I am reading Griffiths' Intro to Electrodynamics (Section 9.1.3 Boundary Conditions: Reflection and Transmission) where he analyzes the case of an incident wave sent down a string that is tied to another string. The first string has uniform mass per unit length $\mu_1$ and the second has uniform mass per unit length $\mu_2$.
He says the incident wave is a sinusoidal oscillation that extends (in principle) all the way back to $z=-\infty$ (defining $z$ the axis along the length of the string) and that the same goes for the reflected wave and transmitted wave (except $z=+\infty$ for the transmitted wave).
I'm not entirely sure what's confusing me, but I'll start with this question:
What guarantees that the transmitted and reflected waves are sinusoidal?