Complex sound speed

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. 

The elastic moduli of a lossy material can also be represented as complex quantities [3,4]. The complex dynamic modulus M is related to the corresponding complex sound speed c as follows [3],... 

Techniques for measuring the complex sound speeds and moduli of polymers are described in the section on test methods. The data shows that the real and imaginary components of the elastic moduli are frequency dependent. The frequency dependence is strongest for materials with high values of the loss factor r. Materials with frequency-dependent elastic moduli are called dispersive, and measurements and theory show that sound absorption mechanisms lead to dispersion. The real and imaginary part of an elastic modulus are related by the Kramers-Kronig relations, which are presented in the next section. 

Eg.8 has the same form as the relation between the (real) elastic modulus and sound speed in a lossless medium. From Eqs.6-8 it follows that the sound attenuation coefficient is related to the complex elastic modulus,... 

Similarly, if one or both of the media are absorptive the analysis becomes more complicated. As we will now demonstrate, the sound speed then becomes a complex quantity. Hence the impedance of the media will also become a complex quantity, and the reflection and transmission coefficients will similarly be complex. 

The complex nature of the sound speed can be derived as follows For an absorptive media, the amplitude of the acoustic wave is exponentially reduced as it propagates. Hence... 

Hence attenuation is considered as an imaginary part of the wave number. (There are also fundamental physical justifications for this, but these need not be addressed here.) This correspondingly forces sound speed to also become a complex quantity ... 

Since sound speed is a frequency dependent complex quantity, it therefore follows that the characteristic impedance of the media will also be frequency dependent and complex. If the frequency dependence of sound speed is not known, it can be estimated from the attenuation coefficient as follows. For the rubber composites of interest here, usually a A is essentially independent of frequency. Using Kramers-Kronig relationships (5) it can then be shown that ... 

Sound-speed measurement is especially critical for characterizing emulsions. There are considerable data indicating that the sound speed of various liquids is extremely dependent on small traces of contamination. Examples of such complex behavior of different mixed liquids is given... 

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. 

Figure 3 illustrates the geometry. There are a total of N layers, where the semi-infinite media to the left and to the right of the layer stack have the indices 0 and N + 1. Later on, the crystal will usually be layer 1 and the index 1 will be replaced by q. Each layer j is characterized by the thickness, dj, an acoustic impedance, Zj, and a speed of sound, Cj. Both the impedance Zj and the speed of sound cj are complex. They are given by ... 

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