The parallel propagating waves (Langmuir, R and L waves) are studies in different plasma environments by choosing the different values of $\eta =({mc^{2}}/{k_{B}T})$ and $({\omega ^{2}_{\mathrm{ pe}}}/{\omega ^{2}_{\mathrm{ ce}}})$ . The dispersion relations of these waves are derived under a weak magnetic field limit for a fully relativistic plasma environment. The analytical solution of the integrals in the dispersion relations is not possible, so we use a numerical quadrature approach for their solution. We observe that in both relativistic and weakly relativistic plasmas, these modes shifted to lower values of frequency. The cutoff points of R- and L-waves shifted to further lower values in high-density plasma compared to low density.

中文翻译：

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弱磁化相对论电子等离子体中的平行传播波

平行传播波（Langmuir、R 和 L 波）是在不同的等离子体环境中通过选择不同的值来研究的 $\eta =({mc^{2}}/{k_{B}T})$ 和 $({\omega ^{2}_{\mathrm{ pe}}}/{\omega ^{2}_{\mathrm{ ce}}})$ . 这些波的色散关系是在完全相对论的等离子体环境的弱磁场限制下导出的。色散关系中积分的解析解是不可能的，因此我们使用数值求积方法来解决它们。我们观察到，在相对论和弱相对论等离子体中，这些模式转移到较低的频率值。与低密度等离子体相比，R 波和 L 波的截止点在高密度等离子体中移动到更低的值。