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