Thermodynamics: deriving the quantum volume

[iScience]

so we used an equation in class to basically describe the relationship between some quantities of a vibrating string held fixed at both ends. and i noticed (just now...) that this was.. well.. i don't want to say that it's 'wrong' yet but, the expression does not make sense to me; and just to make sure that i haven't gone dumb or anything i checked online for the expression of the same scenario and they had the same equation that i derived.

here is a picture of the page in my thermo book using the equation (circled in green) that i disagree with.

http://i.imgur.com/gcBrguW.jpg

even at the bottom of the page with the diagram, λ₃ is NOT (2/3)L it should be 1.5L or, (3/2)L. basically my question is... why in the world is the 2 and the n inverted??


thanks

 

[256bits]

Are you asking about how many wavelengths fit in the box, or how long is the wavelength?

 

[jtbell]

 

[iScience]

either way, the wavelength is in terms of L, so wouldn't the answer to one be the answer to the other? for instance, the very top wave in the diagram in the picture, n=3. there are 1.5L waves in this harmonic, or.. there are 1.5 wavelengths in the length L. (this answers "how many wavelengths fit in the box")

as far as how long the wavelength is.. that can't really be determined numerically if that's what you're asking, since we are leaving the wavelength in terms of L so.. however many wave units in length L IS 'how long the wave is'.

so.. either way, i don't see how the equation circled in green is correct.

 

[iScience]

jtbell: how are you getting the 2/3 ??.. it should be 3/2

http://www.physicsclassroom.com/class/waves/u10l4e.cfm

 

[256bits]

If one and a half wavelengths fit in a box of length L, then the wavelength is 2/3 the length L of the box.

For λ1, half a wavelength fits in the box, and the wavelength is twice the box length.

 

[jtbell]

http://www.physicsclassroom.com/class/waves/u10l4e.cfm

That page has $L = \frac{3}{2} \lambda$ which is equivalent to $\lambda = \frac {2}{3} L$.

 

[Philip Wood]

n lots of \frac{\lambda}{2} fit into length L.
So \frac{n \lambda}{2} = L.
Re-arrange this to get the equation you've circled in green.