# Math Help - Modelling the shape of a vibrating guitar string

1. ## Modelling the shape of a vibrating guitar string

Hello, I am Marckutt and I teach at an underprivelleged school outside Buenos Aires. We are trying to model the shape of a guitar string as it vibrates. More specifically, I need to be able to determine the displacement of any point along the string from the restpoint as the string oscillates. This requires incorporating two independent variables - the point along the string and time. What type of model could I use to model this scenario?

2. Well for a start lets say displacement is a function of two variables the point x horizontally along the sting and t time, the time a note is held?

$\displaystyle d(x,t) = \dots \sin (\dots) +\dots$

What other information do you have?

$$\frac{\partial^2 y}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}$
$

where $v$ is the speed of propagation.

We assume the at $x=0$ and $x=L$, the displacement $y=0$.

The solution to the equation is separable, which means

$y(x,t)=X(x)T(t)$

Solving the wave equation with this general solution, be the method of separation of variables, we find that

$X(x)=sin(\frac{n\pi x}{L})$

and

$T(t)=sin(\frac{n\pi vt}{L})$

where $n$ is an integer corresponding to the frequencies of vibrations (modes)

Thus the displacement is

$y(x,t)=sin(\frac{n\pi x}{L})sin(\frac{n\pi vt}{L})$

The most general solution is an superposition of the solutions for all values of $n$

$y(x,t)=\sum_{n=1}^{\infty}B_{n}sin(\frac{n\pi x}{L})sin(\frac{n\pi vt}{L})$