Summary In this paper, we study semi-infinite-spatial-domain wave equations modeling the real-world problem of longitudinal waves propagating along a long, slender, homogeneous elastic rod of finite length. A practical conclusion from the paper is that precision of the downhole card can be increased by improving the accuracy of the friction law in the wave equation. To conclude, we provide a rigorous proof of Gibbs' theorem and illustrate its validity with an existing well.
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DaCunha et al. (2009) studied this question.
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