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Sine wave

Characteristics of Sine Waves: – A sine wave represents a single frequency with no harmonics – Adding sine waves of different frequencies results in a […]

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Characteristics of Sine Waves:
– A sine wave represents a single frequency with no harmonics
– Adding sine waves of different frequencies results in a different waveform
– Higher harmonics alter the timbre of the sound
– Different instruments sound distinct due to varied harmonics
– Sine waves are considered acoustically pure tones

Properties of Sinusoids:
– Sinusoids are sine waves of arbitrary phase and amplitude
– General form includes amplitude, time, angular frequency, frequency, and phase
– Phase shift affects the waveform in time
– Adjusting phase by 2π results in an equivalent wave
– Combination of two sine waves of different phases yields an arbitrary phase sine wave

Applications of Sine Waves:
– Displacement of an undamped spring-mass system oscillating is a sine wave
– Wave number relates angular frequency to linear speed and wavelength
– Sine waves moving left or right depend on wave number and angular frequency
– Sine waves maintain form in distributed linear systems for wave analysis

Standing Waves and Wave Patterns:
– Superposition of waves creates standing waves
– Standing wave patterns occur when waves reflect at fixed endpoints
– Resonant frequencies of a string are multiples of the fundamental wavelength
– Standing wave formation explained by superimposing waves

Fourier Analysis and Its Significance:
– Joseph Fourier discovered sinusoidal waves can approximate any periodic waveform
– Fourier series are used in signal processing and time series analysis
– Fourier transform extends Fourier series for general functions
– Fourier analysis is a key concept in signal processing and mathematics
– Fourier analysis helps in analyzing and decomposing complex functions

Sine wave (Wikipedia)

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.

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.

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.

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