# Standard Notation

From WikiWaves

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

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

## Greek letters

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

## Other notation, style etc.

- We prefer [math]\partial_x\phi[/math] etc. for all derivatives or [math]\frac{\partial\phi}{\partial x}[/math]. Try to avoid [math]\phi_x\,[/math] or [math]\phi^{\prime}[/math]
- We prefer [math]\mathrm{d}x\,\![/math] etc. for differentials. Avoid [math]dx\,\![/math]
- [math]\mathrm{Re}\,\![/math] and [math]\mathrm{Im}\,\![/math] 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.