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When making the transition from the dispersion relation for a beaded string to the relation for a continuous string, I'm confused about the following issue. Take a to be the spacing between beads, m the mass of each bead, and T the tension in the string. We assume these to be constant.

For the beaded string, we have that [itex]\omega_{n}=2 \omega_{o} \sin(\frac{k_{n} a}{2}).[/itex] Clearly, the value of [itex] \omega_{n} [/itex] can never exceed [itex]2 \omega_{o} [/itex].

When we transition to a continuous string, by taking the limit as [itex] a \rightarrow 0 [/itex], we end up seeing that [itex] \omega_{n} = \frac{n \pi}{L} \sqrt{\frac{T}{\mu}} [/itex]. But if we let [itex] n [/itex] get very, very large, this should be able to surpass [itex]2 \omega_{o} [/itex], assuming we allow n to become large enough. So why is it that if we had a billion (or trillion or googol...) masses on a string there would be no way for the frequency of the highest-frequency normal mode to exceed a certain value, but as soon as we take the continuum limit, the frequencies can no get arbitrarily large?

For the beaded string, we have that [itex]\omega_{n}=2 \omega_{o} \sin(\frac{k_{n} a}{2}).[/itex] Clearly, the value of [itex] \omega_{n} [/itex] can never exceed [itex]2 \omega_{o} [/itex].

When we transition to a continuous string, by taking the limit as [itex] a \rightarrow 0 [/itex], we end up seeing that [itex] \omega_{n} = \frac{n \pi}{L} \sqrt{\frac{T}{\mu}} [/itex]. But if we let [itex] n [/itex] get very, very large, this should be able to surpass [itex]2 \omega_{o} [/itex], assuming we allow n to become large enough. So why is it that if we had a billion (or trillion or googol...) masses on a string there would be no way for the frequency of the highest-frequency normal mode to exceed a certain value, but as soon as we take the continuum limit, the frequencies can no get arbitrarily large?

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