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 cosinecomponents, respectively.
A sine wave represents a single frequency with no harmonics and is considered an acousticallypure 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]
where:
, amplitude, the peak deviation of the function from zero.
, phase, specifies (inradians) where in its cycle the oscillation is at t = 0.
When is non-zero, the entire waveform appears to be shifted backwards in time by the amount seconds. A negative value represents a delay, and a positive value represents an advance.
Adding or subtracting (one cycle) to the phase results in an equivalent wave.
As a function of both position and time
Sinusoids that exist in both position and time also have:
a spatial variable that represents the position on the dimension on which the wave propagates.
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 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.
Differentiating any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle:
Anintegrator has a pole at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1st order low-pass filter's stopband, although an integrator doesn't have a cutoff frequency or a flat passband. A nth-order low-pass filter approximately performs the nth time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency.