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In case of a home experiment about string vibration under the boundary condition $$y(l,t)=y(0,t)=0$$ Where $y=$ the displacement of the string at spatial co-ordinate $x$ and at time $t$, I observed that under any kind of arbitrary initial condition as the wave damps out with time ultimately it vibrates in the first normal mode for some time after all other modes having been damped out before stopping.

Physically it can be argued that as the frequency corresponding to higher normal modes are higher the velocity of the different portions of the string due to the higher normal modes will also be higher and due to high velocity it will be damped very fast as the damping in real life is proportional to velocity. But I am failing to mathematically model this problem to have a quantitative idea about the damping of the individual normal modes. Now,how can I mathematically model this damping?

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The free vibration of a string can be modelled with the partial differential equation:

$$T\frac{\partial^2 y(x,t)}{\partial x^2}-\rho \frac{\partial^2 y(x,t)}{\partial t^2} = 0$$

Where $T$ is the axial tension on the string, $\rho$ is the mass density, $y$ is your transverse displacement, and $x$ is the position along the string length. The solution of this equation can be obtained using, for instance, the method of separation of vibrations ($y(x,t) = Y(x)\exp(\omega t)$), and later applying your boundary conditions.

Damping can be very difficult to predict, but a viscous type damping model can be incorporated in this model as

$$T\frac{\partial^2 y(x,t)}{\partial x^2} + \beta\frac{\partial y(x,t)}{\partial t}-\rho \frac{\partial^2 y(x,t)}{\partial t^2} = 0$$.

Where $\beta$ is a viscous damping coefficient, and since the damping forces are proportional to the velocity, the higher is the frequency, the higher are the damping forces. So each mode of vibration is affected differently by damping.

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Try the following: 1. compare the test between vertical and horizontal settings, to see the gravitation effect, which is most likely in play in your test; 2. plug in different initial modes: first mode, plug the midpoint; second mode, plug two points with an offset but in opposite direction; so on; 3. change the rigidity of your boundary condition: your BC may be not rigid enough

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