Editing Sine wave (section)

== As a function of both position and time ==
[[Image:Animated-mass-spring.gif|right|The displacement of an undamped [[Spring mass system|spring-mass system]] oscillating around the equilibrium over time is a sine wave.|thumb|246x246px]]
Sinusoids that exist in both position and time also have:

* a spatial variable <math>x</math> that represents the ''position'' on the dimension on which the wave propagates. 
* a [[wave number]] (or angular wave number) <math>k</math>, which represents the proportionality between the [[angular frequency]] <math>\omega</math> and the linear speed ([[phase velocity|speed of propagation]]) <math>v</math>:
** wavenumber is related to the angular frequency by <math display="inline"> k {=} \frac{\omega}{v} {=} \frac{2 \pi f}{v} {=} \frac{2 \pi}{\lambda}</math> where <math>\lambda</math> ([[lambda]]) is the [[wavelength]]. 

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

*<math>y(x, t) = A\sin(kx - \omega t + \varphi)</math>, if the wave is moving to the right, or
*<math>y(x, t) = A\sin(kx + \omega t + \varphi)</math>, if the wave is moving to the left.
Since sine waves propagate without changing form in ''distributed linear systems'',{{Definition needed|date=August 2019}} they are often used to analyze [[wave propagation]].

=== Standing waves ===
{{Main|Standing wave}}
When two waves with the same [[amplitude]] and [[frequency]] traveling in opposite directions [[superposition principle|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 <math>y</math> of the wave at a position <math>x</math> at time <math>t</math> 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 <math>x</math> and wavenumber <math>k</math> 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 ====
{{excerpt|Sinusoidal plane wave}}