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Concept
Sine wave
Summary
A sine wave, sinusoidal wave, or sinusoid is a mathematical curve defined in terms of the sine trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in mathematics, as well as in physics, engineering, signal processing and many other fields.
Its most basic form as a function of time (t) is:
where:
A, amplitude, the peak deviation of the function from zero.
f, ordinary frequency, the number of oscillations (cycles) that occur each second of time.
ω = 2pif, angular frequency, the rate of change of the function argument in units of radians per second.
phase, specifies (in radians) where in its cycle the oscillation is at t = 0. When is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance.
The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.
In general, the function may also have:
a spatial variable x that represents the position on the dimension on which the wave propagates, and a characteristic parameter k called wave number (or angular wave number), which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν;
a non-zero center amplitude, D
which is
if the wave is moving to the right
if the wave is moving to the left.
The wavenumber is related to the angular frequency by:
where λ (lambda) is the wavelength, f is the frequency, and v is the linear speed.
This equation gives a sine wave for a single dimension; thus the generalized equation given above gives the displacement of the wave at a position x at time t along a single line.
This could, for example, be considered the value of a wave along a wire.
Official source
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