# Airbox Calculations

It is possible to modify or design an airbox for improved airflow while also keeping your intake quiet. Aside from removing restrictive internal snorkels, you can also enlarge the box or fit it with a different size inlet. Enlarging the box can usually be done with no ill effects, but successfully changing the inlet will require some complicated calculations. Remember any modification will require you to upjet.

## Airbox Volume

The larger the box, the better the noise attenuation AND the better the flow, to a point.

In order to avoid choking the engine, an airbox should be at least 10x your Engine displacement, but doesn't need to be 20x more. Stock airboxes will sometimes be less than 10x displacement for aesthetic or restriction reasons. So, for 50cc, you need an airbox between 0.5 - 1 litre; and for 75cc, 0.75 - 1.5 litre. The stock A55 airbox, for example, is only about 0.5 L: the minimum size necessary for a 50cc but inadequate for running kits.

## Inlet length and diameter

The longer the inlet, the better the noise attenuation BUT the more restricted the flow. The wider the inlet, better the flow BUT the noisier the box.

To calculate the length and diameter of the pipe drawing air in from the outside it gets complicated. Airboxes work as "low-pass" noise filters, they cancel out sound waves above a specific frequency. Your intake creates it's largest pulse of sound once per revolution, other noises twice or three or four times depending on the engine.

To translate this into frequency, divide your RPM by 60 to get the lowest note. For example, 6000 RPM / 60 s/m = 100 Hz. To get an idea of what that sounds like, have a listen on Wikimedia: https://commons.wikimedia.org/wiki/Sine_wave_tones#10Hz_.E2.88.92_100Hz

Once you know how low you want your box to resonate, we can take a look at an equation for Heimholtz resonators:

**f = a/2π * √(A/((L+πD/4)*V)**

where f is your target frequency, a is acoustic velocity (343 m/s @ 20ºC), π is pi 3.141592654, √ is the square root, A L D are the area, length, and diameter of the inlet pipe, and V is the airbox volume.

## A Simple Example

To familiarise ourselves with this equation, we can without any algebra use it to estimate what frequency a stock airbox was designed to cancel. Let's say we have a 500cc airbox with a 12mm diameter inlet that is 5cm long. In terms of meters, that is:

pipe area: 0.0001131 m2, pipe length of 0.05 m, pipe diameter 0.012m, & box volume 0.0005 m3.

Placing those numbers into the equation, we get

343m/s / 2π * √( .0001131m2 /(( .05m + π * .012m / 4 ) * .0005m3 ))

Which comes out to 73 Hz. This airbox would deaden the primary intake pulses above 4380 RPM, and may have some effect on minor noise above 2190 RPM.

## Cooking Up a New Box

Now, let's say we want to silence your performace bike at 6000 RPM and above (6000 / 60 = 100 Hz), using a 1-litre (0.001 m3) shampoo bottle and a length of 3/4" PVC pipe. The inner diameter of the PVC pipe is 20.9mm (0.0209m), the area is 343 mm2 (.000343 m2).

100 Hz = 343m/s / 2π * √( .000343m2 /(( L + π * .0209m / 4 ) * .001m3 ))

100 = 50.59 * √( .343 /( L + .01641 ))

1.977 = √( .343 /( L + .01641 ))

3.909 = .343 / ( L + .01641)

L + .01641 = .343 / 3.909

L + .01641 = .08775

L = .07134 m

If I've done my maths properly you'd want your pipe to be 7.1 cm long. Oh but my bike makes plenty noise already at 3000 RPM and I want to be super stealthy, you say. Well if you're bothered by those super low notes below 100Hz you could TRY:

50 Hz = 343m/s / 2π * √( .000343m2 /(( L + π * .0209m / 4 ) * .001m3 ))

50 = 50.59 * √( .343 /( L + .01641 ))

.9883 = √( .343 /( L + .01641 ))

.9767 = .343 / ( L + .01641)

L + .01641 = .343 / .9767

L + .01641 = .3512

L = .3348 m

Yikes! Halving the frequency resulted in a crazy 33 cm long inlet! Let's try a 1/2" PVC pipe, which is 16.81 mm wide and 222.1 mm2 in area.

50 Hz = 343m/s / 2π * √( .0002221m2 /(( L + π * .01681m / 4 ) * .001m3 ))

50 = 50.59 * √( .2221 /( L + .0132 ))

.9883 = √( .2221 /( L + .0132 ))

.9767 = .2221 / ( L + .0132)

L + .0132 = .2221 / .9767

L + .0132 = .2274

L = .02142 m

That helped, but 21 cm is still quite long. Maybe we can alleviate that with a huge Costco-size shampoo bottle of 1.5L:

50 Hz = 343m/s / 2π * √( .0002221m2 /(( L + π * .01681m / 4 ) * .0015m3 ))
50 = 50.59 * √( .1481 /( L + .0132 ))

.9883 = √( .1481 /( L + .0132 ))

.9767 = .1481 / ( L + .0132)

L + .0132 = .1481 / .9767

L + .0132 = .1516

L = .01384 m

a slightly more reasonable, but still quite long 14 cm.