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Qmechanic ♦
Qmechanic ♦
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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=$y=$ the displacement of the string at spatial co-ordinate x$x$ and at time t$t$, 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?

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?

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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user157588
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String vibration and damping

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?