Final answer:
In standing waves on a string, the wave velocity remains the same across different harmonics due to the medium's static properties such as mass per unit length and tension, not the waves' frequency or wavelength.
Step-by-step explanation:
The relationship between wave velocity, wavelength, and frequency is fundamental to understanding wave behavior. In standing waves, such as those seen in a vibrating string fixed at both ends, the wave velocity remains consistent across different harmonics. This constant velocity is because the wave's velocity is dependent on the static properties of the medium, like the mass per unit length and the tension of the string, and not on the frequency or wavelength. Therefore, despite the harmonic containing different wavelengths and frequencies, the velocity of each wave remains the same.
In standing waves, only waves that match the boundary conditions, such as having an integer number of half-wavelengths fitting within the string, can form. These standing wave patterns consist of nodes where there is no motion and antinodes where the motion is greatest. The energy of the standing waves increases with the number of half-wavelengths—or nodes—present in the string at a given amplitude.