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author | Prefetch | 2024-07-17 10:01:43 +0200 |
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committer | Prefetch | 2024-07-17 10:01:43 +0200 |
commit | 075683cdf4588fe16f41d9f7b46b9720b42b2553 (patch) | |
tree | f200b341b35e9c89aa1030a2f6da94cd0e1e958b /source/know/concept/korteweg-de-vries-equation/index.md | |
parent | c4d597e8d695eb145755464cffbf88a68fd0c88a (diff) |
Improve knowledge base
Diffstat (limited to 'source/know/concept/korteweg-de-vries-equation/index.md')
-rw-r--r-- | source/know/concept/korteweg-de-vries-equation/index.md | 31 |
1 files changed, 17 insertions, 14 deletions
diff --git a/source/know/concept/korteweg-de-vries-equation/index.md b/source/know/concept/korteweg-de-vries-equation/index.md index 2857e23..e8035d1 100644 --- a/source/know/concept/korteweg-de-vries-equation/index.md +++ b/source/know/concept/korteweg-de-vries-equation/index.md @@ -162,11 +162,11 @@ $$\begin{aligned} = q_0 - \frac{g}{q_0} \Big( \eta(x, t) + \alpha + \gamma(x, t) \Big) \end{aligned}$$ -Where $$\alpha$$ is a constant parameter -(which we will use to handle velocity discrepancies -between the linear and nonlinear theories). +Where $$\alpha$$ is a constant parameter, +which we will use to handle velocity discrepancies +between the linear and nonlinear theories. The correction represented by $$\gamma$$ is much smaller, -i.e. $$\eta \sim \alpha \gg \gamma$$. +i.e. $$\eta \gg \alpha \gg \gamma$$. We insert this ansatz into the above equations, yielding: $$\begin{aligned} @@ -265,14 +265,15 @@ $$\begin{aligned} \equiv \frac{h^3}{3} - \frac{h T}{g \rho} \end{aligned}$$ -What about $$\alpha$$? +But what about $$\alpha$$? Looking at the ansatz for $$f$$, we see that -the body of water is already assumed to be moving at $$q_0$$, -minus $$g \alpha / q_0$$, so by varying $$\alpha$$ -we are modifying the water's velocity. -The term in the KdV equation simply corrects for our chosen value of $$\alpha$$. -It has no deeper meaning than that: for any value of $$\alpha$$, -the full range of KdV solutions can still be obtained. +the body of water is assumed to be moving at $$q_0 - g \alpha / q_0$$, +and $$q_0$$ is set to $$\pm \sqrt{g h}$$ by almost all authors, +so $$\alpha$$ controls the velocity of our reference frame. +Nonlinear waves do not travel at the same speed as linear waves, +so we can choose $$\alpha$$ to make the wave stationary +without breaking the $$q_0$$ "tradition". +That term in the KdV equation simply corrects for our chosen value of $$\alpha$$. @@ -383,14 +384,16 @@ These are the final scale parameter values, leading to the desired dimensionless form: $$\begin{aligned} - 0 - &= \tilde{\eta}_{\tilde{t}} - 6 \tilde{\eta} \tilde{\eta}_{\tilde{x}} + \tilde{\eta}_{\tilde{x} \tilde{x} \tilde{x}} + \boxed{ + 0 + = \tilde{\eta}_{\tilde{t}} - 6 \tilde{\eta} \tilde{\eta}_{\tilde{x}} + \tilde{\eta}_{\tilde{x} \tilde{x} \tilde{x}} + } \end{aligned}$$ Recall that $$\alpha$$ sets the background fluid velocity, and $$v_c$$ controls the coordinate system's motion: our choice of $$v_c$$ simply cancels out the effect of $$\alpha$$. -This reveals the point of $$\alpha$$: +This demonstrates the purpose of $$\alpha$$: the KdV equation has solutions moving at various speeds, so, for a given $$\eta$$, we can always choose $$\alpha$$ (and hence $$v_c$$) such that the wave appears stationary. |