Wave propagation

Although part of Chapter 16 is devoted to wave propagation in viscoelastic materials and some specific simple cases are studied in detail as part of the engineering applications of viscoelasticity, it is useful to mention here that there are several experimental methods to determine the dynamic response 

A final comment seems to be pertinent. In most cases actual measurements are not made at the frequencies of interest. However, one can estimate the corresponding property at the desired frequency by using the time (fre-quency)-temperature superposition techniques of extrapolation. When different apparatuses are used to measure dynamic mechanical properties, we note that the final comparison depends not only on the instrument but also on how the data are analyzed. This implies that shifting procedures must be carried out in a consistent manner to avoid inaccuracies in the master curves. In particular, the shape of the adjacent curves at different frequencies must match exactly, and the shift factor must be the same for all the viscoelastic functions. Kramers-Kronig relationships provide a useful tool for checking the consistency of the results obtained. 

Indentation methods as a special type of analysis of viscoelastic stress or displacement are studied in Chapter 16. It should be noted that indentation 

Strictly speaking, there are no static viscoelastic properties as viscoelastic properties are always time-dependent. However, creep and stress relaxation experiments can be considered quasi-static experiments from which the creep compliance and the modulus can be obtained (4). Such tests are commonly applied in uniaxial conditions for simphcity. The usual time range of quasi-static transient measurements is limited to times not less than 10 s. The reasons for this is that in actual experiments it takes a short period of time to apply the force or the deformation to the sample, and a transitory dynamic response overlaps the idealized creep or relaxation experiment. There is no limitation on the maximum time, but usually it is restricted to a maximum of 10 s. In fact, this range of times is complementary, in the corresponding frequency scale, to that of dynamic experiments. Accordingly, to compare these two complementary techniques, procedures of interconversion of data (time frequency or its inverse) are needed. Some of these procedures are discussed in Chapters 6 and 9. 

An important aspect of any structure is its ability to store energy when work has been done on it by imposing a strain. One result of this storage ability is that an impulse can be transmitted over significant distances, i.e. a wave can travel through the material over distances comparable to the wavelength or greater. Loss processes will inevitably occur and at low Deborah numbers the viscous processes rapidly damp out the oscillation 

The solution to this equation can be written in terms of sines and cosines  

Now using the initial condition that a = a0 at t = 0 in Equations (2.16) and (2.17) we have the following results  

The time taken for one complete oscillation is the period of the oscillation Tw  

The above discussion will be familiar to all physical scientists, especially in the context of microwave spectroscopy and atomic vibrations. We should now consider what would happen if the mass m was in the middle of a long chain and the masses either side were free to move. As the mass moves, the force on the adjacent masses increases until cat = n/2. Following Tabor,6 we can make the approximation that at this point the adjacent masses move in turn and a wave propagates through the material. This gives the wave velocity, vw, as 

If light is reflected at the boundary of two different optical media, the polarisation of the electromagnetic vibration is changed according to the Fresnel equations [104]. The change of the status of polarisation is characterised by 

Recent developments have extended the ultrasonic techniques to the characterisation of thin layers of metals and polymers deposited on substances to obtain measurements of the thickness/density product. Using techniques where the film are immersed in a fluid, such as water, measurements have been made, by the low frequency normal incidence double through-transmission method, with film thickness ranging from 20 to 200 pm [112] a range which is of particular relevance to membrane systems. 

Surface acoustic waves (SAW), which are sensitive to surface changes, are especially sensitive to mass loading and theoretically orders of magnitude more sensitive than bulk acoustic waves [43]. Adsorption of gas onto the device surface causes a perturbation in the propagation velocity of the surface acoustic wave, this effect can be used to observe very small changes in mass density of 10 g/cm (the film has to be deposited on a piezoelectric substrate). SAW device can be useful as sensors for vapour or solution species and as monitors for thin film properties such as diffusivity. They can be used for example as a mass sensor or microbalance to determine the adsorption isotherms of small thin film samples (only 0.2 cm of sample are required in the cell) [42]. 

