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Increases in pipe wall response and external radiation are largest at frequencies at which higher order acoustic modes can propagate, in which case coincidence effects lead to strong excitation of supersonic pipe modes.
Meanwhile, for the compressible hyperelastic models which are reformulated into deviatoric and hydrostatic parts, only quasi modes can propagate, with abnormal ray directions that can be observed for longitudinal waves.
A numerical mapping technique is evoked to transform the solution domain into a rectangular geometry and the procedure is used to solve for frequencies where cross modes can propagate in the straight sections of duct.
Both theories are in reasonable agreement with experimental data at low frequencies, though discrepancies are noted in the region where higher order acoustic modes can propagate within the duct.
Two modes can propagate in the forward direction (x>0).
Three possible modes can propagate in the forward direction in region VI ((vert yvert < d), (x>0)) from ((pi /2< k<3pi /5), (3pi /5< k<4pi /5), (4pi /5< k< pi)) for (a=1).
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Band diagram shows that only transverse magnetic (TM) mode can propagate through the waveguides.
Band diagram also shows that only TE mode can propagate through the waveguides.
According to the characteristic equation, three longitudinal waves namely, elastodiffusive (ED), mass diffusion (MD-mode) and thermodiffusive (TD-mode), can propagate in such solids in addition to transverse waves.
In addition, only the whistler mode can propagate against the solar wind with a group velocity larger than the solar wind speed.
In this case, for the working frequency of 0.26 (a/λ) (see the red dash line in the Fig. 4b), the TM polarization mode can propagate through photonic crystal waveguide W2 at the localized temperature of 25 °C while it can not pass through W2 at the localized photonic crystal temperature of 650 °C.
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