Sine wave

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in both pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

The graphs of the sine (solid red) and cosine (dotted blue) functions are sinusoids of different phases


  • 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 = 2πf, 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 oscillation of an undamped spring-mass system around the equilibrium is a sine wave.

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.