How to plot harmonics

     %PLOTTING TIME DEPENDENT MOTION OF A STANDING WAVE
     %Define Parameters
  a_n = 1;
  k = 5;
  w_n = 5;
  x = 0:0.05:4;
  t = 1:0.05:20;  
      for j=1:length(t)                       
            for i=1:length(x)                     
                    u(i) = a_n*sin(k.* x(i)).* cos(w_n.*t(j)); %Formula for displacement
               end
      pause (0.1)
      plot(u)
      axis([1,65,-1.5,1.5])
      grid on
        end

Above is the code for a standing wave along a string. I need to create a few more of these, to end up with a series of standing waves which I can then calculate the sum of, which gives me the net displacement of the wave. I then need to plot this net displacement, which will result in a travelling wave.

Equation 1 on the formula sheet attached is the equation that governs the source that generates the series of standing waves. Tau is a 'tuning parameter'. I can just assign it a value. It determines the angular frequency. Imagine plucking a guitar. That is the source. We pluck the string of a guitar at a certain distance along the string. That distance is denoted as X_s (X subscript s). That can just be a value.

Equation 2 is the displacement for the nth standing wave (harmonic) where L is the length of the string, x is displacement and w_n (omega subscript n) is the angular frequency n is the harmonic number. I need to incorporate this somehow into the loop I have used to generate the wave in my code, in order to plot, lets say, the first 5 harmonics as subplots.

Equation 3 is then used to sum all of these harmonics together, to generate a final plot.

Any ideas on how to do this using the code and formula I have written

[JDilla]

I was thinking of using symsum to calculate the sum of the standing waves,but this did not work.

[JDilla]

Any ideas??

Answer 1

[Guillaume]

Learn to use vectorisation. You can calculate all the u(x,t) at once:

Using ndgrid (or meshgrid):

x = 0:0.05:4;
t = 1:0.05:20;  
[tt, xx] = ndgrid(t, x);
u = a_n*sin(k*xx).*cos(w_n*tt);
%now you can plot u
for j = 1:numel(t)
   pause(0,1);
   plot(u(j, :));
   %...

Using bsxfun:

x = 0:0.05:4;
t = 1:0.05:20;  
u = bsxfun(@(x, t) a_n*sin(k*x).*cos(w_n*t), x, t');
%now you can plot u
for j = 1:numel(t)
   pause(0,1);
   plot(u(j, :));
   %...

If you want to add a third variable L, just add an extra dimension to u:

x = 0:0.05:4;
t = 1:0.05:20;  
L = 1:5
[tt, xx, LL] = ndgrid(t, x, L);
u = a_n*sin(k*xx./LL).*cos(w_n*tt);
%sum all the L together:
u = sum(u, 3);

[JDilla]

I think meshgrid is a good option. I don't understand your code though. How can I use vectorisation to plot the individual harmonics?

[Guillaume]

You use vectorisation to generate u, for the plotting, you still have to use a loop:

[xx, tt, LL] = ndgrid(x, t, L);
u = a_n*sin(k*xx./LL).*cos(w_n*tt);
for j = 1:numel(t)
   pause(0.1)
   plot(squeeze(u(:, j, :))) %time is 2nd dimension
   axis([1,65,-1.5,1.5])
   grid on
end

[JDilla]

Thank you for trying, but that code just gives me what I already have. I need to make 5 different waves, involving n inside the loop maybe? n=1, n=2, n=3, n=4,n=5 etc, and then subplot them all. Then sum them all together (but instead of 5, I sum 50) and then plot this, the wave then instead of being a standing wave that oscillates I guess "upwards and downwards* actually, if my equations are correct, becomes a travelling wave and moves left to right.

[Guillaume]

I'm afraid I don't understand what you're asking anymore. The u that is generated has three dimensions, the first one is x, the second one is the time and the third one is the harmonic. Thus if you want to see the 5 harmonics at time th, you just plot

plot(squeeze(u(:, th, :)); %plot all harmonics at time |th|.

If you want to sum these harmonics:

sumharmonic = sum(u, 3);
%to plot that sum at time |th|:
plot(sumharmonic(:, th));

[JDilla]

Thank you for your help

I want to plot the harmonics (first 5 harmonics individually, as subplots) and then sum a specified number (so I want to find the sum of 50) and plot it. Does that make sense? The third dimension you said was L? but L was length, which is just a variable.

[Guillaume]

Right, I misread your image, I thought the harmonic was L. It does not change much anyway. You have, u as a function of three variables x, t and n:

u = f(x, t, n)  %the specific of f don't matter.

You generate u all at once, for whatever combination of values:

x = 0:0.05:4;
t = 1:0.05:20;  
n = 1:50;
[xx, tt, nn] = ndgrid(x, t, n);
u = f(xx, tt, nn);

And if you want to fix a variable, you just fix that dimension. For example to plot the first 5 harmonics at time t0

figure;
for nh = 1:5
   subplot(nh, 1, 5);
   plot(squeeze(u(:, t0, nh));
end

The sum of the 50 harmonics (3rd dimension) is still:

sum(u, 3)