Vibrations of a circular membrane
A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame. Due to the phenomenon of resonance, at certain vibration frequencies, its resonant frequencies, the membrane can store vibrational energy, the surface moving in a characteristic pattern of standing waves. This is called a normal mode. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental mode. There exist infinitely many ways  in which a membrane can vibrate, each depending on the shape of the membrane at some initial time, and the transverse velocity of each point on the membrane at that time. The vibrations of the membrane are given by the solutions of the two-dimensional wave equation with Dirichlet boundary conditions which represent the constraint of the frame. It can be shown that any arbitrarily complex vibration of the membrane can be decomposed into a possibly infinite series of the membrane's normal modes. This is analogous to the decomposition of a time signal into a Fourier series. The study  of vibrations on drums led mathematicians to pose a famous mathematical problem on whether the shape of a drum can be heard, with an answer being given in 1992 in the two-dimensional setting. Motivation. Analyzing the vibrating drum head problem explains percussion instruments such as drums and timpani. However, there is also a biological application in the working of the eardrum. From an educational point of view the modes of a two-dimensional object are a convenient way to visually demonstrate the meaning of modes, nodes, antinodes and even quantum numbers. These concepts are important to the understanding of  the structure of the atom. The problem. Consider an open disk formula_1 of radius formula_2 centered at the origin, which will represent the "still" drum head shape. At any time formula_3 the height of the drum head shape at a point formula_4 in formula_1 measured from the "still" drum head shape will be denoted by formula_6 which can take both positive and negative values. Let formula_7 denote the boundary of formula_8 that is, the circle of radius formula_2 centered at the origin, which represents the rigid frame to which the drum head is attached. The mathematical equation that governs the  vibration of the drum head is the wave equation with zero boundary conditions, Due to the circular geometry of formula_1, it will be convenient to use cylindrical coordinates, formula_13 Then, the above equations are written as Here, formula_16 is a positive constant, which gives the speed at which transverse vibration waves propagate in the membrane. In terms of the physical parameters, the wave speed, c, is given by where formula_18, is the radial membrane resultant at the membrane boundary (formula_19), formula_20, is the membrane thickness, and formula_21 is the membrane density. If the membrane has uniform tension, the uniform tension  force at a given radius, formula_22 may be written where formula_24 is the membrane resultant in the azimuthal direction. The axisymmetric case. We will first study the possible modes of vibration of a circular drum head that are axisymmetric. Then, the function formula_25 does not depend on the angle formula_26 and the wave equation simplifies to We will look for solutions in separated variables, formula_28 Substituting this in the equation above and dividing both sides by formula_29 yields The left-hand side of this equality does not depend on formula_31 and the right-hand side does not depend on formula_3  it follows that both sides must be equal to some constant formula_33 We get separate equations for formula_34 and formula_35: The equation for formula_34 has solutions which exponentially grow or decay for formula_39 are linear or constant for formula_40 and are periodic for formula_41. Physically it is expected that a solution to the problem of a vibrating drum head will be oscillatory in time, and this leaves only the third case, formula_42 so we choose formula_43 for convenience. Then, formula_34 is a linear combination of sine and cosine functions, Turning to the equation for formula_46 with the observation that formula_47  all solutions of this second-order differential equation are a linear combination of Bessel functions of order 0, since this is a special case of Bessel's differential equation: The Bessel function formula_49 is unbounded for formula_50 which results in an unphysical solution to the vibrating drum head problem, so the constant formula_51 must be null. We will also assume formula_52 as otherwise this constant can be absorbed later into the constants formula_53 and formula_54 coming from formula_55 It follows that The requirement that height formula_25 be zero on the boundary of the drum head results in the condition The  Bessel function formula_59 has an infinite number of positive roots, We get that formula_61 for formula_62 so Therefore, the axisymmetric solutions formula_25 of the vibrating drum head problem that can be represented in separated variables are where formula_66 The general case. The general case, when formula_25 can also depend on the angle formula_26 is treated similarly. We assume a solution in separated variables, Substituting this into the wave equation and separating the variables, gives where formula_71 is a constant. As before, from the equation for formula_34 it follows that formula_43 with formula_74 and From the equation we obtain, by multiplying  both sides by formula_77 and separating variables, that and for some constant formula_80 Since formula_81 is periodic, with period formula_82 formula_83 being an angular variable, it follows that where formula_85 and formula_86 and formula_87 are some constants. This also implies formula_88 Going back to the equation for formula_46 its solution is a linear combination of Bessel functions formula_90 and formula_91 With a similar argument as in the previous section, we arrive at where formula_95 with formula_96 the formula_97-th positive root of formula_98 We showed that all solutions in separated variables of the vibrating drum head problem are of the  form for formula_100 Animations of several vibration modes. A number of modes are shown below together with their quantum numbers. The analogous wave functions of the hydrogen atom are also indicated as well as the associated angular frequencies formula_101.