One-dimensional waves

Recall from Chapter 2, Section VI.A, that waves are described by periodic functions, and that simple wave equations can be written in the form 

I also stated in Chapter 2 that any wave, no matter how complicated, can be described as the sum of simple waves. This sum is called a Fourier series and each simple wave equation in the series is called a Fourier term. Either Eq. (5.1) or (5.2) could be used as a single Fourier term. For example, we can write a Fourier series of n terms using Eq. (5.1) as follows 

According to Fourier theory, any complicated periodic function can be approximated by this series, by putting the proper values of h, Fh, and ah in each term. Think of the cosine terms as basic wave forms that can be used to build any other waveform. Also according to Fourier theory, we can use the sine function or, for that matter, any periodic function in the same way as the basic wave for building any other periodic function. 

With the terms written in this fashion, a Fourier series looks like this  

In words, this series is the sum of n simple Fourier terms, one for each integral value of h beginning with zero and ending with n. Each term is a simple wave with its own amplitude Fh, its own frequency h, and (implicitly) its own phase a. 

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Lawrence, R.J., Announncement of WONDY IIIA—A Computer Program for One-Dimensional Wave Propagation (SC-DR-70-315), Sandia Corporation Memorandum Report No. SC-M-70-587, Albuquerque, NM, 6 pp., August 1970. 

The Chapman-Jongnet (CJ) theory is a one-dimensional model that treats the detonation shock wave as a discontinnity with infinite reaction rate. The conservation equations for mass, momentum, and energy across the one-dimensional wave gives a unique solution for the detonation velocity (CJ velocity) and the state of combustion products immediately behind the detonation wave. Based on the CJ theory it is possible to calculate detonation velocity, detonation pressure, etc. if the gas mixtnre composition is known. The CJ theory does not require any information about the chemical reaction rate (i.e., chemical kinetics). 

Crowley, C. J., G. B. Wallis, and J. J. Barry, 1992, Validation of One-Dimensional Wave Model for the Stratificd-to-Slug Regime Transition, lnt. J. Multiphase Flow 18 249 271. (3)... 

It is often possible to write the solution of a partial differential equation as a sum of terms, each of which is a function in one of the variables only. This procedure is called solution by separation of variables. The one-dimensional wave equation... 

Since the one-dimensional wave equation is linear, the general solution periodic in x with period 2n is the linear superposition... 

The real part of the general solution (1-22) to the one-dimensional wave equation is... 

The general solution of the linear one-dimensional wave equation describes two waves that move in opposite directions without change in shape or mutual interaction. The two components therefore move apart in finite time and, for most purposes, it is sufficient to consider only one of the components. More specifically, the discussion may be restricted to solutions10 of... 

In order to understand these observations it is necessary to resort to quantum mechanics, based on Planck s postulate that energy is quantized in units of E = hv and the Bohr frequency condition that requires an exact match between level spacings and the frequency of emitted radiation, hv = Eupper — Ei0wer. The mathematical models are comparatively simple and in all cases appropriate energy levels can be obtained from one-dimensional wave equations. 

Consider the thermal wave given in Fig. 4.4. If a differential control volume is taken within this one-dimensional wave and the variations as given in the figure are in the x direction, then the thermal and mass balances are as shown in Fig. 4.5. In Fig. 4.5, a is the mass of reactant per cubic centimeter, Cj is the rate of reaction, Q is the heat of reaction per unit mass, and p is the total density. Note that alp is the mass fraction of reactant a. Since the problem is a steady one, there is no accumulation of species or heat with respect to time, and the balance of the energy terms and the species terms must each be equal to zero. 

Evans Ablow (Ref 66) described the following one-dimensional waves "One-dimensional Steady-State Reaction Waves with Instantaneous Reaction (pp 137-45) wOne-Dimensional Steady-State Reaction Waves with Finite Reaction Rate ... 

The theories of transient processes leading to steady detonation waves have been concerned on the one hand with the prediction of the shape of pressure waves which will initiate, described in Section VI, A of Ref 66, and on the other hand with the pressure leading to the formation of such.an initiating pulse, described in Section VI, B. In Section V it was shown that the time-independent side boundary conditions are important in determining the characteristics of steady, three-dimensional waves. It now becomes necessary to take into consideration time-dependent rear boundary conditions. For one-dimensional waves, the side boundary conditions are not involved... 

We consider an infinitely long thin tube (—oo < r < + oo) along which the concentrations of A and B vary. The sides of the tube are impermeable, and we imagine that there are no concentration gradients along the axes perpendicular to the r axis (so we have a one-dimensional wave). A general picture of the system at some time after initiation is that corresponding to Fig. 11.2(c). 

The plot of represents the locus of all possible solutions for a one-dimensional wave whether the process is adiabatic or nonadiabatic. To clarify some of the properties of this function consider a stationary combustion front possessing an incoming initial Mach number and a value for Q/CPT. Then from Equation 7, it follows that there will be a solution for the final Mach number, i /2, as long as the value of the left side of the equation lies between 0 and + 1)). In the range of values from (y - l)/(2y )... 

The simplest, but not necessarily complete, description of a propagating wave is to be found in the one-dimensional wave equation, which is in essence a form of Newton s second law Force = Mass x Acceleration,... 

PROBLEM 5.7.1. Demonstrate the validity of the one-dimensional wave equation for longitudinal waves, Eq. (5.7.1). 

Let us explain the implication of dispersion relation through a simple example of one-dimensional wave propagation whose governing equation is given by. 

The group velocity for multi-dimensional problem is a vector, decided by the variation of circular frequency with the wave number vector. It is only the real part of Eqn. (1.4.8) that is termed the group velocity - as discussed in Whitham (1978). Thus, this is the velocity at which the energy of a group of waves travel, centered about the middle of the wave number group. It is noted that the one-dimensional wave given by Eqn. (1.4.3), is non-dispersive with Vg = c. 

The solution was interpreted in terms of the transverse vibrations of a string with a variable length. A few years later, these results were published [7], and were extended to the case of electromagnetic field. A similar treatment was reported by Havelock [8] in connection with the problem of radiation pressure. About 25 years later, the one-dimensional wave equation in the time-dependent interval interval 0 < x < a + bt was considered [9] under the name Spaghetti problem. ... 

Moore s approach is based on the decomposition (6) of the field over the mode functions satisfying automatically the (one-dimensional) wave equation (1). There exists another approach (proposed in the framework of the classical... 

The vibrational energy of the puckering mode may be calculated from the one-dimensional wave equation... 

M.A. Nunez, Computation of expectation values with Dirichlet one-dimensional wave functions, Int. J. Quant. Chem. 53 (1) (1995) 15-25. 

The one-dimensional wave equation in an infinite domain can be solved by a change of independent variables to reduce the wave equation that can be integrated to produce d Alembert s solution. Consider a one-dimensional wave equation. 

The phenomena of interference and diffraction of light cannot be understood without introducing the wave concept. In fact, the wave properties of light were established precisely from these phenomena. Here, we will introduce the essential aspects of a propagating wave and the formulae needed to explain the optical effects described throughout this section. Let us first recall the one dimensional wave formula that we met for the first time during the physics classes in senior high school. 

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