Why does decreasing the tension on a string causes the numbers of half-wavelengths to increase?

Answer 1

Per the ref., the wave velocity along a taut wire or string
v = sqrt(T/(m/L)), where T is tension and m/L is the mass of the string per unit length. So if you decrease T, v decreases also. This means that for a given frequency f, wavelength v/f becomes shorter, so the number of half-wavelengths in a given length of string increases.