A paper popper like the one shown in this YouTube video makes a loud sound when moved quick through the air.The air pressure inside the 2 conical structures in the popper would be higher when this popper is moved fast. But how exactly is this difference in pressure converted to sound? Will having different paper types produce different frequencies?

# [Physics] What makes a origami paper popper produce loud sound

acousticsshock-waves

#### Related Solutions

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.

The question has more degrees of complexity.

**The first approximation** could be the model of many points of mass connected by springs. In this case the point of mass returns to its equilibrium position as the wave passes, cause it's the state of minimal energy. The derivation of wave equation is simply a proof, that the air molecules are governed by such a "spring forces" as well.

**The more detailed answer** could be very broad, too broad for just one question. Let me just show some cases:

If we consider viscosity as well than there are cases (especially in boundary layers) where the wave is polarized and particles are "orbiting" around the equilibrium state.

In many cases there are complex motion of the air convection and local oscillation (wind instruments, sound waves in tubes with flow...). These cases are highly nonlinear and I would suggest you further reading before detailed question will be formulated. Start with the Aeroacoustics in wikipedia.

## Best Answer

Whilst physLads answer is correct, it is perhaps too succinct to be helpful with those unversed in gas dynamics, so I will attempt to explain what a 'blast wave' is in this case.

This much is correct. However, as the flap of paper moves away, the high pressure gas pocket that had built up must rush in after it, pealing away layer by layer in a 'rarefaction' that means that, as the paper is moving it is followed by a moderately low pressure, but crucially very fast moving air. The paper is then brought to a very sudden halt (in the frame of reference of the persons hand), but still has a mass of fast moving air piling in towards it. This creates a 'blast wave', in this case a region of stationary high pressure air between the paper the region of fast moving low pressure air. The blast wave is the discontinuity in velocity and pressure between these two regions of air. As more air piles in across the blast wave the region of high pressure air adjacent to the paper increases in size, and the blast wave moves outwards through the air. This 'blast wave' or 'pressure discontinuity' is the loud pop heard.

Edit: you state in comments that you believe the sound to be produced by the region above the paper popper as the paper accelerates into it. It is true that this will also create a blast wave. However the acceleration of the paper is much slower than its deceleration, thus this wave will be smaller. In addition, the rarefaction caused when the paper halts will reduce the size of it yet further. Recordings of a paper popper being used may contain an ear splitting pop followed or preceeded by a much quieter one, the quieter one will be produced by this method.