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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;^14.3 it 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:

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