Logarithmic plots of the maximum amplitude obtained from Fig. 6-8 vs. oscillation number, to determine logarithmic decrement. 

Automatic recording of free torsional oscillations with logarithmic amplitudes. (Kop-pclmann. ) 

Automatic recording of longitudinal wave propagation along thin strips. (Nolle. ) 

An alternative is to use flexural waves, excited by transverse motions of an electromechanical driver. A flexural (bending) deformation of a rod or strip measures Young s modulus because one side of the sample is stretched and the other compressed at a bend. For traveling flexural waves, the analogs of equations 39 and 40 of Chapter 5 are (for small damping, r 1)  

In this case, one dimension of the sample must be known with considerable accuracy, in contrast to the extensional waves described above and the shear waves described in Chapter 5 where no dimensions are needed at all. The frequency range is from 100 to 10,000 Hz. 

The definition of the Poynting vector, Eq. (1.2.3), requires that S be orthogonal to both E and H. In order to better visualize the relative orientation of these three vectors, we align a Cartesian coordinate system so that the r-axis coincides with the direction of the Poynting vector. The components of S along the y- and z-axes, as well as the components of E and H in the direction of the a -axis, must then be zero Sy = = Ej = Hj = 0. The vectors E and H do not have components 

Except for a static field, which is not of interest in this context, Eq. (1.2.5) indicates that Hy must be zero if E vanishes and, conversely, must disappear when Ey is zero. These conditions require E and H to be at right angles to each other E, H, and S form a right-handed, orthogonal system of vectors. 

In an isotropic, stationary medium, the material constants cr, e, and pt are uniform and constant scalars. The first pair of Maxwell s equations may then be stated  

If one differentiates Eq. (1.3.1) with respect to time and multiplies by /r, one obtains 

Eor a medium at rest the order of differentiation with respect to space and time may be interchanged. Applying the vector identity 

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Before the performance of the loading we have to apply 5 up to 12 sensors, according their size, on the cylindrical part of the drums and after a short check of the required sensitivity and the wave propagation the pneumatic pressure test monitored by AE can be performed. The selection of the sensors and their positions was performed earlier in pre-tests under the postulate, that the complete cylinder can be tested with the same sensitivity, reliability and that furthermore the localisation accuracy of defects in the on-line- and the post analysis is sufficient for the required purpose. For the flat eovers, which will be tested by specific sensors, the geometrical shape is so complicated, that we perform in this case only a defect determination with a kind of zone-location. 

Numerical Modeling of Elastic Wave Propagation in Inhomogeneous Anisotropic Media. 

The benefit of such a model is that better understanding of the wave propagation process may be gained. Also, it is possible to make controlled parameter studies in order to understand the influence of e.g. defect orientation, probe angle and frequency on the test results. Results may be presented as A-, B- or C-scans. 

We used the concept of sound velocity dispersion for explanation of the shift of pulse energy spectrum maximum, transmitted through the medium, and correlation of the shift value with function of medium heterogeneity. This approach gives the possibility of mathematical simulation of the influence of both medium parameters and ultrasonic field parameters on the nature of acoustic waves propagation in a given medium. 

After amplification both signals change their initial phases due to the delay r of the amplifier unblank (r = 0.1 - 0.5 ms), phase shift in it and wave propagation in passive vibrator s elements. All the mentioned phase changes are proportional to the frequency. The most contribution of them has unblank delay z. Thus frequency variations changes the initial phases) f/, and j(/c) of both signals and their difference A - Vi ... 

The common civil engineering seismic testing techniques work on the principles of ultrasonic through transmission (UPV), transient stress wave propagation and reflection (Impact Echo), Ultrasonic Pulse Echo (UPE) and Spectral Analysis of Surface Waves (SASW). 

Zaikin A N and Zhabotinsky A M 1970 Concentration wave propagation in two-dimensional liquid-phase self-oscillating system Nature 225 535-7... 

Kosloff R and Kosloff D 1986 Absorbing boundaries for wave propagation problems J. Comput. Phys. 63 363... 

