Waes: normal mode frequencies for 1 fixed extremity

1. The problem statement, all variables and given/known data

A String of length L has one of its extremities fixed and the other one loose.

A. What's the equation for the normal mode frequencies?
B. Draw a snapshot of the string for the 1st 3 normal modes

2. Relevant equations
wave equation

3. The attempt at a solution

My idea was to follow the same line of thinking for a string with both extremities fixed. Then we can assume that

y(x, t) = g(x).cos(wt + d) --(1)

and (1) must be solution of the wave equation, and after some math we get the general solution

A(x)'' + A.k^2 = 0, k = w/v --(2)

we know that A(0) = 0, since the x=0 extremity is fixed and the general solution for 2 is

A(x) = b.cos(kx) + c.sin(kx)

so, b = 0 and A(x) = c.sin(kx)

since A(x) != 0, we know that c != 0, but there's no other known condition in order to compute possible values for k

any ideas? and.. tbh i dunno if i can assume everything i did since 1 of the extremities is loose. help pls! :D

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