Complex wavenumber

In each case the attenuation of sound can be formally represented by defining a complex wavenumber, where the sound attenuation coefficient is the imaginary part of the wavenumber. The complex wavenumber also leads to the definition of a complex sound speed and a complex dynamic elastic modulus. 

The expression for the sound wave now has the same form, p = p exp[i(k x-cjt) ], as in a lossless medium. The complex wavenumber, substituted in Eg.4, defines a complex sound speed for the material as follows. 

A wave can be characterized by an amplitude, frequency, and wavelength which may change with time or distance traveled from the source. We can express both the storage and loss properties of a sonic wave moving in a material concisely as the real and imaginary parts of a complex wavenumber k = co/c + ia, where c is the speed of sound, co is the angular frequency (=2 Jt/),/is frequency, / = V - 1, and a is the attenuation coefficient. Ultrasonic properties are often frequency dependent so it is necessary to define the wavelength at which k is reported. The dependency of k on frequency is the basis of ultrasonic spectroscopy. 

If a single sharp absorption occurs at a wavenumber v, as shown in the wavenumber domain spectmm in Figure 3.15, the cosine wave corresponding to is not cancelled out and remains in the I 5) versus 5 plot, or interferogram, as it is often called. For a more complex set of absorptions the pattern of uncancelled cosine waves becomes more intense and irregular. 

Rising of CO partial pressure resulted in the appearance of a series of absorption bands with v (CO) 2176, 2156 and 2136 cm" in the area of CO stretching vibrations. The band at v (CO) 2176 cm can be attributed to the interaction of CO with weak Bronsted acid sites while the band at v (CO) 2156 cm can be attributed to the complex of CO with non-acid hydroxyl groups. The band with wavenumber v (CO) 2136 cm" can be related to physical-adsorbed CO over Silica. 

Fig. 2 The experimentally determined potential energy V(), expressed as a wavenumber for convenience, as a function of the angle in the hydrogen-bonded complex H20- HF. The definition of

Fig. 15. Thermal denaturation of triosephosphate isomerase with FTIR (upper left), second-derivative FTIR (upper right), and VCD (bottom) showing irreversible aggregation effects. The IR shift from a simple maximum at 1650-1640 cm-1 to a lower frequency distorted to low wavenumber is seen to be irreversible when the original spectrum is not recovered. The second-derivative result makes the changes more dramatic and shows the original native state spectrum to be more complex (negative second derivatives correspond to peak positions). Loss of structure is even more evident in the VCD, which loses most of its intensity at 60°C.

Carbonyl absorptions in the neutral, five-coordinate Ru and Os carbynes occur at unusually low wavenumbers, suggesting that it is appropriate to regard these molecules as zerovalent complexes. A significant increase in v(CO) for the d8 dicarbonyl cations is noted. IR data for selected carbyne complexes are given in Table IX. 

It is known [4] that methylacetylene can be adsorbed dissociatively or not. The dissociation of methylacetylene is characterized by the appearance of a typical v(OH) vibration. Therefore, the absence of any zeolitic v(OH) band for the different LSX samples indicates that methylacetylene adsorbs mainly without dissociation The presence of non-dissociated adsorbed methylacetylene is also evidenced by the detection of specific v(C=C) and v(=CH) vibrations band. As expected, their wavenumbers decrease with increasing basicity. Moreover, the complex shape of the v(=CH) band reveals different environments of basic sites. 

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