The range of energies of sound waves

Question: The amplitude of vibration of your eardrum at the threshold of pain is about 10⁶ times greater than the amplitude with which it vibrates in response to the softest sound you can hear. How many times greater is the energy with which your ear has to cope for the painfully loud sound, compared to the soft sound?

Solution: The amplitude is 10⁶ times greater, and energy is proportional to the square of the amplitude, so the energy is greater by a factor of 10¹². This is a phenomenally large factor!