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A crucial step in deriving the wave equation for waves in
Chapter 16, Problem 16.38(choose chapter or problem)
A crucial step in deriving the wave equation for waves in a fluid was the neglect of the first term on the right of Equation (16.134).
(a) Justify this by using (16.139) to rewrite the right side of (16.125) as \(\varrho^{\prime} \mathbf{g}-\mathrm{BM} \nabla \varrho^{\prime} / \varrho_{0}\). Argue that the ratio of the first to the second term is of order \(g \varrho_{0} \lambda / \mathrm{BM}\), where \(\lambda\) is a typical distance over which \(\varrho^{\prime}\) varies. (A good choice for \(\lambda\) would be the wavelength of the proposed wave — of order a centimeter or at most a few meters.) Using the values for water (BM = 2 GPa, etc.) show that the first term is negligible. (You would also reach the same conclusion for air.)
(b) Show with a similar argument that the second term on the right of (16.136) is negligible.
Questions & Answers
QUESTION:
A crucial step in deriving the wave equation for waves in a fluid was the neglect of the first term on the right of Equation (16.134).
(a) Justify this by using (16.139) to rewrite the right side of (16.125) as \(\varrho^{\prime} \mathbf{g}-\mathrm{BM} \nabla \varrho^{\prime} / \varrho_{0}\). Argue that the ratio of the first to the second term is of order \(g \varrho_{0} \lambda / \mathrm{BM}\), where \(\lambda\) is a typical distance over which \(\varrho^{\prime}\) varies. (A good choice for \(\lambda\) would be the wavelength of the proposed wave — of order a centimeter or at most a few meters.) Using the values for water (BM = 2 GPa, etc.) show that the first term is negligible. (You would also reach the same conclusion for air.)
(b) Show with a similar argument that the second term on the right of (16.136) is negligible.
ANSWER:Step 1 of 6
(a) From the topic of wave in a fluid, the equation (16.134), the expression can be given as,
\(\rho_{0} \frac{\partial \vec{v}}{\partial t}=\rho^{\prime} \vec{g}-\nabla p\)
(1)