How can a sine wave represent a longitudinal wave? I envision a longitudinal wave as a series of vertical lines like that drawn on the board in an introductory physics class.  This image contains no angles.  Sound is a longitudinal wave.
Some audiovisual references claim that sound is composed of sine waves.
I am trying to reconcile how longitudinal waves (no angles) can be composed of sine waves (dependent on angles).  What maths or theory explains the relationship?
 A: The "lines" are areas where the fluid is compressed or expanded.  Plot the position of any one of the particles in that fluid, and you'll see that it moves in a sine wave back and forth over time.

See 


*

*Longitudinal and Transverse Wave Motion from Dan Russell's Acoustics and Vibration Animations

*This Flash animation
A: Just because a sine wave looks like a side-to-side wave when you plot it, it doesn't mean that  anything is actually oscillating sideways.
In general, a wave is just a pattern in some physical quantity that propagates through space. The physical quantity could be something visible, like the transverse displacement of a string or the longitudinal compression of a slinky, or it could be something invisible, like a force or the electromagnetic field. It can be something which has a direction, like the examples in the previous sentence, or it can be something which has no direction, like pressure or density. It depends on the medium and the type of wave.
However, we can mathematically describe waves in a way that is independent of the medium by expressing the physical quantity that is "waving" as a function $Q(x,t)$ of position and time. Regardless of which medium or which physical quantity is involved, the function will satisfy the wave equation,
$$\frac{\partial^2 Q}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 Q}{\partial t^2}$$
In a sound wave, $Q$ could be pressure or density or longitudinal displacement. In a light wave, $Q$ can be any of the components of the electromagnetic field, or any of the components of the EM potential. And so on.
Now, if you're familiar with Fourier analysis, you'll know that any periodic function can be expressed as a sum of (co)sine waves, or equivalently of complex exponentials
$$f(t) = \sum_{n = 0}^{\infty}F_s(n\omega_0)\sin(n\omega_0 t) + F_c(n\omega_0)\cos(n\omega_0 t) = \sum_{n = -\infty}^{\infty}F(n\omega_0)e^{-in\omega_0 t}$$
for some value of $\omega_0$. And this generalizes to non-periodic functions as the Fourier transform,
$$f(t) = \frac{1}{\sqrt{2\pi}}\int F(\omega)e^{i\omega t}\mathrm{d}\omega$$
This is particularly convenient for analyzing waves because it's very easy to analyze what happens when $Q(x,t)$ takes the form of a sine wave. And once you know how sine waves behave, you can use the Fourier transform to add them up to reconstruct the behavior of any general wave.
There is another reason we like sine waves, namely resonance. As waves (sound, light, etc.) travel through space they are eventually going to run into objects and interact with them. In many cases, these objects are sensitive only to certain frequencies, and they will "pick out" only those particular frequencies from the incoming wave, letting the rest pass through untouched. For example, this is how absorption spectra are produced in stars (with light waves, of course). If you imagine the wave as being constructed by adding together sine waves of different frequencies, in the sense of the Fourier transform, then it's easy to understand how an object can pick out only certain frequencies of the wave, and easy to analyze what the effect of any given physical system is on a wave.
A: Let's get simple.
Hang a weight from a string, over the edge of a table, and set it swinging.
As a function of time, it is making a sine wave, right? (Roughly speaking).
Now, set it swinging not just side to side, but in a circle. Same idea.
Now, imagine a wheel underneath it, turning at the same rate as the weight is swinging in a circle, so a point on the edge of the wheel follows the swinging weight.
Let the wheel have a radial line marked on it going from the center to the point on the edge.
That line, compared to due North, say, is making an angle which is rotating at the same rate as the wheel and as the swinging weight.
The East-West position of that point on the wheel is swinging back and forth in a sine wave, for which the angle at any moment is the angle between the line and due North.
That's how anything that moves with a sine wave has an angle.
It is the constantaly turning angle of the underlying, possibly imaginary, rotary motion that the sine wave represents.
The difference between the longitudinal wave and the swinging weight is just that the longitudinal wave consists of a lot of molecules of air (or whatever) that are all swinging back and forth, according to all their little wavy laws.
