Standing Waves (tension vs. wavelength)

  • #1

Homework Statement



I'm trying to decode a graph of Tension vs. [tex]\lambda^2[/tex] of standing waves on a string to understand the actual meaning of the slope. I also need to derive the equation for the line from two expressions:

[tex]v= \sqrt{T/\mu} [/tex]
[tex] v= f\lambda [/tex]

Homework Equations



[tex] \lambda=2L/n [/tex]
[tex] \mu = m/L [/tex]

The Attempt at a Solution



I'm starting to think I'm completely missing something in my initial approach.

[tex] v = \sqrt {T/\mu} [/tex]

[tex] v^2 = T/\mu [/tex]

[tex]v = f\lambda[/tex]

[tex] v^2 = (f\lambda)^2[/tex]

[tex]T/\mu = f^2\lambda^2 [/tex]

[tex]T = \mu f^2\lambda^2 [/tex]

[tex]T = \frac {\mu f^2 4L^2}{n^2} [/tex]

I feel like it should lead to a clearer solution than this one, but I'm not sure what. I first thought that the slope was simply a surface tension (dynes/cm^2) on the string but I don't see how that relates to the equation or slope I'm trying to find.

I also tried this:

[tex] f = \frac {v}{\lambda} [/tex]

[tex] f = \frac {\sqrt {T/\mu}}{\lambda} [/tex]

[tex] f^2 = \frac {T/\mu}{\lambda^2} [/tex]

[tex] \mu f^2 = \frac {T}{\lambda^2} [/tex]

which seems closer, but I don't know how to apply it in a usable manner. Any help would be appreciated.
 

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