Final answer:
The standing wave on a guitar string and the sound wave in the air both oscillate with the same frequency, but they can have different wavelengths due to differences in propagation speed. They are linked by the oscillation of the guitar string, which generates the sound wave through the vibration of air molecules.
Step-by-step explanation:
Both the standing wave on a guitar string and the sound wave that propagates through the air share common characteristics. First, they both involve oscillations: the guitar string vibrates up and down, while the surrounding air molecules vibrate back and forth. The frequencies of these two waves are also the same, meaning that if a string vibrates at a certain frequency, the generated sound wave in the air will oscillate at that same frequency. However, the wavelengths of the two waves can be different since the speed of sound in air and the speed of the wave on the string differ.
Standing waves on a guitar string are formed by the reflection of waves from the fixed ends of the string. At the ends, nodes are created where there is no movement (the wave disturbance is zero), while antinodes occur at points of maximum amplitude. Because of the boundary conditions, only waves with an integer number of half-wavelengths can exist, leading to discrete resonant frequencies. The physical length of the string and the tension determine the resonant frequencies and wavelengths of these standing waves.