The permittivity of a medium \(\varepsilon(\vec{k}, \omega) \), where ω and \(\vec{k}\) are the frequency and wavevector, respectively, has no imaginary part. For a longitudinal wave, \(\vec{k}\) is parallel to the electric field such that \(\vec{k} \times \vec{E}=0 \) , while for a transverse wave \(\vec{k} \cdot \vec{E}=0 \). In the absence of free charges and free currents, the medium can sustain
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longitudinal waves with \(\vec{k}\) and ω when \(\varepsilon(\vec{k}, \omega)>0 \)
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transverse waves with \(\vec{k}\) and ω when \(\varepsilon(\vec{k}, \omega)<0 \)
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longitudinal waves with \(\vec{k}\) and ω when \(\varepsilon(\vec{k}, \omega)=0\)
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both longitudinal and transverse waves with \(\vec{k}\) and ω when \(\varepsilon(\vec{k}, \omega)>0 \)