Standard Notation

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This is a list of standard notation with definitions. If you find notation which does not appear here or non-standard notation please feel free to highlight this, or better still try and fix it. The material on these webpages was taken from a variety of sources and we know the notation is currently not always consistent between pages.

Latin Letters

  • A is the wave amplitude
  •  c \,(=\omega / k)  or sometime c_p is the wave phase velocity
  •  c_g = \frac{\mathrm{d} \omega}{\mathrm{d} k} is the wave group velocity
  • d is a water depth parameter
  • D is the modulus of rigidity for a plate
  • e^{i\omega t} is the time dependence in frequency domain
  • E is the Young's modulus
  •  \mathcal{E}(t) is the energy density
  • g is the acceleration due to gravity
  • h is the water depth (with the bottom at z=-h)
  • \mathbf{i} is the unit vector in the x direction
  • \mathrm{Im} is the imaginary part of a complex argument
  • \mathbf{j} is the unit vector in the y direction
  • \mathbf{k} is the unit vector in the z direction
  •  k is the wave number
  • k_n are the roots of the dispersion eqution
  • \mathcal{L} is the linear operator at the body surface
  • \mathcal{M} is the momentum
  • \mathbf{n} is the outward normal
  • \frac{\partial\phi}{\partial n} is \nabla\phi\cdot\mathbf{n}
  • P is the pressure (P_1, P_2 etc are the first, second order pressures)
  • \mathcal{P}(t) the energy flux is the rate of change of energy density  \mathcal{E}(t)
  • \mathbf{r} vector in the horizontal directions only (x,y)
  • R is the radius of a cylinder
  • \mathrm{Re} is the real part of a complex argument
  • S_F is the free surface
  • t is the time
  •  T \,(= 2\pi / \omega) is the wave period
  • U is the forward speed
  • U_n is the normal derivative of the moving surface of a volume
  •  V_n = \mathbf{n} \cdot \nabla \Phi is the flow in the normal direction for potential \Phi
  • \mathbf{v} is the flow velocity vector at \mathbf{x}
  • \mathbf{x} is the fixed Eulerian vector (x,y,z)
  • x and y are in the horizontal plane with z pointing vertically upward and the free surface is at z=0
  • \bar{x} is the x coordinate in a moving frame.
  • X_n(x) is an eigenfunction arising from separation of variables in the x direction.
  • Z(z) is an eigenfunction arising from separation of variables in the z direction.

Greek letters

  • \alpha is free surface constant \alpha = \omega^2/g
  • \mathcal{E} is the energy
  • \zeta is the displacement of the surface
  • \xi any other displacement, most usually a body in the fluid
  • \eta any other displacement, most usually a body in the fluid
  •  \lambda \,(= 2\pi/k) is the wave length
  • \rho is the fluid density (sometimes also string density).
  • \rho_i is the plate density
  • \phi\, is the velocity potential in the frequency domain
  • \phi^{\mathrm{I}}\, is the incident potential
  • \phi^{\mathrm{D}}\, is the diffracted potential
  • \phi^{\mathrm{S}}\, is the scattered potential (\phi^{\mathrm{S}}
 = \phi^{\mathrm{I}}+\phi^{\mathrm{D}}\,)
  • \phi_{m}^{\mathrm{R}}\, is the radiated potential (for the m mode
  • \Phi\, is the velocity potential in the time domain
  • \bar{\Phi}\, is the velocity potential in the time domain for a moving coordinate system
  • \omega is the wave/angular frequency
  • \Omega\, is the fluid region
  • \partial \Omega is the boundary of fluid region, \partial\Omega_F is the free surface, \partial\Omega_B is the body surface.

Other notation, style etc.

  • We prefer \partial_x\phi etc. for all derivatives or \frac{\partial\phi}{\partial x}. Try to avoid \phi_x\, or \phi^{\prime}
  • We prefer \mathrm{d}x\,\! etc. for differentials. Avoid dx\,\!
  • \mathrm{Re}\,\! and \mathrm{Im}\,\! for the real and imaginary parts.
  • We use two equals signs for the first heading (rather than a single) following wikipedia style, then three etc.