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I wonder what is the physical meaning of Dirichlet, Neumann and Robin boundary conditions for a vibrating string? Or link to other applications?

epimorphic
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AhSh
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1 Answers1

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Let $u(x,t)$ be the transverse position of the string in question, and denote $\frac{du}{dx}$ by $u_x$.

Dirichlet

Dirichlet boundary conditions are ones in which the value of $u$ itself is given at the ends of the string. Many times $u$ on the boundaries will be specified as a constant value. In this case, the Dirichlet condition physically corresponds to the situation in which the ends of the vibrating string are held fixed at a constant position.

One may also specify a Dirichlet condition in which $u$ is not constant on the boundaries, but is actually given by some function of time. This case physically corresponds to moving the ends of the string up or down in a specific way while the string vibrates.

In either case, the Dirichlet condition physically means that you are holding the ends of the string at a certain value (or values) as it vibrates.

Neumann

Neumann boundary conditions are ones in which the value of $u_x$ is specified at the ends of the string. Identifying $u_x$ with the "slope" of the string at the boundaries, a Neumann condition can be physically achieved by imagining that the ends of the string are attached to frictionless tracks that are free to move up and down.

To intuitively see this, imagine that the boundary conditions are given as $u_x=0$ on both ends of the string. This means that the string must have a horizontal tangent line at the ends. As the string vibrates, if the ends are on movable tracks, then this "flatness" at the ends can be preserved as the string vibrates.

Of course, values other than zero can be specified, and $u_x$ can even be specified as a function, but this latter case may be more difficult to physically picture.

Robin

The Robin condition comes from specifying the value $u_x + au$ (where $a$ is just a constant) on the ends of the string. The $u_x$ term suggests that this case might be physically similar to attaching the ends of the string to tracks, as in the Neumann case. And the $au$ term suggests a physical intuition of "holding" the ends of the string similar to the Dirichlet condition.

The truth is that this intuition points us in the right direction. The PDE with the Robin condition is a dampened form of the Neumann condition. Physically, it can be achieved by allowing the ends of the string to move on tracks with friction. This is in contrast to the frictionless tracks of the Neumann case.

wgrenard
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