C2.15.2 a right circularly polarized wave is illustrated. As tire wave propagates, Eq sweeps out a circle in tire x-y plane. It is clear tliat, given a well characterized light source, tliere are many attributes we can attempt to control (wavelengtli, polarization, etc.) tire question is how to generate well-characterized light ... 

Figure C2.15.2. Right circularly polarized light. As tire wave propagates tire resultant E sweeps out a circle in tire x-y plane.

Increase Sound- Transmission Loss. The only significant iacreases ia sound-transmission loss that can be achieved by the appHcation of dampiag treatments to a panel occur at and above the critical frequency, which is the frequency at which the speed of bending wave propagation ia the panel matches the speed of sound ia air. AppHcation of dampiag treatment to 16 ga metal panel can improve the TL at frequencies of about 2000 H2 and above. This may or may not be helpful, depending on the appHcation of the panel. 

It should be possible to achieve greater penetration into a load material using the fringing field of a slow-wave than can be achieved by plane-wave propagation (68). There are, however, no reports of practical appHcation of these principles. 

Transverse electromagnetic waves propagate in plasmas if their frequency is greater than the plasma frequency. For a given angular frequency, CO, there is a critical density, above which waves do not penetrate a plasma. The propagation of electromagnetic waves in plasmas has many uses, especially as a probe of plasma conditions. 

Magnetic fields introduce hydromagnetic waves, which are transverse modes of ion motion and wave propagation that do not exist in the absence of an apphed B field. The first of these are Alfven, A, waves and their frequency depends on B and p, the mass density. Such waves move parallel to the apphed field having the following velocity ... 

The description of phenomena in a continuous medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of wave propagation are described by a class of partial differential equations called hyperbolic, and these are essentially different in their properties from other classes such as those that describe equilibrium ( elhptic ) or diffusion and heat transfer ( para-bohc ). Prototypes are ... 

Hyperbolic The wave equation d u/dt = c d u/dx + d u/dy ) represents wave propagation of many varied types. 

Water Hammer When hquid flowing in a pipe is suddenly decelerated to zero velocity by a fast-closing valve, a pressure wave propagates upstream to the pipe inlet, where it is reflected a pounding of the hne commonly known as water hammer is often produced. For an instantaneous flow stoppage of a truly incompressible fluid in an inelastic pipe, the pressure rise would be infinite. Finite compressibility of the flmd and elasticity of the pipe limit the pressure rise to a finite value. The Joukowstd formula gives the maximum pressure... 

A current wave propagating symmetrically about its zero axis, i.e. when the envelopes of the peaks of the current wave are symmetrical about its zero axis, is termed symmetrical (Figure 13.24) and a wave unable to maintain this symmetry is termed asymmetrical (Figure 13.25). 

Expansion waves are the mechanism by which a material returns to ambient pressure. In the same spirit as Fig. 2.2, a rarefaction is depicted for intuitive appeal in Fig. 2.7. In this case, the bull has a finite mass, and is free to be accelerated by the collision, leading to a free surface. Any finite body containing material at high pressure also has free surfaces, or zero-stress boundaries, which through wave motion must eventually come into equilibrium with the interior. Expansion waves are also known as rarefaction waves, unloading waves, decompression waves, relief waves, and release waves. Material flow is in the same direction as the pressure gradient, which is opposite to the direction of wave propagation. 

Figure 2.10. (a) An Eulerian x-t diagram of a shock wave propagating into a material in motion. The fluid particle travels a distance ut, and the shock travels a distance Uti in time ti. (b) A Lagrangian h-t diagram of the same sequence. The shock travels a distance Cti in this system. 

We will be concerned with the interaction of waves with boundaries and with other waves throughout this text. To determine how these interactions take place, it is important to consider that discontinuities in either pressure or particle velocity cannot be sustained in any material. If a discontinuity in either of these variables is created at some point by impact or wave interaction, the resulting motion will be such that the pressure and particle velocity become continuous across the boundary or point of interaction. Unless the material separates at that point, the motion will consist of one or more waves propagating away from the point of the discontinuity. For pressure discontinuities, it is easy to see that waves must propagate by again considering an... 

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