Before systematically exploring the wave equation, it is good to pause and contemplate some basic solutions. We are looking for a function q of 2 variables, x and t, whose second x derivatives and second t derivatives are proportional. You can probably guess such functions with a little thought. But our derivation of the equation from the model of a chain of oscillators gives a strong hint.
elementary solution, wave equation, chapter 6, six
Torre, Charles G., "06 Elementary Solutions to the Wave Equation" (2012). Foundations of Wave Phenomena. Book 17.