Question

The diagram shows three different modes of vibration P, Q and R of the same string. Which vibration will produce a louder sound and why?

Answer 1

Loudness depends on the square of the amplitude of the wave.

That is, loudness $\propto$ $A^2$
Amplitude is the size of the vibration, and this determines how loud the sound is. Larger vibrations make a louder sound.

Hence, wave R produces louder sound because its amplitude is maximum.