Sine wave

One positive frequency component, cosine and sine, from rotating vector (fast animation)

Animated-mass-spring

Sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave and is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields.

Definition

The sine wave is mathematically described by the function:

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

where:

• A is the amplitude, the peak deviation of the function from zero.
• f is the frequency, or the number of oscillations that occur in a unit time.
• \phi is the phase, representing the shift from the origin.
• t represents time.

Characteristics

The sine wave is characterized by its smooth, repetitive oscillation, as it is the graph of the sine function. In terms of sound waves, a sine wave represents a pure tone with a single frequency and no harmonics. In electrical engineering, sine waves are used to describe the voltage or current in alternating current (AC) circuits.

Applications

Sine waves are found in many areas of science and engineering. In physics, they describe the motion of oscillating systems such as pendulums and masses on springs. In electrical engineering, they are used to model alternating current (AC) as well as radio and audio signals. The sine wave is also fundamental in signal processing for analyzing periodic functions.

Generation

Sine waves can be generated by oscillators, including electronic oscillators, which produce alternating current at a given frequency. They can also be generated by rotating a coil within a magnetic field in electrical generators.

Mathematical Properties

The sine wave is the solution to the differential equation:

\frac{d^2y}{dt^2} + \omega^2y = 0

where \(\omega = 2\pi f\) is the angular frequency of the wave. This property makes it a fundamental function in the study of harmonic motion.

See Also

• Wave
• Harmonic Oscillation
• Fourier Series
• Signal Processing