Explain why there are missing harmonics for standing waves on a string with a loose (free) end. (Hint: Is there a pattern to how much of a wavelength fits between the ends of the string with a loose end

It is because there are a odd number of wavelengths fitting onto the string

Explanation:

The general formula for the harmonics on a string loose at one end is f = nv/2L where n = odd integer, v = speed of wave in string and L = length of string.

For a string loose at one end, there are a odd number of wavelengths which can fit onto the string. Since this is the case, we only have harmonics of odd numbers and thus we have even number harmonics missing.