If this isn't the right place for this question, please tell me. My question is how does a person's voice affect how a soundwave looks? For example, if I say the letter "a" it would sound different than if you said the letter "a", but my question is if there is a pattern to this, or maybe if we are able to derive a waveform based on voice (however using more factors than just x and t, due to the difference in a wave based on what you say).
The sound patterns in speech are not easily describable via a wave equation. An acoustic unit of speeh is called a "phoneme". An easy way is to represent each phoneme as a "voiceprint", which is a plot of the time evolution of the frequency components of the phoneme. The meaningful aspect of a voiceprint is the arrangement of the frequency components, rather than the frequency components per se.
It is possible to produce understandable speech by injecting white noise into the mouth, then shaping words with the mouth. The mouth, tongue, soft palate and lips act as a resonant cavity and filter the sound spectrum to create phonemes.
Expanding on the answer given by S. McGraw:
The most common form of introducing white noise in the vocal tract is to whisper. The vocal folds are prevented from vibrating, but they are positioned in such a way that significant turbulance is created. The resulting sound is sufficiently close to white noise to support recognizable speech.
Tongue, mouth and soft palate are reshaped to create a wealth of possible forms of the resonant cavity. For each differnt vowel we adjust the resonant properties of the vocal tract.
Sense of hearing is extraordinarily rich. Let me compare with eyesight.
A technological device that can reproduce color vision needs to accomodate 5 degrees of freedom. The colors of human color vision can to a good approximation be represented with three degrees of freedom (as in the three primary colors of additive color mixing). Images are planar: another two degrees of freedom.
Every sound is a superposition of many frequencies. You can play a note with a particular pitch on many, many different instruments, and an expert familiar with each of those instrument can discern the difference and name each instrument. A synthesizer with a limited number of degrees of freedom kan produce a rough approximation of an instrument, but our hearing can tell the differnce. I guestimate that the information density that our hearing can extract from sound corresponds to many dozens, possibly hundreds of degrees of freedom.
The sound produced by our voice is so rich, and our ability to discern nuances in sound is so refined that we often can tell that two persons are siblings just from the similarity of how they sound. (Just as facial features tend to be similar siblings tend to have similarly shaped vocal tract, giving rise to similarity in the sound of their voice.) That is, human voice is so rich in information that speaking doesn't take up all of the information carrying potential.)
If you would want to seek a spatial representation of sound then you need many dozens of spatial dimensions.
As an indication of the amount of information in sound: both video compression technology and audio compression technology are based on figuring out how much of the information you can discard and still be able to reconstruct the signal to a form where the only the most expert viewer/listener can tell the difference. Under those circumstances the amount of required bandwith for the compressed video and the compressed audio is about the same. You might expect that the video component will be much larger, but that is not the case, and that is because of the quality of our hearing.