During my study for quantum mechanics, I encountered the classical wave equation, which is stated as "$\frac{1}{v^2} \frac {\partial^2\phi}{\partial t^2}=\nabla^2\phi$".
However, when I learnt about wave equations in general, the equation that I am more familiar with is "$\frac{1}{v^2} \frac {\partial^2y}{\partial t^2}=\frac {\partial^2y}{\partial x^2}$".
So I was wondering how these two wave equations are interlinked? Is the first one just a higher dimension wave equation in comparison to the second one?