# What is the relationship between the pressure inside of a balloon and the decibels at which it pops?

Looking for this on the web I was provided with little information, only that as the pressure increases so does tension, and if there is a higher tension it results in a louder pop, how may that be expressed in the form of an equation?

• Check out thin shell membrane sress. – user207455 Jun 17 at 8:05

It is true that tension increases when you inflate a balloon, but the pressure has a peak value. Once the pressure reaches the peak value there is a sudden expansion of the balloon with a decrease of the internal pressure (see reference 1), this is due to the nonlinear behavior of the material.

To obtain a mathematical expression let us assume the following:

• The material behaves as a neo-Hookean hyperelastic material and it behaves nearly incompressible.
• The shape of the balloon is spherical for the whole inflation process. This is not true but it simplifies things, so we can do it by hand.
• It is inflated slowly, so we can neglect inertial effects.

The radius ($$r$$) and thickness ($$t$$) of the balloon at any instant are given by

$$r = \lambda r_0\, ,\quad t = \frac{t_0}{\lambda^2}\, ,$$

where $$r_0$$ is the initial radius, and $$t_0$$ the initial thickness. $$\lambda$$ is termed the stretch, and the thickness should decrease quadratically because of the incompressibility.

Since we have a neo-Hookean material we can say that the tension is given by

$$\sigma_{11} = \sigma_{22} \equiv T = 2\alpha\left(\lambda^2 - \frac{1}{\lambda^4}\right)\, ,$$

and $$\alpha$$ is a material constant measuring how difficult is to inflate the balloon.

Equilibrium between the tension (membrane stress) and the pressure gives

$$P = \frac{2 T t}{r}\, ,$$

and, putting it all together,

$$P = 2\alpha\left(\frac{1}{\lambda} - \frac{1}{\lambda^7}\right) \frac{2t_0}{r_0}\, .$$

And you end up with the behavior shown in the following figure.

Now, we need to know the energy stored to estimate how loud it will sound when popped

$$E = PV = P \frac{4}{3}\pi r^3 = 4\alpha\left(\lambda^2 - \frac{1}{\lambda^4}\right) t_0 r_0^2\, .$$

This expression can be rewritten in terms of the mass of the balloon and density of the material, as well.

### References

1. Jame F. Doyle and C.T. Sun (2007). Theory of Elasticity, Purdue University.