The Problem
Consider an open disk of radius centered at the origin, which will represent the "still" drum head shape. At any time the height of the drum head shape at a point in measured from the "still" drum head shape will be denoted by which can take both positive and negative values. Let denote the boundary of that is, the circle of radius 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, it will be convenient to use cylindrical coordinates, Then, the above equations are written as
Here, 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, is the radial membrane resultant at the membrane boundary, is the membrane thickness, and is the membrane density. If the membrane has uniform tension, the uniform tension force at a given radius, may be written
where is the membrane resultant in the azimuthal direction.
Read more about this topic: Vibrations Of A Circular Drum
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