# Single frequency solutions to wave equation - Two forms? Whats the difference?

I've seen these two forms from multiple sources for solutions to Maxwell's equations: $$\cos(kz - \omega t)$$ and $$\cos(\omega t - \vec{k} \cdot \vec{r})$$

The first one shows a wave travelling in the positive $$z$$ direction. What does the second one show? What's the significance of having swapping the signs on the two terms? i.e .$$\cos(-kz + \omega t)$$

• What is $\cos(-x)$ equal to? – David Aug 13 at 8:30
• Yes, missed that thanks. What about the main part of the question can you assist? – Natalie Johnson Aug 13 at 8:33
• My comment should address your question about swapping the signs no? Or is it the $k \cdot r$ vs $kz$ that bothers you? – David Aug 13 at 8:36
• Yes unclear. Also, if time is increasing are k and z both increasing and positive? – Natalie Johnson Aug 13 at 9:04
• Possible duplicate: physics.stackexchange.com/questions/351975/… – Gulce Kardes Aug 13 at 9:26

$$k = \dfrac {2\pi}{\lambda}$$ where $$\lambda$$ is the wavelength of the wave.

$$\cos(kz - \omega t)$$

is the equation of a (plane wave) travelling in the positive $$z$$ direction.
You can think of it as $$\cos(\vec k \cdot \vec r - \omega t)$$ where $$\vec k= k \,\hat z$$ and $$\vec r= z \,\hat z$$.

The direction of the k-vector gives the direction of travel of the wave or the opposite direction if the form of the equation is $$\cos(kz + \omega t)$$.

$$\cos(\omega t - \vec{k} \cdot \vec{r})$$

is the equation of a wave in three dimensions travelling in the k-vector direction with $$\vec k \cdot \vec r = k_{\rm x} \, x + k_{\rm y} \, y + k_{\rm z} \, z$$

This two-dimensional plane wave visualisation may help you understand what is going on.

$$\cos(kz - \omega t)$$ and $$\cos(-kz + \omega t)= \cos (-[kz - \omega t])$$

describe the same wave travelling in the positive z-direction because $$\cos(-kz + \omega t)= \cos (-[kz - \omega t]) = \cos(kz - \omega t)$$