Free vibrations of stretched strings

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The discussion focuses on a physics problem involving a stretched string with mass m, length L, and tension T, driven by two out-of-phase sources at each end. The smallest normal mode frequency of the string is given by the formula ω=π(T/LM)^(1/2). Participants are analyzing the wave functions for the string, represented as y1(x,t) and y2(x,t), and discussing the differentiation process with respect to time and space. There is a question about whether the approach taken is correct for solving the problem. The conversation emphasizes understanding the fundamentals of wave behavior in stretched strings.
kaamos
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A stretched string of mass m, length L, and tension T is driven by two
sources, one at each end. The sources both have the same frequency  and
amplitude A, but are exactly 180 degrees out of phase with respect to one another.
(Each end is an antinode). What is the smallest normal mode frequency of the
string?

Solution: ω=π(T/LM)^(1/2)

Attempt: y1(x,t)=f(x)cosωt and y2(x,t)=f(x)cosωt
Then differentiating both of them w.r.t t and x. Am I even on the right track??
 
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What's your thinking behind your attempt?

By the way, I'm guessing this problem is from an intro course, so I've moved it to the introductory physics forum.
 

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