Having some free time, I've begun a mini-project attempting to understand how music works in more detail, which has so far only involved music theory and a bit of set theory. I'm now trying to get my head around the physics of sound waves, and in particular, I'm trying to understand how any sound can be analysed as a sum of sin waves. I'm studying for a mathematics degree, so I'm familiar with Fourier transforms, trigonometric functions etc., but I've never really needed to apply them to sound waves in any detail. In this question I'm not really asking how the mathematics of Fourier transforms work, I'm more concerned with the specifics of how a sine (or cosine) function can model a sound wave. Apologies in advance for the long question, it was originally almost twice as long but I've condensed it as much as I could.
I know that a sound wave is an longitudinal oscillation of particles in a medium, and that a wave might be generated by something moving back and forth repeatedly such as a guitar string, or a piece of cardboard inside a speaker. I've had a bit of trouble understanding how these oscillations are passed on, so as context to my questions I'm going to try and explain my current understanding of it (and hopefully this will make it easier to highlight where I've gone wrong).
My conceptual understanding so far: (Having read this question) I'm imagining the first oscillation of a speaker membrane (i.e. a piece of cardboard) in air, but it could be a vocal chord, a guitar string etc. The membrane imparts a certain initial velocity to the particles it hits, but this velocity is quickly absorbed by the surrounding particles of air due to inefficient collisions occurring at angles different to the direction of wave propagation. The nearby particles are pushed very close together, and due to the nature of gases/liquids, spread out, but this causes a gathering of particles somewhere further down the line, and the chain continues until the movement of particles reaches my ear. When the speaker membrane moves backwards, an area of space is freed up for the particles to move into; the particles begin to move in the reverse direction. Since this movement is not at all governed by the initial velocity imparted to the particles, the speed of sound is constant (within a given medium).
If what I have said above is correct, then (as far as I can tell) to define the wavelength $\lambda$ we have to stop time for a moment, then measure the distance between two points where the particles are maximally dense. The frequency $f$ of the speaker membrane is defined to be how many times it oscillates back and forth per unit time, so that if the time taken for one oscillation is $T$, we have $f=\frac{1}{T}$. Clearly the speed of the wave will be $v=\frac{\lambda}{T}=f\lambda$. However we know $v$ is a constant, so increasing the frequency of the speaker membrane means decreasing the wavelength of the wave. Every time the membrane oscillates it sends another chain of dense particles on its way, so that the frequency of the membrane is equal to that of the wave.
Question 1: Does there exist a velocity such that the membrane moves too quickly for the air particles? In order for the above model to work, the air must fill the gap left by the speaker membrane as it oscillates almost instantaneously, else the membrane will have no particles to move on its return journey. The speed of sound in air is $343$ $m/s$. If the membrane moves at say twice this speed, then it will have returned before the air particles have returned to their original positions. Obviously this puts no limit on the frequency, as you can make the velocity of the membrane arbitrarily small by reducing the distance it has to travel, but am I right in saying that this will occur?
Question 2: In which cases can we model sound with a sine function, and why? This is my main problem; I don't understand how a sine function corresponds to a sound wave in reality. Suppose $f(x)=\sin(x)$ where $x$ is the axis of wave propagation. Does $\sin(x)$ (the amplitude) represent the displacement of the speaker membrane from its equilibrium position, or does it represent air pressure (for a given moment)? I'm not quite sure how we even know that air pressure varies in accordance with a sinusoid, I can't seem to find an explanation anywhere. Having googled this quite a lot, I find most courses either just state that the sinusoid will represent air pressure or some other quantity, or worse still they neglect to comment and don't label the axes etc. Also, the function $f(x)=\sin(x)$ is not time dependent, and only represents variation in one axis. If this is the case, how can it fully model the propagation of a wave in space. When you look at wave forms on programs like audacity, they only graph time and amplitude, so a simple pure tone would be a function like $A(t)=\sin(t)$. This function is completely position independent, so I'm assuming it is created by detecting pressure variations at a point and graphing it. It doesn't fully describe the wave though! So I'm wondering what kind of function will represent a pure tone in three dimensions?
Answer
To expand on Xcheckr's answer:
The full equation for a single-frequency traveling wave is $$f(x,t) = A \sin(2\pi ft - \frac{2\pi}{\lambda}x).$$ where $f$ is the frequency, $t$ is time, $\lambda$ is the wavelength, $A$ is the amplitude, and $x$ is position. This is often written as $$f(x,t) = A \sin(\omega t - kx)$$ with $\omega = 2\pi f$ and $k = \frac{2\pi}{\lambda}$. If you look at a single point in space (hold $x$ constant), you see that the signal oscillates up and down in time. If you freeze time, (hold $t$ constant), you see the signal oscillates up and down as you move along it in space. If you pick a point on the wave and follow it as time goes forward (hold $f$ constant and let $t$ increase), you have to move in the positive $x$ direction to keep up with the point on the wave.
This only describes a wave of a single frequency. In general, anything of the form $$f(x,t) = w(\omega t - kx),$$ where $w$ is any function, describes a traveling wave.
Sinusoids turn up very often because the vibrating sources of the disturbances that give rise to sound waves are often well-described by $$\frac{\partial^2 s}{\partial t^2} = -a^2 s.$$ In this case, $s$ is the distance from some equilibrium position and $a$ is some constant. This describes the motion of a mass on a spring, which is a good model for guitar strings, speaker cones, drum membranes, saxophone reeds, vocal cords, and on and on. The general solution to that equation is $$s(t) = A\cos(a t) + B\sin(a t).$$ In this equation, one can see that $a$ is the frequency $\omega$ in the traveling wave equations by setting $x$ to a constant value (since the source isn't moving (unless you want to consider Doppler effects)).
For objects more complicated than a mass on a spring, there are multiple $a$ values, so that object can vibrate at multiple frequencies at the same time (think harmonics on a guitar). Figuring out the contributions of each of these frequencies is the purpose of a Fourier transform.
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