# Engineering Acoustics/Transverse vibrations of strings

## Introduction

This section deals with the wave nature of vibrations constrained to one
dimension. Examples of this type of wave motion are found in objects
such a pipes and tubes with a small diameter (no transverse motion of
fluid) or in a string stretched on a musical instrument.

Stretched strings can be used to produce sound (e.g. music instruments
like guitars). The stretched string constitutes a mechanical system that
will be studied in this chapter. Later, the characteristics of this
system will be used to help to understand by analogies acoustical
systems.

## What is a wave equation?

There are various types of waves (i.e. electromagnetic, mechanical,
etc.) that act all around us. It is important to use wave equations to
describe the time-space behavior of the variables of interest in such
waves. Wave equations solve the fundamental equations of motion in a way
that eliminates all variables but one. Waves can propagate longitudinal
or parallel to the propagation direction or perpendicular (transverse)
to the direction of propagation. To visualize the motion of such waves
click
here
(Acoustics animations provided by Dr. Dan Russell,Kettering University)

## One dimensional Case

Assumptions :

\- the string is uniform in size and density

\- stiffness of string is negligible for small deformations

\- effects of gravity neglected

\- no dissipative forces like frictions

\- string deforms in a plane

\- motion of the string can be described by using one single spatial
coordinate

Spatial representation of the string in vibration:

![](1Dwave_graph1.png "1Dwave_graph1.png"){width="300"}

The following is the free-body diagram of a string in motion in a
spatial coordinate system:

![](string_dwg.jpg "string_dwg.jpg")

From the diagram above, it can be observed that the tensions in each
side of the string will be the same as follows:

![](equations1.jpg "equations1.jpg")

Using Taylor series to expand we obtain:

![](equations2.jpg "equations2.jpg")

## Characterization of the mechanical system

A one dimensional wave can be described by the following equation
(called the wave equation):

$\left( \frac{\partial^2 y}{\partial x^2} \right)=\left( \frac{1}{c^2} \right)\left( \frac{\partial^2 y}{\partial t^2} \right)$

where,

$y(x,t)= f(\xi)+g(\eta)\,$ is a solution,

With $\xi=ct-x\,$ and $\eta=ct+x\,$

This is the D\'Alambert solution, for more information see:
1

Another way to solve this equation is the Method of separation of
variables. This is useful for modal analysis. This assumes the solution
is of the form:

$y(x,t)= f(x)g(t)\$

The result is the same as above, but in a form that is more convenient
for modal analysis.

For more information on this approach see: Eric W. Weisstein et al.
\"Separation of Variables.\" From MathWorld---A Wolfram Web Resource.
2

Please see Wave
Properties
for information on variable c, along with other important properties.

For more information on wave equations see: Eric W. Weisstein. \"Wave
Equation.\" From MathWorld---A Wolfram Web Resource.
3

Example with the function $f(\xi)$ :

Example: Java String
simulation

This show a simple simulation of a plucked string with fixed ends.

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