The intensity of sound is proportional to the square of the amplitude because sound energy is directly related to the square of the amplitude of a sound wave. This relationship arises from the nature of wave propagation and the energy carried by a wave.
When a sound wave travels through a medium, such as air, it causes variations in air pressure. These pressure variations create compressions and rarefactions that propagate as the wave moves through space. The amplitude of a sound wave corresponds to the maximum displacement or variation in air pressure caused by the wave.
The intensity of a sound wave is a measure of the power or energy carried by the wave per unit area. Mathematically, the intensity (I) of a sound wave is defined as the power (P) transmitted by the wave per unit area (A) perpendicular to the direction of wave propagation. Thus, I = P/A.
Now, considering the power (P) of a sound wave, it is directly related to the energy carried by the wave. The energy is determined by the amplitude of the wave. However, since power is also related to time, it is necessary to consider the rate at which energy is transferred.
The rate at which energy is transferred by a sound wave is proportional to the square of the amplitude. When the amplitude of a sound wave is doubled, the energy carried by each cycle of the wave is quadrupled because the energy is proportional to the square of the amplitude. Similarly, if the amplitude is tripled, the energy is nine times greater.
Therefore, when calculating the intensity of a sound wave, which represents the power per unit area, the square of the amplitude is used because the energy carried by the wave is directly related to the square of the amplitude. Consequently, the intensity of sound is proportional to the square of the amplitude.