Resonance – Wave equations & Spectral Method

    PDE Eigenvalue Problems and Chladni Plates
    Blake Keeler
    August 13, 2015
    1 Introduction
    This lab deals with Chladni gures, named after the man who discovered them, German
    musician and physicist Ernst Florence Friedrich Chladni. He xed metal plates at the center
    and excited them with his violin bow, which generated sounds of differing pitch depending
    on where he touched the plate with his bow. More curiously, if there was dust or sand
    on the plate, different beautiful patterns (Chladni gures) would emerge for each pitch.
    This discovery piqued the interest of many a scientist, but it took nearly a century for the
    associated mathematical model to be developed. The primary difficulty in constructing such
    a model was the unusual fact that the boundaries of the plate are not specied in any way,
    but rather are free to move as dictated by the vibration of the plate.
    Figure 1.1: Chladni’s original method of stimulating the plates, from Elementary Lessons
    on Sound (1879) by William Henry Stone.
    After many years and several attempts by different mathematicians (each building on
    his predecessor’s work), a full mathematical description of a vibrating plate with free edges
    was completed by Gustav Kirchhoff in 1850, who showed that the patterns that Chladni
    found corresponded to the eigenpairs of a particular biharmonic operator L, and that those
    eigenpairs must obey the following conditions:
    1
    ? Interior:
    uxxxx + 2uxxyy + uyyyy = u; (x; y) 2 O:
    ? Edges:
    uxx + uyy = 0; uxxx + (2 ?? )uxyy = 0; x = L; y 2 (??H;H);
    uyy + uxx = 0; uyyy + (2 ?? )uxxy = 0; y = H; x 2 (??L;L):
    ? Corners:
    uxy = 0; (x; y) = (L;H):
    In the above, O is the physical domain (the plate itself), which is simply the rectangle
    [??L;L] [??H;H]: Also, is a physical material constant between 0 and 1.
    2 Review of Resonance
    In your coursework, you have seen the solution of wave equations using the spectral method,
    wherein the PDE is written in the form
    utt = Lu + f(x; t);
    where L is a symmetric linear operator and f is a forcing function. Suppose L has eigen-
    functions j(x) with eigenvalues j for j = 1; 2; : : : : Let us also assume, without loss of
    generality, that the eigenfunctions are normalized, i.e. ? j ; j? = 1 for all j: We know that
    the solution u(x; t) can be written as
    u(x; t) =
    S1
    j=1
    cj(t) j(x)
    for some time-dependent coefficients cj(t): Substituting this expression in for u in the original
    PDE, we obtain
    S1
    j=1
    c”
    j (t) j(x) =
    S1
    j=1
    j j(x) + f(x; t):
    Since L is symmetric, we know that ? i; j? = 0 whenever i ?= j: Therefore, we can take the
    inner product of both sides of the above equation with k to obtain
    c”
    k(t) = kck + ?f(; t); k()?:
    Exercise T1. Suppose we have initial conditions u(x; 0) = 0 and ut(x; 0) = 0 for all x,
    which is the case with the Chladni plates since we excite them from a at, stationary
    state. Find a1 and a2 and write down the explicit formula for ck(t):
    Exercise T2. Now we will examine the special case where f(x; t) = (t) m(x): Using the
    formula you derived in the previous exercise, write down the solution u(x; t) to the
    original PDE. What is so interesting about this result? (Hint: what does the solution
    look like for any xed time t?)
    2

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