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Definition. A triangle wave of period p that spans the range [0, 1] is defined as where is the floor function. This can be seen to be the absolute value of a shifted sawtooth wave. For a triangle wave spanning the range [−1, 1] the expression becomes. Triangle wave with amplitude = 5, period = 4. A more general equation for a triangle wave ...
Wave equation. The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics.
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function ...
Helmholtz equation. In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: where ∇2 is the Laplace operator, k2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
Mathieu function. In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation. where a, q are real -valued parameters. Since we may add π/2 to x to change the sign of q, it is a usual convention to set q ≥ 0.
A Green's function, G(x,s), of a linear differential operator L = L(x) acting on distributions over a subset of the Euclidean space , at a point s, is any solution of. (1) where δ is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form.
An eikonal equation (from Greek εἰκών, image [1][2]) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation. The classical eikonal equation in geometric optics is a differential equation of the form. (1)