# 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?

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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.**

**Sources:**