Deriving 1D Wave Equation for Vibrating Guitar String

In summary, the conversation is about a project on a vibrating guitar string and the need to derive the 1 dimensional case of the wave equation for theoretical analysis. The link provided offers a direct derivation for the string, which is considered the most appropriate approach for a wire string.
  • #1
deadstar33
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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.
 
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  • #2


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.
 
  • #3


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.
 
  • #4


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.
 
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  • #5


nasu said:
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.
 
  • #6


I am glad it helped you.
 

What is the 1D wave equation for a vibrating guitar string?

The 1D wave equation for a vibrating guitar string is a mathematical equation that describes the physical behavior of a guitar string when it is plucked or strummed. It is a second-order partial differential equation that takes into account factors such as tension, length, and mass of the string.

How is the 1D wave equation derived for a vibrating guitar string?

The 1D wave equation is derived using the principles of Newton's second law of motion, Hooke's law, and the wave equation. By considering the forces acting on a small segment of the guitar string and applying these principles, the equation can be derived.

What are the assumptions made when deriving the 1D wave equation for a vibrating guitar string?

Some of the assumptions made when deriving the 1D wave equation for a vibrating guitar string include: the string is perfectly elastic, it is under tension, it has a constant cross-sectional area, and it is vibrating in a single plane.

What are the applications of the 1D wave equation for a vibrating guitar string?

The 1D wave equation for a vibrating guitar string has various applications in acoustics, music, and engineering. It can be used to understand the behavior of various musical instruments, design new instruments, and analyze the sound produced by them.

What are some limitations of the 1D wave equation for a vibrating guitar string?

Some limitations of the 1D wave equation for a vibrating guitar string include: it does not take into account the effects of damping, it assumes a linear relationship between tension and displacement, and it does not consider the effects of higher harmonics or non-uniform strings.

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