M066 · A struck membrane

Drum Modes & Strike Position

Strike a three-dimensional circular drum. Isolate a frequency family, find nodal circles and diameters, then compare which vibrations survive a center or off-center strike.

Selected natural frequency $0\,\mathrm{Hz}$

Selected / fundamental frequency $0$

Gold-point displacement $0\,\mathrm{\mu m}$

Gold-point RMS displacement $0\,\mathrm{\mu m}$

Actual height magnification $0$

Displayed mechanical energy $0\,\mathrm{\mu J}$

Displayed / retained strike energy $0\%$

Excitation & geometry

Choose what the strike excites

Nodal diameters

m

2.00\,

Radial index including fixed rim

n

1.00\,

Isolate selected frequency family

b_{\rm isolate}

1.00\,

Uniform tension per unit length

T

100.00\,\mathrm{N/m}

Drum radius

R

0.30\,\mathrm m

Strike radius fraction

r_d/R

0.55\,

Strike azimuth

\theta_d

25.00\,

Height envelope / shorter side

H_{\rm env}/L_{\min}

0.22\,

The circular rim is fixed; tension is uniform and areal density is fixed. The ideal point impulse excites a finite retained basis: four angular orders and three radial roots. Each nonaxisymmetric family retains both orthogonal angular partners.

Isolation filters the selected frequency family from that same strike. Its energy readout is a subset of retained impulse energy, not the energy of the omitted high modes. Center strikes cannot excite angular modes. No nonlinear coupling transfers energy between modes.

Physical coordinates lie in the rest surface; positive displacement points upward. The height envelope sets one fixed-in-time magnification, reported in the readouts. The map shows long-time RMS motion and the trace uses physical units. Slow playback uses fundamental periods, not wall-clock physical time.

Small transverse motion and linear elasticity only. No sand or friction dynamics, sound synthesis, air loading, real material calibration, free-edge plates, clamped plates or contact forces. Finite modal truncation limits spatial detail.

Ideal tensioned membrane with a fixed rim. An exact finite modal expansion, slowed and magnified for display; no acoustic radiation, material damping or complete impact waveform.

Start with the essentials

Focus question: Why does moving the strike change the mode mixture without changing its natural frequencies?
One-sentence intuition: Mode frequencies belong to the membrane. Strike position sets the projection onto each mode.

Core mathematical model

Specified parameters

\sigma=0.3\,\mathrm{kg/m^2},\qquad I=10^{-4}\,\mathrm{N\,s}

The areal mass and point impulse are fixed. Change tension and radius to test frequency scaling.

The tensioned membrane

\sigma\ddot w=T\nabla^2w,\quad w(R,\theta)=0,\quad \omega_{mn}=\frac{j_{mn}}{R}\sqrt{\frac{T}{\sigma}}

Areal mass density and tension per unit length determine wave speed. The roots are zeros of the Bessel function.

Both angular partners

\phi^{c,s}_{mn}=J_m(j_{mn}r/R)\{\cos(m\theta),\sin(m\theta)\}

The angular index counts nodal diameters; the radial index counts nodal circles including the fixed rim. Only the cosine component exists for zero angular order.

A point impulse selects the modes

q_k(t)=\frac{I\phi_k(\mathbf r_d)}{M_k\omega_k}\sin(\omega_kt),\quad E_k=\frac{I^2\phi_k(\mathbf r_d)^2}{2M_k}

A mode whose value vanishes at the strike point receives zero impulse. Energy is conserved within the finite retained basis.

Common difficulties

An amplitude map is not sand

Typical misconceptionEvery pale line predicts a measured sand pattern.

Better mental modelThis map is computed RMS displacement in a linear model. Granular motion, friction, air and real support compliance are outside its scope.

Run the experiment

01
Follow the quiet lines
Compare Two nodal diameters and Two internal nodal circles; inspect top and side views.
What to observe: The nodal map stays fixed while opposite lobes rise and fall.
02
Move the strike
Compare center and off-center strikes using all retained modes.
What to observe: The center excites only axisymmetric families; off-center strikes add angular patterns.
03
Separate physics from display
Quadruple tension, then vary only the height display gain.
What to observe: Fourfold tension doubles frequency. Display gain leaves physical displacement and energy unchanged.

Primary sources: Penn State Acoustics · Circular membrane modes NIST DLMF · Bessel zeros