Question

A steel string of length 2.0 meters and mass 15g/m is stretched with a tension of 120 N and is fixed at both ends.

(i)   Write down the wave equation and find the general solution for string oscillations using the method of variable separation.

(ii)   Find the 1^st three natural frequencies and draw the corresponding mode shapes.

Answer 1

The wave equation is:

where is the speed of the wave in the medium.

To solve it by the method of separation of variables, we assume that:

Let the length of the string be . The boundary conditions in the x-direction are:

The first boundary condition gives us and the second boundary condition gives us the eigenvalues:

So, by using the principle of superposition, a general solution to the wave equation is:

where the constants as defined below can be determined from the initial conditions which are not stated in this case, so the answer is left in its most general form.

First three natural frequencies correspond to respectively and the mode shapes correspond to the shapes of the function . They are plotted as shown below:

The wave velocity is:

In this case we must find the first 3 natural frequencies as: