It is not a sine wave. It is multiple sine waves that form a standing wave. The intensity of those sine waves relative to each other is what creates the tone quality, or basically the timbre in musical terms - what we hear as opposed to what actually exists physically. (They're close, but not identical. Just like pitch and fundamental frequency are not identical, but close.)
The tone quality is affected by the construction of the string and what it's attached too, as well as acoustic reflections of nearby objects. The details (lattice structure of cured wood) of the wood's internal structure will affect the tone quality, and hence timbre of the instrument. (This isn't unique to guitars. The specific materials in a brass instrument will affect it, for example.) The species of wood will definitely affect it's properties for sustaining and supporting standing waves and the intensity of them. Obviously, there will be a range for organic material, but one wood species can be distinguished from another readily.
It's clear with a basic understanding of physics that different wood species will produce different tone quality, or timbre, or what we normally simply refer to as tone in the guitar world. It's reasonable to discuss the extent to which it contributes to the overall sound of a specific instrument, compared to all other factors, but it is absolutely incorrect to suggestion the wood has no affect on the tonal qualities of a vibrating string. Really, everything contributes to some extent.
Also, note that an analysis of standing waves only works for steady-state and does not consider the transients when the vibration is initiated - when the string is plucked.