To which natural frequency does wavelength equal to L1

[jorgegalvan93]

Homework Statement

The length L1, is not the wavelength of the fundamental frequency of the string.
With the tension equal to F1, to which natural frequency does the wavelength equal to L1 correspond?

Homework Equations

I was reading online, and found that when a string vibrates at fundamental frequency; that is,
f1 = 1/2L√(F/μ), the standing wave has a wavelength λ1 equal to twice the length of the string …
2L = λ… L = λ/2…
And that at each higher harmonic, and additional 1/2 of its wavelength is added onto the string. Can someone please clear this up for me?
How does a standing wave have a wavelength equal to twice the length of the string?
And why do you add 1/2 for each successive harmonic?

The Attempt at a Solution

Anyways, my attempt at the solution was this…
L1 = (1/2)λ1 L1 = (2/2)λ2… L1 = λ2
So L1 corresponds to the second natural frequency, or second harmonic.

 

[Simon Bridge]

How does a standing wave have a wavelength equal to twice the length of the string?

A string fixed at both ends may support any shape which has a node at the endpoints.
The harmonic wave amplitudes are all sine functions of position. $A(x)=\sin(kx)$

If the string has length L, so it goes from x=0 to x=L, work out the set of sine waves that will fit on the string and you have answered your own question.