A string, stretched between two fixed posts, forms standing wave resonances at 420 Hz and 490 Hz. What is the greatest possible value of its fundamental frequency?
Lets assume that this resonances appear as n and p harmonic in the string where n>p
We know that a fixed string like this have the wave length `lambda` as following depending on the harmonics.
`lambda` = 2/n(L) where L is the length of the string and n is the number of harmony.
So for n th harmony velocity = 420*2/n(L)
For p th harmony velocity = 490*2/p(L)
Since both locations we have same velocity in string;
420*2/n(L) = 490*2/p(L)
p/n = 490/420=7/6
So n=6 and p=7
In fundamental frequency;
lambda = 2L
So velocity = `f_0` *2L
By comparing with one of the harmonics previously found we get;
`f_0` *2L = 490*2/p*L
`f_0` = 70 Hz
So the fundamental frequency is 70 Hz.