Harmonic Waveforms Propagating to the Left or Right

An interesting special case of this solution is the case of harmonic waves propagating to the left or right. Harmonic waves are simply waves that oscillate with a given harmonic frequency. For example:

$$y(x,t) = y_0 \sin(x - vt)$$ (113)

is one such wave. $y_0$ is called the amplitude of the harmonic wave. But what sorts of parameters describe the wave itself? Are there more than one harmonic waves?

This particular wave looks like a sinusoidal wave propagating to the right (positive $x$ direction). But this is not a very convenient parameterization. To better describe a general harmonic wave, we need to introduce the following quantities:

• The frequency $f$. This is the number of cycles per second that pass a point or that a point on the string moves up and down.
• The wavelength $\lambda$. This the distance one has to travel down the string to return to the same point in the wave cycle at any given instant in time.

To convert $x$ (a distance) into an angle in radians, we need to multiply it by $2\pi$ radians per wavelength. We therefore define the wave number:

$$k = \frac{2\pi}{\lambda}$$ (114)

and write our harmonic solution as:

$$y(x,t) = y_0 \sin( k (x-vt) )$$ (115)

$$= y_0 \sin( kx - kvt )$$ (116)

$$= y_0 \sin( kx - \omega t)$$ (117)

where we have used the following train of algebra in the last step:

$$kv = \frac{2\pi}{\lambda} v = 2 \pi f = \frac{2 \pi}{T} = \omega$$ (118)

and where we see that we have two ways to write $v$:

$$v = f \lambda = \frac{\omega}{k}$$ (119)

As before, you should simply know every relation in this set of algebraic relations between $\lambda, k, f, \omega, v$ to save time on tests and quizzes. Of course there is also the harmonic wave travelling to the left as well:

$$y(x,t) = y_0 \sin( kx + \omega t).$$ (120)

A final observation about these harmonic waves is that because arbitrary functions can be expanded in terms of harmonic functions (e.g. Fourier Series, Fourier Transforms) and because the wave equation is linear and its solutions are superposable, the two solution forms above are not really distinct. One can expand the ``arbitrary'' $f(x - vt)$ in a sum of $\sin(kx - \omega t)$'s for special frequencies and wavelengths. In one dimension this doesn't give you much, but in two or more dimensions this process helps one compute the dispersion of the wave caused by the wave ``spreading out'' in multiple dimensions and reducing its amplitude.