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:



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



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



Using Taylor series to expand we obtain:



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:

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. 

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. 

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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