# 4.3.10. pde.pdes.wave module¶

A simple diffusion equation

class WavePDE(speed: float = 1, bc: Union[Dict[str, Union[Dict, str, BCBase]], Dict, str, BCBase, Tuple[Union[Dict, str, BCBase], Union[Dict, str, BCBase]], Sequence[Union[Dict[str, Union[Dict, str, BCBase]], Dict, str, BCBase, Tuple[Union[Dict, str, BCBase], Union[Dict, str, BCBase]]]]] = 'auto_periodic_neumann')[source]

Bases: PDEBase

A simple wave equation

The mathematical definition,

$\partial_t^2 u = c^2 \nabla^2 u$

is implemented as two first-order equations:

$\begin{split}\partial_t u &= v \\ \partial_t v &= c^2 \nabla^2 u\end{split}$

where $$u$$ is the density field that and $$c$$ sets the wave speed.

Parameters
• speed (float) – The speed $$c$$ of the wave

• bc – The boundary conditions applied to the field. Boundary conditions are generally given as a list with one condition for each axis. For periodic axis, only periodic boundary conditions are allowed (indicated by ‘periodic’ and ‘anti-periodic’). For non-periodic axes, different boundary conditions can be specified for the lower and upper end (using a tuple of two conditions). For instance, Dirichlet conditions enforcing a value NUM (specified by {‘value’: NUM}) and Neumann conditions enforcing the value DERIV for the derivative in the normal direction (specified by {‘derivative’: DERIV}) are supported. Note that the special value ‘natural’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation.

evolution_rate(state: FieldCollection, t: float = 0) [source]

evaluate the right hand side of the PDE

Parameters
• state (FieldCollection) – The fields $$u$$ and $$v$$ distribution

• t (float) – The current time point

Returns

Scalar field describing the evolution rate of the PDE

Return type

FieldCollection

explicit_time_dependence: Optional[bool] = False

Flag indicating whether the right hand side of the PDE has an explicit time dependence.

Type

bool

property expressions: Dict[str, str]

the expressions of the right hand side of this PDE

Type

dict

get_initial_condition(u: ScalarField, v: Optional[ScalarField] = None)[source]

create a suitable initial condition

Parameters
• u (ScalarField) – The initial density on the grid

• v (ScalarField, optional) – The initial rate of change. This is assumed to be zero if the value is omitted.

Returns

The combined fields u and v, suitable for the simulation

Return type

FieldCollection