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. Based on the applied boundary condition, at certain vibration frequencies, its natural frequencies, the surface moves 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 frequency. 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 (it cannot) being given in 1992 in the two-dimensional setting.
Practical significance 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 Ω {\displaystyle \Omega } of radius a {\displaystyle a} centered at the origin, which will represent the "still" drum head shape. At any time t , {\displaystyle t,} the height of the drum head shape at a point ( x , y ) {\displaystyle (x,y)} in Ω {\displaystyle \Omega } measured from the "still" drum head shape will be denoted by u ( x , y , t ) , {\displaystyle u(x,y,t),} which can take both positive and negative values. Let ∂ Ω {\displaystyle \partial \Omega } denote the boundary of Ω , {\displaystyle \Omega ,} that is, the circle of radius a {\displaystyle a} 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 fixed boundary conditions,
∂ 2 u ∂ t 2 = c 2 ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) for ( x , y ) ∈ Ω {\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}=c^{2}\left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right){\text{ for }}(x,y)\in \Omega \,}
u = 0 on ∂ Ω . {\displaystyle u=0{\text{ on }}\partial \Omega .\,}
Due to the circular geometry of Ω {\displaystyle \Omega } , it will be convenient to use polar coordinates ( r , θ ) . {\displaystyle (r,\theta ).} Then, the above equations are written as
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