# Steel Guitar String: Find First 5 Modes, Sketch Eigenfunctions

• scott_uca03
In summary, the conversation involves finding the frequencies of the first five modes of vibration and sketching a graph of the associated eigenfunctions for a steel guitar string with given dimensions and tension. The density of steel is needed to find the linear mass density of the string, which can be found by calculating the volume per unit length and converting it to mass per unit length. The density of steel can be easily found through a simple online search.
scott_uca03

## Homework Statement

A steel guitar string with a diameter of .3 mm and 65 cm long has a tension of 100 N. Find the frequencies of the first five modes of vibration and sketch a graph of the associated eigenfunctions. The density of steel,7700 kg/m^3 is needed to find $$\mu$$.

## Homework Equations

I know c^2=T/$$\mu$$. I also know that Fn=nc/2L. I also know the solution to the problem, but I am unsure how to find $$\mu$$ .

## The Attempt at a Solution

Isn't that just the linear mass density of the string? You need to calculate the volume per unit length of the string, then convert it to a mass per unit length. This should be easy because the string is cylindrical.

Yea, I've found the volume of the string. I'm unsure how to find the mass though. If I could find the mass I could find the density and that's what I'm looking for to finish the problem.

Use google to find what you don't know! Since you said this is a steel wire, I googled on "density of steel" and found that the density of steel, while it varies slightly with the type of steel, averages around 7.9 grams per cubic centimeter. Multiply by the cross-section area of the string to find the mass per centimeter.

Last edited by a moderator:
Thanks a lot. Put me on the write track and I figured it out.

## 1. What is a steel guitar string?

A steel guitar string is a musical instrument string made of steel that is used to produce sound on a steel guitar. It is typically made of high carbon steel and can vary in thickness depending on its purpose and desired tone.

## 2. How do you find the first 5 modes of a steel guitar string?

The first 5 modes of a steel guitar string can be found by solving the wave equation, which takes into account the tension, length, and mass of the string. The first mode is the fundamental frequency, and the following modes are integer multiples of the fundamental frequency.

## 3. What are eigenfunctions in relation to a steel guitar string?

Eigenfunctions are the solutions to the wave equation for a steel guitar string. They represent the standing waves that can be produced on the string and are important in determining the different modes of vibration for a given string.

## 4. How can you sketch eigenfunctions for a steel guitar string?

To sketch eigenfunctions for a steel guitar string, you can use a mathematical model of the string's vibration, such as a sine or cosine function, and plot it for different modes of vibration. Alternatively, you can use a computer program or app that can simulate the vibration of a steel guitar string.

## 5. Why is it important to know the modes and eigenfunctions of a steel guitar string?

Understanding the modes and eigenfunctions of a steel guitar string is important for musicians and engineers alike. It allows musicians to create different pitches and tones by plucking or strumming the string in specific places. Engineers can also use this knowledge to design and optimize steel guitar strings for different purposes and applications.

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