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• Method of Characteristics for Linear Equations
  \partial_t u + \partial_x u = 0,\,\,-\infty < x < \infty,\,\,t>0, \left. u \right|_{t=0} = f(x)
  5 KB (913 words) - 05:25, 7 August 2020
• Introduction to the Inverse Scattering Transform
  \partial_{t}u+6u\partial_{x}u+\partial_{x}^{3}u & =0\\ u(x,0) & =f\left( x\right)
  2 KB (297 words) - 03:26, 15 September 2020
• Conservation Laws for the KdV
  \partial _{t}T\left( u\right) +\partial _{x}X\left( u\right) =0 \int_{-\infty }^{\infty }\partial _{t}T\left( u\right) \mathrm{d} x = -\int_{-\infty
  5 KB (830 words) - 02:59, 7 September 2020
• Burgers Equation
  \partial _{t}u+u\partial _{x}u=\nu \partial _{x}^{2}u (changing variables to <math>u</math> and this equation is known as Burgers equation.
  8 KB (1,308 words) - 02:14, 24 October 2020
• Numerical Solution of the KdV
  \partial _{t}u+\partial _{x}^{3}u=0,\ \ -\infty < x <\infty u\left( x,0\right) =f\left( x\right)
  7 KB (1,244 words) - 02:58, 17 August 2023
• Example Calculations for the KdV and IST
  u\left( x\right) =\left\{ We cannot work with a hat function numerically, because the jump in <math>u</math> leads
  3 KB (426 words) - 03:38, 1 October 2020
• Long Wavelength Approximations
  <math> U(x,t):\ \mbox{Velocity of ambient unidirectional flow} \,</math> <math> P(x,t):\ \mbox{Pressure corresponding to} \ U(x,t) \,</math>
  9 KB (1,545 words) - 01:55, 12 February 2010
• Connection betwen KdV and the Schrodinger Equation
  where <math>Q=\partial_{t}w+\partial_{x}^{3}w-3\left( \lambda-u\right) and <math>u\left( x,t\right) </math> evolves according to the KdV. Many other
  6 KB (1,181 words) - 05:30, 14 September 2023
• Rankine Intergral Equations For Ship Flows
  <u>Rankine integral equations for ship flow problems with forward speed</u> ...ves when the free surface condition is more complex than that of the <math>U=0\,</math> frequency domain problem.
  6 KB (949 words) - 23:36, 16 October 2009
• Nonlinear Shallow Water Waves
  <math>u(x,t)</math>, the velocity of the water, and <math>h(x,t)</math> the water d ...Delta x} \rho h(s,t) \mathrm{d}s = \rho h(x,t)u(x,t) - \rho h(x+\Delta x,t)u(x+\Delta x,t)
  19 KB (3,336 words) - 04:31, 17 August 2020
• Seakeeping In Random Waves
  <u>Spectral analysis with forward-speed</u> <center><math> \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos \beta \right| </math></center>
  3 KB (376 words) - 15:26, 6 August 2010
• Boundary Element Method for a Fixed Body in Finite Depth
  ...mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying \mathbf{M}(k_p)\mathbf{u}_{k_p}=0
  8 KB (1,400 words) - 21:32, 10 February 2010
• Forward-Speed Ship Wave Flows
  - 2 U \frac{\partial^2\bar{\Phi}}{\partial x \partial t} + U^2 \frac{\partial \bar{\Phi}}{\partial x^2}
  12 KB (1,969 words) - 11:28, 23 February 2011
• Ship Kelvin Wake
  which is moving in the <math>x</math> direction with speed <math>U</math>. We denote the moving coordinate \bar{x} = x + U t
  9 KB (1,578 words) - 11:37, 6 November 2010
• Properties of the Linear Schrodinger Equation
  has two kinds of solutions for <math>u\rightarrow0</math> as <math>x\rightarrow\pm\infty.</math> The there are at most a finite number of bound solutions (provided <math>u\rightarrow0</math>
  8 KB (1,327 words) - 01:03, 24 September 2020
• Weakly Nonlinear Wave Theory for Periodic Waves (Stokes Expansion)
  Particle velocity <math> \vec{V} = \vec{i} u + \vec{k} w \, </math> <center><math> u = \frac{akg}{\sigma} \frac{\cosh[k(z+h)]}{\cosh kh} \cos\theta + \frac{3}{4
  20 KB (3,008 words) - 10:28, 10 September 2009
• Wave Scattering by Submerged Thin Bodies
  \oint_{-1}^{1} \frac{(1-v^2)^{1/2}U_m(v)}{(u-v)^2}dv = -\pi (m+1) U_m(u) \mathbb{Y} &= \mathbb{U} \ \mathbb{W}^{(1)}\mathbb{W}^{(2)} \\
  8 KB (1,267 words) - 22:54, 29 September 2009
• Solution of Wave-Body Flows, Green's Theorem
  ==Frequency-domain radiation-diffraction. U = 0== <u>Rankine source</u>: <math> \nabla_x^2 G = 0 \, </math>
  13 KB (2,103 words) - 19:55, 26 July 2012
• Strip Theory Of Ship Motions. Heave & Pitch
  Ship advances in the positive x-direction with constant speed <math>U\,</math>. Regular waves with absolute frequency <math>\omega_0\,</math> and <center><math> \vec{U}=U\vec{i}; \quad \mbox{Forward speed} \,</math></center>
  21 KB (3,361 words) - 16:05, 10 September 2010
• Fundamental Solution for thin plates
  \left(\Delta^2 - k^2\right) u =0, where <math>u \,</math> is the displacement, <math>k \,</math> is the wavenumber, and <ma
  6 KB (900 words) - 21:14, 27 April 2010

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