Semi-infinite rod

Case 1. his very large (A - oo). This case is equivalent to considering a semi-infinite rod. In this situation, Eq. (16.255) the wave propagation equation has the form... 

The temperature distribution in a semi-infinite rod with one end of the rod kept at the prescribed temperature is described by the solution of the following initial-boundary-value problem ... 

The temperatnre distribntion in a semi-infinite rod with prescribed heat flnx appfied at one end can be obtained as follows. The temperature distribution in the rod is governed by the following equation ... 

Uniform internal energy is generated in a semi-infinite rod of cross-sectional area A and periphery P (Fig. 2P-11). The peripheral and tip heat transfer coefficients are the same, say h. The ambient temperature is T. If heat loss from the tip is neglected, the temperature of the rod is uniform, T — Tx = u A/hP. Find the distance 5 over which the effect of heat loss from the tip is appreciable. 

Here, a semi-infinite rod with one end at x = 0 and extending to infinity along the positive x-axis will be a typical physical model. For example, consider the following model ... 

There is some work on TRS materials which does not take the temperature field as given, but seeks to solve for it. Hunter (1967) manages to obtain solutions for the temperature field in a semi-infinite rod subjected to forced oscillations. It is necessary however to assume that the loss tangent depends linearly on the temperature field. Lockett and Morland (1967) assume that temperature and mechanical fields are decoupled, so that the temperature field may be solved for independently and specified as input to the mechanical equations. 

Thin semi-infinite rod is charged uniformly with linear density r = 1 rC/m. At a distance Tq = 20 cm from the end of the rod perpendicular to it a point O is located (Figure E4.15). Calculate the electric field strength created by the charged rod in point O. 

The procedure in use here involves the deposition of a radioactive isotope of the diffusing species on the surface of a rod or bar, the length of which is much longer than tire length of the metal involved in the diffusion process, the so-called semi-infinite sample solution. 

Another viable method to compare experiments and theories are simulations of either the cell model with one or more infinite rods present or to take a solution of finite semi-flexible polyelectrolytes. These will of course capture all correlations and ionic finite size effects on the basis of the RPM, and are therefore a good method to check how far simple potentials will suffice to reproduce experimental results. In Sect. 4.2, we shall in particular compare simulations and results obtained with the DHHC local density functional theory to osmotic pressure data. This comparison will demonstrate to what extent the PB cell model, and furthermore the whole coarse grained RPM approach can be expected to hold, and on which level one starts to see solvation effects and other molecular details present under experimental conditions. 

Consider now a perfect crystal truncated by a sharp surface (or semi-infinite crystal). It can be obtained by the product of a step function describing the electron density variation perpendicular to the surface, and an infinite lattice. The diffraction pattern is then the convolution of the 3D reciprocal lattice with the Fourier transform of the step function. An infinity of Fourier components is necessary to build this latter, so that there remains non zero intensity in between Bragg peaks as a function of / the reciprocal space is made of rods of intensity, called crystal truncation rods (CTR), extending perpendicular to the surface, and connecting bulk Bragg peaks [24, 25]. The intensity variation as a function of (or Qj or /) is found by stopping the summation at n3 = 0 in Eq. (1) and (2), yielding ... 

Figure 15-3. Configuration of contacting bodies in rubbing. 1 Stationary body. 2 Moving body. (a) Protuberance on a semi-infinite non-emissive body. (b) Long emissive rod.

TABLE 15-1. FORMULAS FOR THE INTERFACIAL TEMPERATURE OF A LONG, SQUARE ROD SLIDING ON A SEMI-INFINITE BODY... 

The crystal truncation rod (CTR). We now calculate the scattering intensity for a semi-infinite lattice, i.e., which has only one reflecting interface. This sum is nearly... 

The calculations in the preceding section have been performed for a simple planar slab waveguide of infinite extent but finite thickness sandwiched between semi-infinite substrate and cover layers. Other waveguide configurations are possible, most notably the cyhndricaUy symmetrical case of a high refractive index rod surrounded by a lower index cladding. This is the fiber-optic configuration used widely in telecommunications. 

The exponential expression has the form of the Gaussian normal distribution curve. If an amount s of diffusing substance diffuses only into one side of a semi-infinite medium -that is, if the diffusing material is placed on the end of a rod with all the other above conditions applying - then the solution (5-40) needs only be multiplied by a factor 2. (The superposition principle for solutions of linear differential equations has been used here.)... 

Another nonsteady state conductive heat transfer problem involves a semi-infinite body (Fig. 11.10) where not enough time has elapsed for thermal equilibrium to be established. A typical problem of this sort is where the top surface of the semi-infinite body is heated to a constant temperature (7 ), and the problem is to find the temperature at an interior point (A) a distance (y) beneath the surface (Fig. 11.10), after a certain elapsed time (t). Before the temperature of the surface is brought to temperature (Tg) at (0 equals 0, the body is at uniform temperature (T ). This is similar to the problem of heating one end of a long rod whose sides are perfectly insulated so that heat can flow only along the axis of the rod. 

Hunter, S. C. (1967) The transient temperature distribution in a semi-infinite viscoelastic rod, subject to longitudinal oscillations. Int. J. Eng. Sci. 5, 119-143 Hunter, S.C. (1968) The motion of a rigid sphere embedded in an adhering elastic or viscoelastic medium . Proceedings of the Edinburgh Mathematical Society 16 (Series II), Part I, pp. 55-69 Hunter, S.C. (1983) Mechanics of Continuous Media, 2nd edition (Wiley, New Ycrk)... 

As discussed in Chapter 2, xanthan has a structure that is not quite a rigid rod since it has some degree of flexibility. This type of structure was described by Porod and Kratky as the worm-like chain model (Richards, 1980, p. 88). Although this may be visualised intuitively to be rather like a semi-flexible string of plastic pop-in beads, it requires the definition of the persistence length, /p, in order to develop the idea in a more quantitative way. This quantity is defined for an infinite polymer chain as follows ... 

ILS experiments indicate a semi-rigid behavior for the PDA chains. Therefore we can expect to observe the form factor of the Porod-Kratky chain. More precisely, the q scattering behavior of a rod like molecule should be measured since the normalize form factor P( of an infinitely long worm-like chain has the asymptotic form ... 

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