Finding the Longest Wavelength of Standing Waves on a 254.0 cm String

[kathyt.25]

Homework Statement

What is the longest wavelength for standing waves on a 254.0 cm long string that is fixed at both ends?

Homework Equations

The Attempt at a Solution

When x = L (length of string):

k(n) = n*pi / L
wavelength (n) = n*pi / k(n)

I got the answer, I just don't understand why these equations are used in my textbook to find this particular answer.

 

[ideasrule]

"how can a wave be made?"

Imagine pulling on the rope hard, to make it taut. You could then make the rope vibrate like a guitar string.

A wave has to be set up in the rope to make it vibrate. Obviously, the fixed ends can't be vibrating, which sets restrictions on the permitted wavelengths. So: what's the longest possible wavelength?

 

[LowlyPion]

Maybe this is what they are getting at?

http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html#c1

http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html#c3