I think that your difficulty arises from the fact that you are trying to think about this in terms of a static problem, whereas any "extreme" properties of metamaterials result from the essentially dynamic phenomena. I.e., you cannot really have a material with a negative mass or a negative module (although, admittedly, there are some ingenious mechanisms that engineers can construct for you to achieve the semblance of either).
The extreme properties of metamaterials are usually manifested in the vicinity of inclusion resonances. Suppose that you have a doubly-periodic composite. You can take an external excitation (think time-harmonic for the sake of simplicity) and tune it to excite a mode in one of the corners of the Brillouin zone. This would result in a standing wave, with all inclusions oscillating, some perfectly in phase, some perfectly out of phase. If you now de-tune your excitation a little bit, you will obtain a strong beating motion, which can have an extremely long characteristic wavelength. If you are interested in modelling the envelope of this beat, then you will find that its “effective properties” can often be described by the models with very unusual properties, such as the ones you described in your question. However, if the entire motion is considered, your mass remains reassuringly positive (and constant) and your modulae satisfy all of the usual constitutive requirements.
All interesting properties of (not necessarily acoustic) metamaterials are just interference phenomena. This situation is a bit similar to the famous experiments on superluminal light propagation through the strongly absorbing media: http://www.nature.com/nature/journal/v406/n6793/abs/406277a0.html
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.
Best Answer
The air pressure inside the (intact) bubble is larger than in the surrounding. This pressure difference is called Laplace pressure and is caused by the surface tension between the soap film and the air. When the bubble pops the compressed air expands, thus creating a pressure wave, which you ultimately hear as the typical popping sound.