Sine wave

Tracing the y component of a circle while going around the circle results in a sine wave (red). Tracing the x component results in a cosine wave (blue). Both waves are sinusoids of the same frequency but different phases.

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.

Audio example

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.

Sinusoid form

Sine waves of arbitrary phase and amplitude are called sinusoids and have the general form:^[1]

y(t)=A\sin(\omega t+\varphi )=A\sin(2\pi ft+\varphi )

where:

$A$ , amplitude, the peak deviation of the function from zero.
$t$ , the real independent variable, usually representing time in seconds.
$\omega$ , angular frequency, the rate of change of the function argument in units of radians per second.
$f$ , ordinary frequency, the number of oscillations (cycles) that occur each second of time.
$\varphi$ $\varphi$ , phase, specifies (in radians) where in its cycle the oscillation is at t = 0.
- When $\varphi$ is non-zero, the entire waveform appears to be shifted backwards in time by the amount ${\tfrac {\varphi }{\omega }}$ seconds. A negative value represents a delay, and a positive value represents an advance.
- Adding or subtracting $2\pi$ (one cycle) to the phase results in an equivalent wave.

As a function of both position and time

The displacement of an undamped spring-mass system oscillating around the equilibrium over time is a sine wave.

Sinusoids that exist in both position and time also have:

a spatial variable $x$ that represents the position on the dimension on which the wave propagates.
a wave number (or angular wave number) $k$ $k$ , which represents the proportionality between the angular frequency $\omega$ $\omega$ and the linear speed (speed of propagation) $v$ $v$ :
- wavenumber is related to the angular frequency by ${\textstyle k{=}{\frac {\omega }{v}}{=}{\frac {2\pi f}{v}}{=}{\frac {2\pi }{\lambda }}}$ where $\lambda$ (lambda) is the wavelength.

Depending on their direction of travel, they can take the form:

$y(x,t)=A\sin(kx-\omega t+\varphi )$ , if the wave is moving to the right, or
$y(x,t)=A\sin(kx+\omega t+\varphi )$ , if the wave is moving to the left.

Since sine waves propagate without changing form in distributed linear systems, they are often used to analyze wave propagation.

Standing waves

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).

Multiple spatial dimensions

The earlier equation gives the displacement $y$ 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.

In two or three spatial dimensions, the same equation describes a travelling plane wave if position $x$ and wavenumber $k$ 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.

Sinusoidal plane wave

In physics, a sinusoidal plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. It is also called a monochromatic plane wave, with constant frequency (as in monochromatic radiation).

Fourier analysis

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.

Differentiation and integration

Differentiation

Differentiating any sinusoid will phase shift the sinusoid backwards by ${\tfrac {\pi }{2}}$ radians (or ${\tfrac {1}{4}}$ of a cycle) and multiply its amplitude by its frequency:

${\begin{aligned}{\frac {d}{dt}}[A\sin(\omega t+\varphi )]&=A\omega \cos(\omega t+\varphi )\\&=A\omega \sin(\omega t+\varphi +{\tfrac {\pi }{2}})\,.\end{aligned}}$

Differentiation is effectively a 1^st order high-pass filter without a cutoff frequency.

Integration

Integrating any sinusoid will phase shift the sinusoid forwards by ${\tfrac {\pi }{2}}$ radians (or ${\tfrac {1}{4}}$ of a cycle) and divide its amplitude by its frequency:

${\begin{aligned}\int A\sin(\omega t+\varphi )dt&=-{\frac {A}{\omega }}\cos(\omega t+\varphi )+C\\&=-{\frac {A}{\omega }}\sin(\omega t+\varphi +{\tfrac {\pi }{2}})+C\\&={\frac {A}{\omega }}\sin(\omega t+\varphi -{\tfrac {\pi }{2}})+C\,.\end{aligned}}$

Integration is effectively a 1^st order low-pass filter without a cutoff frequency. The constant of integration $C$ will be zero if the interval of integration is an integer multiple of the sinusoid's period.

References

↑ Smith, Julius Orion. "Sinusoids". ccrma.stanford.edu. Retrieved 2024-01-05.

External links

"Sine Wave". Mathematical Mysteries. 2021-11-17. Retrieved 2022-09-30.

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[1] Smith, Julius Orion. "Sinusoids". ccrma.stanford.edu. Retrieved 2024-01-05.

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