of space, called its domain. For example, the seismic waves generated by earthquakes are significant only in the interior and surface of the planet, so they can be ignored outside it. However, waves with infinite domain, that extend over the whole space, are commonly studied in mathematics, and are very valuable tools for understanding physical waves in finite domains. A plane wave is an important mathematical idealization where the disturbance is identical along any (infinite) plane normal to a specific direction of travel. Mathematically, the simplest wave is a sinusoidal plane wave in which at any point the field experiences simple harmonic motion at one frequency. In linear media, complicated waves can generally be decomposed as the sum of many sinusoidal plane waves having different directions of propagation and/or different frequencies. A plane wave is classified as a transverse wave if the field disturbance at each point is described by a vector perpendicular to the direction of propagation (also the direction of energy transfer); or longitudinal wave if those vectors are aligned with the propagation direction. Mechanical waves include both transverse and longitudinal waves; on the other hand electromagnetic plane waves are strictly transverse while sound waves in fluids (such as air) can only be longitudinal. That physical direction of an oscillating field relative to the propagation direction is also referred to as the wave's polarization, which can be an important attribute. Mathematical description Single waves A wave can be described just like a field, namely as a function where is a position and is a time. The value of is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a vector in the Cartesian three-dimensional space . However, in many cases one can ignore one dimension, and let be a point of the Cartesian plane . This is the case, for example, when studying vibrations of a drum skin. One may even restrict to a point of the Cartesian line — that is, the set of real numbers. This is the case, for example, when studying vibrations in a violin string or recorder. The time , on the other hand, is always assumed to be a scalar; that is, a real number. The value of can be any physical quantity of interest assigned to the point that may vary with time. For example, if represents the vibrations inside an