Find the velocity of a travelling wave like behaviour

2 min readAug 16, 2023

Hi everyone, I have a travelling wave solution drawn at each time step for a set of data obtained from a previous simulation.

Tmax = 10000;
m =load('solution.mat');
SS = m.sol;
N = m.N1;
for i = 1:50:Tmax
    Xval = SS(i,N);      
    hold on 
    plot(N,Xval,'Linewidth',2);
end
hold off

and below is the out put I got from the above code where x axis represents a distance (N) from 0 to 200 and y axis takes values between 0–1. .Each coloured line is drawn at a different time step.

Next, I’d like to calculate the velocity at each time step when y=0.5. I.e. I want to identify the x value when y=0.5 for each colour line and calculate the velocity as

. The main question here is sometimes there are no exact points in Xval with value 0.5 or no exact N value associated to 0.5.

I’d be happy if someone could help me in creating a loop for this calculation.

NOTE:-

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A simple intersection approach works —

t = linspace(0, 25, 450);
x = sin(2*pi*t+rand(size(t)))/2+0.5;
L = numel(t);
xval = 0.5;
tidx = find(diff(sign(x-xval)));                        % Approimate Intersections
for k = 1:numel(tidx)
    idxrng = max(1,tidx(k)-1) : min(tidx(k)+1,L);
    tv(k,:) = interp1(x(idxrng), t(idxrng), xval);      % Exact Interseections
end
xv = ones(size(tv)) * xval;
Velocity = 1 ./ diff(tv);
Results = table(tv(2:end), Velocity, 'VariableNames',{'Time','Velocity'})

Results = 61×2 table
     Time      Velocity
    _______    ________

    0.86599     2.0118 
     1.4494     1.7142 
     1.9056     2.1916 
     2.4067     1.9959 
     2.8642     2.1857 
     3.3668     1.9895 
     3.8813     1.9438 
     4.3854     1.9837 
     4.8913     1.9769 
     5.4618     1.7527 
     5.8731     2.4316 
     6.4307     1.7934 
     6.8758     2.2465 
     7.3968     1.9195 
     7.8853     2.0469 
     8.3841     2.0048

figure
plot(t, x, 'DisplayName','Original Data')
hold on
plot(tv, xv, 'xr', 'DisplayName','Exact Intersections')
hold off
grid
legend('Location','best')

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