Vibrating string - 1D

In geophysics we always encounter with the waves such as earthquake waves, sound waves, oscilating electric and magnetic field in electromagnetic radiation. Thus we need to understand the nature of the this physical phenomena. "Wave equation" can be derived basic equation of motion and restoring forces of differential equation. This equation describes the waves in time and space. Let's consider an ocean wave and assume that the displacement given by u(x,t), then the wave equation can be written as;


The factor of the V is speed of the wave. Solution of this linear partial differential equation is;

A: amplitude
k: constant
w: circular frequency
'phi'(greek letter): phase

Now if we fixed the boundary conditions there is no displacement at the end points;

u(0,t)=u(L,t)=0

Finally we have the standing wave that has nodes at the location of two endpoints. New solution for this specific problem is;

Implementation this problem to matlab takes two steps, first create a function which calculates nodes, secondly create another function for plotting purpose;

function u=unodes(x,dt,n,w,phase)
u=cos(n*w*dt+phase).*sin(n*pi*x);
end

plotting function is;

function vs1d(xmin,xmax,n,w,phase)

%%%%%%%%%%%%%%%%%%%%
% Author: Mustafa DENIZ                    %
% Contact: mustafdeniz@itu.edu.tr        %
%%%%%%%%%%%%%%%%%%%%%

tmax=2*pi/w; %maximum time

dt=0.05; %timestep

frms=round(tmax/dt); %number of frame

x=xmin:0.01:1; %grid x

for ii=1:frms-1;
   plot(x,unodes(x,ii*dt,n,w,phase));
   axis([xmin xmax -1.5 1.5])
   m(ii)=getframe;
end
movie(m,3)



Subsequently we can change the nodes,phase length of string arbitrarily by typing this command, for example;

>>vs1d(0,1,1,1,2)

if we want see the third node just change value of n variable in command;

>>vs1d(0,1,3,1,2)


REFERENCES:

Farlow, S.J., 1993. Partial Differential Equations Scientists and Engineers, Dover Publications, INC. New York

Myint-U, T., 2006. Linear Partial Differential Equations for Scientist and Engineers, Birkhäuser, Boston