Loudness of sound is proportional to the ______ of the amplitude of the vibration producing the sound.

Square

The correct answer is option 3. Square.

The loudness of a sound is related to the amplitude of the vibrations that produce it. When an object vibrates, it creates pressure waves in the surrounding medium, such as air. The greater the amplitude of these vibrations, the more energy is transferred to the medium, resulting in a louder sound.

Mathematically, the relationship between the loudness of a sound and the amplitude of its vibrations is not linear but rather follows a quadratic relationship. This means that if the amplitude of the vibration doubles, the loudness of the sound does not simply double; instead, it quadruples. This relationship is expressed by the square of the amplitude, which is why the loudness of sound is proportional to the square of the amplitude of the vibration producing it.