, functions of s and oic the transfer functions relating inputs and outputs. Since the system is linear, the output is the sum of the effects of each individual input. This is called the principle of superposition. 

The calculation method and equations presented in the previous sections are for Newtonian fluids such that the flow due to screw rotation and the downstream pressure gradient can be solved independently, that is, via the principle of superposition. Since most resins are highly non-Newtonian, the rotational flow and pressure-driven flow in principle cannot be separated using superposition. That is, the shear dependency of the viscosity couples the equations such that they cannot be solved independently. Potente [50] states that the flows and pressure gradients should only be calculated using three-dimensional (3-D) numerical methods because of the limitations of the Newtonian model. 

As discussed in Section 7.4 and using the principle of superposition, the flow components were separated into rotational flow and pressure flows. The equation for the total flow and the components are as follows ... 

The principle of superposition is used to break the complicated flow into the component velocities. These component velocities will be derived in the next sections. 

By using the principle of superposition and separation of variables, the solution to... 

Several general methods are available for solving the diffusion equation, including Boltzmann transformation, principle of superposition, separation of... 

Using the principle of superposition, following the same procedure above, several other general solutions can be derived. For example, the solution for arbitrary initial distribution C t o = f(x) for one-dimensional diffusion in an infinite medium with constant D can be found by integration ... 

The above derivation has not made use of the initial and boundary conditions yet, and shows only that A may take any constant value. The value of A can be constrained by boundary conditions to be discrete Ai, A2,..., as can be seen in the specific problem below. Because each function corresponding to given A is a solution to the diffusion equation, based on the principle of superposition, any linear combination of these functions is also a solution. Hence, the general solution for the given boundary conditions is... 

The following solutions for instantaneous sources are useful in conjunction with the principle of superposition to derive solutions to other diffusion problems. 

Consider a complex scalar product space V that models the states of a quantum system. Suppose G is the symmetry group and (G, V, p) is the natural representation. By the argument in Section 5.1, the only physically natural subspaces are invariant subspaces. Suppose there are invariant subspaces Gi, U2, W c V such that W = U U2. Now consider a state w of the quantum system such that w e W, but w Uy and w U2. Then there is a nonzero mi e Gi and a nonzero M2 e U2 such that w = ui + U2. This means that the state w is a superposition of states ui and U2. It follows that w is not an elementary state of the system — by the principle of superposition, anything we want to know about w we can deduce by studying mi and M2. 

When motion of the fluid consists of only small fluctuations about a state of near-rest, Lhe continuum equations are linearized by neglecting nonlinear terms and they become lhe equalions of acoustics. A large variety of fluid motions are described as sound waves when the small-motion or acoustic description can be used, the principle of superposition is valid. This powerful principle allows addition of simple simultaneous motions to represent a more complex motion, such as the sound reaching lhe audience from the instruments of a symphony orchestra. The superposition principle does not apply to large-scale (nonacoustical) motions, and the subject of fluid dynamics (in distinction from acoustics) treats nonlinear flows. i.e.. those that cannot be described as superpositions of other flows. 

A number of the systems described in the remainder of this chapter are assumed to be linear with respect to time, i.e. their time-dependent properties can be described by linear differential equations. Such systems follow the principle of superposition. This property is such that if the individual output of a system is... 

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