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A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge. The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain D in m-dimensional x space, with boundary B.
Any linear combination of plane waves with the same normal vector is also a plane wave. For a scalar plane wave in two or three dimensions, the gradient of the field is always collinear with the direction ; specifically, (,) = (,), where is the partial derivative of with respect to the first argument.
A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue). Electric force vector (E) and magnetic force vector (H) of a standing wave. Standing waves in a string – the fundamental mode and the first 5 harmonics. A standing wave on a circular membrane, an example of standing waves in ...
A standing wave, also known as a stationary wave, is a wave whose envelope remains in a constant position. This phenomenon arises as a result of interference between two waves traveling in opposite directions. The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standing waves commonly arise when ...
Vibrations of a circular membrane. One of the possible modes of vibration of an idealized circular drum head (mode with the notation below). Other possible modes are shown at the bottom of the article. A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by ...
The above form shows that the KP equation is a generalization to two spatial dimensions, x and y, of the one-dimensional Korteweg–de Vries (KdV) equation. To be physically meaningful, the wave propagation direction has to be not-too-far from the x direction, i.e. with only slow variations of solutions in the y direction.
Gaussian function. In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form and with parametric extension for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape.
1-dimensional corollaries for two sinusoidal waves The following may be deduced by applying the principle of superposition to two sinusoidal waves, using trigonometric identities. The angle addition and sum-to-product trigonometric formulae are useful; in more advanced work complex numbers and fourier series and transforms are used.