Bonus Web Problem--2

[Victor Ivrii]

Oscillations of the beam  (with both its ends  having fixed positions but not directions, imagine beam lying on supports) are described by an equation
\begin{equation*}
u_{tt} + K u_{xxxx}=0, \qquad 0<x<l
\end{equation*}
with $K>0$ and the boundary conditions
\begin{equation*}
u(0,t)=u_{xx}(0,t)=u(l,t)=u_{xx}(l,t)=0.
\end{equation*}

• (a) Find  equation describing frequencies and corresponding  eigenfunctions
  (You may assume that all eigenvalues are real and positive).
• (b) Solve  this equation graphically.
• (c) Prove  that eigenfunctions corresponding to different eigenvalues are orthogonal.
• (d) Bonus  Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional.

Compare with eigenvalues of Problem 2 of HA2

[Victor Ivrii]

Actually my comments are related to this problem (so I shuffled posts -- splitting and merging)


 One can see easily the relation to $X''+\lambda X=0$ with the Dirichlet b.c.: the same eigenfunctions and squared eigenvalues. Reason is simple: if $A$ denotes $\frac{d^2\ }{dx^2}$ with Dirichlet b.c. then our operator $B$ is just $A^2$ including b.c.: $u\in \mathfrak{D}(B)$  iff $u\in \mathfrak{D}(A) $ and also $Au\in \mathfrak{D}(A) $. Here $\mathfrak{D}(A) $ denotes domain of definition of $A$.

[Calvin Arnott]

Hmm, interesting. It seems as if that perspective gives some insight as to why the result was an even and odd eigenfunction for the respective problems. What does the generalization to this problem in several spacial dimensions look like?

[Victor Ivrii]

Hmm, interesting. It seems as if that perspective gives some insight as to why the result was an even and odd eigenfunction for the respective problems. What does the generalization to this problem in several spacial dimensions look like?

Oscillations of membranes is described by $u_{tt}-K\Delta u=0$ while oscillations by plates by
$u_{tt}+K\Delta^2 u=0$.

The simplest b.c. for membrane are Dirichlet $u|_S=0$ and Neumann $\partial_nu|_S=0$ where $n$ is a normal to the boundary which is $S$; so $\partial_n u:= \sum_i u_{x_i}n_i$.

The simplest b.c. for plate  are Dirichlet $u|_S=\partial_n u |_S=0$ and free
$\sum_{i,j} u_{x_ix_j}n_in_j|_S =\partial_n\Delta u|_S=0$.

THis particular problem generalizes to $u|_S=\Delta u|_S=0$ and again we have a square of Laplacian.

[Calvin Arnott]

Suppose we somehow have several time-like variables. Does it make sense to consider this kind of system? How would we represent such a structure in the language of equations we're developing in this course?- perhaps in this specific case a Laplacian on time as well? What kind of properties do systems of this kind have that don't appear with a single time-like variable?

[Victor Ivrii]

Suppose we somehow have several time-like variables. Does it make sense to consider this kind of system? How would we represent such a structure in the language of equations we're developing in this course?- perhaps in this specific case a Laplacian on time as well? What kind of properties do systems of this kind have that don't appear with a single time-like variable?

There are so called ultra-hyperbolic equations http://forum.math.toronto.edu/index.php?topic=37.msg176#msg176 but they are much less developed and needed

[Victor Ivrii]

So, when we have this problem we can see easily that general solution
$$
X= A\cosh (cx)+  B\cos (cx) +C\sinh (cx)  +D\sin (cx)
$$
satisfies conditions at $0$ as $A=B=0$ as
$$
c^{-2}X''= A\cosh (cx)-  B\cos (cx) +C\sinh (cx)  -D\sinh (cx)
$$
and therefore
$$
X=C\sinh (cx)  +D\sinh (cx)
$$
satisfies conditions at $0$ as $A=B=0$ as
$$
c^{-2}X''= C\sinh (cx)  -D\sin (cx).
$$
To satisfy conditions on the right end we need
\begin{align*}
& C\sinh (cl)  +D\sin  (cl) =0,\\
& C\sinh (cl)  -D\sin (cl) =0
\end{align*}
which is possible for $(X,D)\ne 0$ iff $C=0$, $\sin (cl)=0$ i.e. we get $\lambda_n= n^2 \pi^2/l^2$, $X_n=\sin (n\pi x/l)$.