Deriving 1D Wave Equation for Vibrating Guitar String

[deadstar33]

I'm doing a project on a vibrating guitar string and I have completed all the simulation and experimental work, but I do not fully understand the theory behind it. I need to derive the 1 dimensional case of the wave equation, as the 1 dimensional case is considered to be the most convenient approach for a wire string due to the fact that its diameter is almost negligible relative to its length. So it is basically treated as a line segment in the theoretical analysis. However, I do not know how to take the generalised form of the wave equation and apply it to this 1 dimensional problem. Anyone have any experience in this area? Thanks very much.

 

[Studiot]

Not sure what you are asking here.

A plucked guitar string vibrates as a standing wave. The actual wave depends upon where it is plucked along its length. By the actual wave I mean the distribution of the fundamental and the harmonics.

 

[deadstar33]

Yeah I understand that. Basically what I have to do is support the work I have done theoretically by mathematically deriving from first principles the governing equation for wave propagation in a vibrating string. Something akin to this derivation: http://en.wikipedia.org/wiki/Wave_equation#Derivation_of_the_wave_equation

Except, I'm not sure if that's exactly the right one. It may well be, but it would take a bit of time to go through that and understand it enough to say whether it is what I'm looking for or not.

 

[nasu]

The derivation in your link is more appropriate for waves in a 1D crystal. It is true that at the end they look at the continuous case (string) as a limit.

A direct derivation for the string is for example here:
http://www.math.ubc.ca/~feldman/apps/wave.pdf
This is quite standard derivation.

 

[deadstar33]

The derivation in your link is more appropriate for waves in a 1D crystal. It is tru that at the end they look at the continuous case (string) as a limit.

A direct derivation for the string is for example here:
http://www.math.ubc.ca/~feldman/apps/wave.pdf
This is quite standard derivation.

Thank you very much, that's just what I was looking for. Appreciate it.

 

[nasu]

I am glad it helped you.