The loudness of a sound is typically related to the amplitude of vibration. To increase the loudness of a sound by a certain factor, the amplitude of vibration needs to be changed accordingly.
In general, the relationship between the loudness of a sound and its amplitude is not linear but logarithmic. The perception of loudness follows a logarithmic scale known as the decibel (dB) scale. A doubling of sound intensity corresponds to an increase of approximately 3 dB.
To increase the loudness of a sound by four times, we need to consider the sound intensity, which is proportional to the square of the amplitude. Since the relationship is logarithmic, doubling the intensity corresponds to an increase of 3 dB. Therefore, to increase the loudness by four times, we would need to increase the sound intensity by 6 dB.
Now, let's consider the amplitude. Since the sound intensity is proportional to the square of the amplitude, increasing the intensity by 6 dB corresponds to increasing the amplitude by a factor of √10 (approximately 3.16).
So, to increase the loudness of a sound by four times, the amplitude of vibration should be changed approximately 3.16 times.