Синусоидалы толқын және оның қолданыстары

Wave shaped like the sine function

A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is simple harmonic motion; as rotation, it corresponds to uniform circular motion. Sine waves occur often in physics, including wind waves, sound waves, and light waves, such as monochromatic radiation. In engineering, signal processing, and mathematics, Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes. When any two sine waves of the same frequency (but arbitrary phase) are linearly combined, the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the sine and cosine components, respectively.

A sine wave represents a single frequency with no harmonics and is considered an acoustically pure tone. Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, which is the reason why the same musical pitch played on different instruments sounds different.

When two waves with the same amplitude and frequency traveling in opposite directions superpose each other, then a standing wave pattern is created. On a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string's resonant frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the fundamental frequency) and integer divisions of that (corresponding to higher harmonics).

The earlier equation gives the displacement of the wave at a position at time along a single line. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling plane wave if position and wavenumber are interpreted as vectors, and their product as a dot product. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed.

French mathematician Joseph Fourier discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including square waves. These Fourier series are frequently used in signal processing and the statistical analysis of time series. The Fourier transform then extended Fourier series to handle general functions, and birthed the field of Fourier analysis.