A report on Standing wave and Wave equation

Animation of a standing wave ( red ) created by the superposition of a left traveling ( blue ) and right traveling ( green ) wave
French scientist Jean-Baptiste le Rond d'Alembert discovered the wave equation in one space dimension.
Longitudinal standing wave
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Transient analysis of a damped traveling wave reflecting at a boundary
1-d standing wave as a superposition of two waves traveling in opposite directions
Standing wave in stationary medium. The red dots represent the wave nodes.
Swiss mathematician and physicist Leonhard Euler (b. 1707) discovered the wave equation in three space dimensions.
A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source.
Electric force vector (E) and magnetic force vector (H) of a standing wave.
Figure 1: Three consecutive mass points of the discrete model for a string
Standing waves in a string – the fundamental mode and the first 5 harmonics.
A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge.
A standing wave on a circular membrane, an example of standing waves in two dimensions. This is the fundamental mode.
A higher harmonic standing wave on a disk with two nodal lines crossing at the center.

The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves).

- Wave equation

Equivalently, this boundary condition of the "free end" can be stated as ∂y/∂x = 0 at, which is in the form of the Sturm–Liouville formulation.

- Standing wave
Animation of a standing wave ( red ) created by the superposition of a left traveling ( blue ) and right traveling ( green ) wave

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Surface waves in water showing water ripples

Wave

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Propagating dynamic disturbance of one or more quantities.

Propagating dynamic disturbance of one or more quantities.

Surface waves in water showing water ripples
Example of biological waves expanding over the brain cortex, an example of spreading depolarizations.
Wavelength λ, can be measured between any two corresponding points on a waveform
Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that f(x,t) + g(x,t) = u(x,t)
Sine, square, triangle and sawtooth waveforms.
Amplitude modulation can be achieved through f(x,t) = 1.00×sin(2π/0.10×(x−1.00×t)) and g(x,t) = 1.00×sin(2π/0.11×(x−1.00×t))only the resultant is visible to improve clarity of waveform.
Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.
The red square moves with the phase velocity, while the green circles propagate with the group velocity
A wave with the group and phase velocities going in different directions
Standing wave. The red dots represent the wave nodes
Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism
Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results.
Identical waves from two sources undergoing interference. Observed at the bottom one sees 5 positions where the waves add in phase, but in between which they are out of phase and cancel.
Schematic of light being dispersed by a prism. Click to see animation.
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Formation of a shock wave by a plane.
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A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.
Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles
One-dimensional standing waves; the fundamental mode and the first 5 overtones.
A two-dimensional standing wave on a disk; this is the fundamental mode.
A standing wave on a disk with two nodal lines crossing at the center; this is an overtone.

When the entire waveform moves in one direction, it is said to be a traveling wave; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave.

Waves are often described by a wave equation (standing wave field of two opposite waves) or a one-way wave equation for single wave propagation in a defined direction.