# $\psi(\vec{\mathbf{r}}, t) = C_1 f(\vec{\mathbf{r}} \cdot \vec{\mathbf{k}}/k - vt) + C_2 g(\vec{\mathbf{r}} \cdot \vec{\mathbf{k}}/k + vt)$

I am currently studying Optics, fifth edition, by Hecht. I was presented with the three-dimensional differential wave equation

$$\dfrac{\partial^2 \psi}{\partial x^2} + \dfrac{\partial^2 \psi}{\partial y^2} + \dfrac{\partial^2\psi}{\partial z^2} = \dfrac{1}{v^2} \dfrac{\partial^2 \psi}{\partial t^2}.$$

I am then told that it can be shown that

$$\psi(x, y, z, t) = f(\alpha x + \beta y + \gamma z - vt)$$

and

$$\psi(x, y, z, t) = g(\alpha x + \beta y + \gamma z + vt)$$

are both plane-wave solutions of the differential wave equation. Finally, I am told that a linear combination of these solutions is also a solution, and we can write this as

$$\psi(\vec{\mathbf{r}}, t) = C_1 f(\vec{\mathbf{r}} \cdot \vec{\mathbf{k}}/k - vt) + C_2 g(\vec{\mathbf{r}} \cdot \vec{\mathbf{k}}/k + vt),$$

where $$C_1$$ and $$C_2$$ are constants.

My understanding is that $$\vec{\mathbf{r}} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}}$$ and $$\vec{\mathbf{k}} = k_x \hat{\mathbf{i}} + k_y \hat{\mathbf{j}} + k_z \hat{\mathbf{k}}$$, and $$k$$ is the magnitude of $$\vec{\mathbf{k}}$$. But I'm still confused about how $$C_1 f(\vec{\mathbf{r}} \cdot \vec{\mathbf{k}}/k - vt)$$ and $$C_2 g(\vec{\mathbf{r}} \cdot \vec{\mathbf{k}}/k + vt)$$ are equivalent to $$\psi(x, y, z, t) = f(\alpha x + \beta y + \gamma z - vt)$$ and $$\psi(x, y, z, t) = g(\alpha x + \beta y + \gamma z + vt)$$, respectively.

I would greatly appreciate it if someone would please take the time to clarify this.

• $\mathbf r\cdot\mathbf k = xk_x + yk_y + zk_z$ Mar 11, 2020 at 7:43
• @JohnRennie And the magnitude of $\vec{\mathbf{k}}$ is $\sqrt{k_x^2 + k_y^2 + k_z^2}$, right? So how does $\vec{\mathbf{r}} \cdot \vec{\mathbf{k}}/k$ get us those equations? That's what I'm confused about. Mar 11, 2020 at 7:49
• Titles with just MathJax in them are a little problematic (for me anyway). If I try and right-click on them to open a new tab in the browser to view them, it's interfered with by the Mathjax handling. Some text would be preferred (by me at least). Mar 11, 2020 at 10:47
• This is the same math as your previous question. Mar 11, 2020 at 16:11
• @G.Smith Hmm? They're different questions, and the math in that question does not answer this question. Mar 11, 2020 at 16:14

If $$\begin{equation} \alpha = k_x/k, \beta = k_y/k, \gamma = k_z/k, \end{equation}$$ then $$\begin{equation} \alpha x + \beta y + \gamma z = (k_x x + k_y y + k_z z)/k = \vec{\mathbf{k}}\cdot \vec{\mathbf{x}}/k. \end{equation}$$
It is worth noting that strictly speaking $$f(\alpha x + \beta y + \gamma z \pm vt)$$ is a solution of wave equation only if $$\alpha^2 + \beta^2 + \gamma^2 = 1$$, which is easy to check by substituting these solutions into the equation (left as an exercize).