Rayleigh lines

Figure 2.5. Relationship of the P-V Hugoniot to the Rayleigh line and a graphical illustration of kinetic and internal energy increase.

The left-hand side of the inequality is the slope of the Rayleigh line, and the right-hand side is the slope of the isentrope centered on the initial state. We showed in Section 2.5 that the isentrope and Hugoniot are tangent at the initial state. Thus, the stability condition which requires that the shock wave be supersonic with respect to the material ahead of it is equivalent to the statement that the Rayleigh line must be steeper than the Hugoniot at the initial state. 

The stability condition that the shock wave is subsonic with respect to the shocked material behind it is equivalent to the statement that the Hugoniot must be steeper than the Rayleigh line at the final state. 

Rayleigh line A chord that connects the initial state of a material on its Hugoniot curve to the final state on the curve. Most frequently drawn in the P-F plane. 

To demonstrate that the Rayleigh line actually represents the thermodynamic path to which material is subjected on being shocked from state p = 0, F = Fq to P = Pi, F = Fi, we demonstrate below that the shock wave sketched in Fig. 4.1 must be steady. Moreover, the Rankine Hugoniot equations ((4.1)-(4.3)) not only describe the conservation of mass, momentum. 

Figure 4.7. Internal reflection of a shock wave from a free surface, (a) Reflection of a shock wave from a free surface causes a reflected rarefaction wave. As indicated in (b), this increases the velocity of the shocked material from u, to Uf. The path upon shocking is Rayleigh line 0-1, whereas unloading occurs along release isentrope curve I -O, (c) Release isentrope path in P- V plane is indicated.

Figure 4.10 indieates that if the Rayleigh line (0-2) from the initial to the final shoek state interseets the Hugoniot at an intermediate shoek state, two shoek waves will form, one from 0-1 and one from 1-2. Thus two shoeks form for final shoek states between 1 and 3. For states at stresses higher than state 3, only one shoek forms. The deformations whieh aeeompany elastie shoeks (discussed in Seetion 4.5, below), deformational shoek and phase transition shock, are sketehed in Fig. 4.11. 

The line connecting the initial state to the shocked state is termed the Rayleigh line characterized in shock velocity as [/ = Io[P — Pq/Vq — F]. Equations (2.1) represent propagation into undisturbed matter, but can be... 

Fig. 2.3. Experimental determination of shock-stress versus volume compression from propagating shock waves is accomplished by a series of experiments carried out at different loading pressures. In the figure, the solid lines connect individual pressure-volume points with the initial condition. These solid straight lines are Rayleigh lines. The dashed line indicates an extrapolation into an uninvestigated low pressure region. Such extrapolation is typical of much of the strong shock data.

Fig. 5.17. The relative change in magnetization for a 3% silicon-iron alloy shows clear indications for a transition at 14 GPa, the end of the mixed phase region, 22.5 GPa, and the overdrive pressure at the Rayleigh line at 37.5 GPa (after Duvall and Graham [77D01]).

Figure 5.4 Resonance Raman spectrum of [Au2(dcpm)2](CI04)2 in acetonitrile solution at room temperature taken with 282.4 nm excitation, after intensity corrections and subtractions of the Rayleigh line, glass bands, and solvent bands. Reproduced with permission from [7a]. Copyright (1999) American Chemical Society.

The ratio of the intensity of anti-Stokes and Stokes lines is primarily determined by the Boltzmann population of the excited vibrational states. For mid-IR frequencies this fractional population is very low (seIO-4 at 2000cm-1). As a result, Raman spectra are usually taken from the Stokes side of the Rayleigh line as these are generally very much more intense and are not broadened by emissions from hot states. 

Figure 3. Energy schemata of transitions involving vibrational states (a excitation of 1st vibrational state - mid-IR absorption b excitation of overtone vibrations - near-IR absorptions c elastic scattering - Rayleigh lines d Raman scattering - Stokes lines e Raman scattering - Anti-Stokes lines f fluorescence).

In either case, the information on the vibrational transition is contained in the energy difference between the excitation radiation and the inelastically scattered Raman photons. Consequently, the parameters of interest are the intensities of the lines and their position relative to the Rayleigh line, usually expressed in wavenumbers (cm 1). As the actually recorded emissions all are in the spectral range determined by the excitation radiation, Raman spectroscopy facilitates the acquisition of vibrational spectra through standard VIS and/or NIR spectroscopy. 

When a compound is irradiated with monochromatic radiation, most of the radiation is transmitted unchanged, but a small portion is scattered. If the scattered radiation is passed into a spectrometer, we detect a strong Rayleigh line at the unmodified frequency of radiation used to excite the sample. In addition, the scattered radiation also contains frequencies arrayed above and below the frequency of the Rayleigh line. The differences between the Rayleigh line and these weaker Raman line frequencies correspond to the vibrational frequencies present in the molecules of the sample. For example, we may obtain a Raman line at 1640 cm-1 on either side of the Rayleigh line, and the sample thus possesses a vibrational mode of this frequency. The frequencies of molecular vibrations are typically 1012—1014 Hz. A more convenient unit, which is proportional to frequency, is wavenumber (cm-1), since fundamental vibrational modes lie between 4000 and 50 cm-1. 

Figure 2 Allowed thermodynamic states in detonation are constrained to the shock Hugoniot. Steady-state shock waves follow the Rayleigh line.

To understand the difference in stagnation pressure losses between subsonic and supersonic combustion one must consider sonic conditions in isoergic and isentropic flows that is, one must deal with, as is done in fluid mechanics, the Fanno and Rayleigh lines. Following an early NACA report for these conditions, since the mass flow rate (puA) must remain constant, then for a constant area duct the momentum equation takes the form... 

The Rayleigh line is defined by the condition which results from heat exchange in a flow system and requires that the flow force remain constant, in essence for a constant area duct the condition can be written as... 

Note that Eq. (5.5) is the equation of the Rayleigh line, which can also be derived without involving any equation of state. Since (p u )2 is always a positive value, it follows that if p2 > p, P2 > Pi and vice versa. Since the sound speed c can be written as... 

Figure 13. Left panel Schematic representation of a shock compression in the PV diagram. Right panel Building up of a Hugoniot by the Rayleigh lines (Rp obtained from different shock experiments.

---