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Traveling Waves

How do we represent this behaviour mathematically? Well,  $A$  is a function of position $\Vec{r}$ and time $t$:  $A(\Vec{r},t)$. At any fixed position $\Vec{r}$, $A$ oscillates in time at a frequency  $\omega$. We can describe this statement mathematically by saying that the entire time dependence of $A$ is contained in [the real part of] a factor   ${\displaystyle e^{-i \omega t}}$  (that is, the amplitude at any fixed position obeys SHM).^14.2

The oscillation with respect to position $\Vec{r}$ at any instant of time $t$ is given by the analogous factor   ${\displaystyle e^{i \sVec{k} \cdot \sVec{r} } }$  where $\Vec{k}$ is the wave vector;^14.3it points in the direction of propagation of the wave and has a magnitude (called the "wavenumber") $k$ given by

$$k \; = \; {2 \pi \over \lambda}$$ (14.1)

where $\lambda$ is the wavelength. Note the analogy between $k$ and

$$\omega \; = \; {2 \pi \over T}$$ (14.2)

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

We may simplify the above description by choosing our coordinate system so that the $x$ axis is in the direction of $\Vec{k}$, so that^14.4  $\Vec{k} \cdot \Vec{r} \; = \; k \, x$. Then the amplitude $A$ no longer depends on $y$ or $z$, only on $x$ and $t$.

We are now ready to give a full description of the function describing this wave:

$$A(x,t) \; = \; A_{_0} \; e^{ikx} \cdot e^{-i\omega t}$$

or, recalling the multiplicative property of the exponential function,   $e^a \cdot e^b = e^{(a+b)}$,

$$A(x,t) \; = \; A_{_0} \; e^{i(kx - \omega t)} .$$ (14.3)

To achieve complete generality we can restore the vector version:

$$\mbox{ \fbox{ \rule[-1.0\baselineskip]{0pt}{2.5\baselineski . . . . . . eft( \sVec{k} \cdot \sVec{r} - \omega t \right)} }$~ }}$$ (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

$$A(x,t) \; = \; A_{_0} \; \exp \left[ 2\pi i \left( {x \over \lambda} - {t \over T} \right) \right]$$ (14.5)

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

Footnotes

. . . SHM).^14.2

Note that ${\displaystyle e^{+i \omega t}}$ would have worked just as well, since the real part is the same as for ${\displaystyle e^{-i \omega t}}$. 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 $k$ and the wavelength $\lambda$ [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 $k$ of $\Vec{k}$, namely the "wavenumber."

. . . that^14.4

In general $\Vec{k} \cdot \Vec{r} = x k_x + y k_y + z k_z$. If $\Vec{k} = k \, \Hat{\imath}$ then $k_x = k$ and $k_y = k_z = 0$, giving $\Vec{k} \cdot \Vec{r} = k \, x$.