Problem 4. [question purpose - to check the understanding of the method of separation of variables and to show that this can explain some fascinating real-world eects] A stringed instrument typically has a fundamental tone {the lowest frequency mode), and the overtones (the other terms that show up from separation of variables) have frequencies that are integer multiples of the frequency of the fundamental. A typical (Western-style) drum has a \"head\" (the part you strike to make sound) with constant thickness. For such a drum, the vibrations obey the 2D waue equation U\"; = (:2sz with U = 0 on the boundary (\"clamped conditions\"). For a 2D circular drum, this does not have the property that ouertone frequencies {that is, the frequencies of the other modes) are integer multiples of the fundamental frequency {see https: //mr.w. acs. psu. edu/ drusse! l/demos/membranecircle/circle. html ifyou are interested in what the ouertones and their corresponding modes are like). As a result the typical person would not say the drum has a \"musical \" sound. In classical Indian music, the \"tabla\" has a head which is thickened in the middle. By doing this, the craftsman who builds the tabla is able to modify how the overtones relate to the funda- mental frequency. Here is the result: bites: //mw. goutube. com/watch? v=r310eTSmOuI (please watch this video). For the tabla [using polar coordinates that is cylindrical coordinates with z = 0] we haue e2 U\" am where d[p, 3) is the mass per unit area of the drum { and is larger near the center). This equation captures the fact that where the mass is larger, the same force produces less acceleration. If it helps your calculations, you can note that in polar coordinates, V2 is the same as in cylindrical coordinates, except we ignore the terms involving .2 deriuatiues. V2U (a) By writing U (p,9,t] = F(p,9]G[t), show that G"(t} = 7G(t}. We could separate F further into a function of p times a function of 9, and we would need to do this if we wanted to find the function F explicitly, but we do not look for that in this question, so there is no need for the extra effort. (b) Explain why 1r above is a constant. (c) Find a PDE for F{_o,3]. The equation should depend on c, d(p,3), and 7 f\"depends on\" means those variables should appear in the equation]. You do not need to solve, and whether to expand VZF is optional. By changing the form of d(p, 8] the person constructing the tabla can modify the The coefcients in the PDE for P, which changes the spatial modes and thus aects the values of 7 that occur. But your answer in part (a) shows that it does not affect the ODE for G, except by changing the values of '7. (d) Find a solution for G(t) in terms of ,8 = 'fry. You should assume 7