Near-Field Acoustic Holography: The Frame Drum

A phase-sensitive setup for near-field acoustic holography was established in order to determine the characteristic resonance modes of a vibrating membrane. The experiment was carried out using a frame drum as the vibrating system of interest. By exciting the membrane at certain fundamental frequencies and recording the complex signals from the pressure and particle velocity fields across the drumhead, we modeled 3-D representations of the vibrating surface. Correlations between an eigen-frequency and its corresponding eigen-mode of resonance are described throughout this report. Introduction and Theoretical Description A percussion instrument played by striking a tightly stretched membrane attached to a circular frame is called a membranophone. The transient nature of its sound reveals tones with non-periodic waveforms which will therefore have non-harmonic partials and pitches more or less indeterminate. To understand and represent such unique acoustical behavior, we conducted our experimental work using a frame drum as the vibrating system of interest. For a circular membrane with no displacement at the boundary, the ideal vibrational model is well known and can be derived mathematically. ( ) ( ) ( ) ( ) r t i X t r X t v t r X 2 0 2 2 2 2 exp , , 1 . , , δ ω φ φ = ∂ ∂ − ∇ This is the expression corresponding to the membrane motion where X is the displacement of the membrane from its rest position, v is the velocity, ω is the angular frequency, r is the position along the radius and φ is the angle to the x-axis. The left hand