#### BELIEVE   ME   NOT!    - -     A   SKEPTIC's   GUIDE   Next: Speed of Propagation Up: Wave Phenomena Previous: Wave Phenomena

## Traveling Waves

How do we represent this behaviour mathematically? Well, is a function of position and time : . At any fixed position , oscillates in time at a frequency . We can describe this statement mathematically by saying that the entire time dependence of is contained in [the real part of] a factor (that is, the amplitude at any fixed position obeys SHM).14.2

The oscillation with respect to position at any instant of time is given by the analogous factor where is the wave vector;14.3it points in the direction of propagation of the wave and has a magnitude (called the "wavenumber") given by (14.1)

where is the wavelength. Note the analogy between and (14.2)

where is the period of the oscillation in time at a given point. You should think of as the "period in space."

We may simplify the above description by choosing our coordinate system so that the axis is in the direction of , so that14.4 . Then the amplitude no longer depends on or , only on and .

We are now ready to give a full description of the function describing this wave: or, recalling the multiplicative property of the exponential function, , (14.3)

To achieve complete generality we can restore the vector version: (14.4)

This is the preferred form for a general description of a PLANE WAVE, but for present purposes the scalar version (3) suffices. Using Eqs. (1) and (2) we can also write the plane wave function in the form (14.5)

but you should strive to become completely comfortable with and - we will be seeing a lot of them in Physics!

#### Footnotes

. . . SHM).14.2
Note that would have worked just as well, since the real part is the same as for . The choice of sign does matter, however, when we write down the combined time and space dependence in Eq. (4), which see.
. . . vector;14.3
The name "wave vector" is both apt and inadequate - apt because the term vector explicitly reminds us that its direction defines the direction of propagation of the wave; inadequate because the essential inverse relationship between and the wavelength [see Eq. (1)] is not suggested by the name. Too bad. It is at least a little more descriptive than the name given to the magnitude of , namely the "wavenumber."
. . . that14.4
In general . If then and , giving .   Next: Speed of Propagation Up: Wave Phenomena Previous: Wave Phenomena
Jess H. Brewer - Last modified: Sun Nov 15 17:57:49 PST 2015