M066 · A struck membrane

Drum Modes & Strike Position

Why does striking a drum in a different spot change its vibration but not its frequencies? The frequencies belong to the membrane: its size, tension and mass per area fix them. The strike only sets how strongly each natural pattern is excited; a pattern that is still at the strike point gets nothing. Patterns with quiet diameters are all still at the center, so a center hit excites only the ring-shaped patterns. The overtones are not whole-number multiples of the lowest frequency.

Strike a circular drum in three dimensions. Pick out one family of frequencies, find its nodal circles and diameters, then compare which vibrations a center strike and an off-center strike set ringing.

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, the tension is uniform and the areal density is constant. The ideal point impulse excites only a finite set of retained modes: four angular orders and three radial roots. Every mode family without axial symmetry keeps both of its orthogonal angular partners.

Isolation picks out the selected frequency family from the same strike. Its energy readout is part of the retained impulse energy; it does not include the energy of the high modes left out. A strike at the centre cannot excite angular modes. No nonlinear coupling moves energy between modes.

Physical coordinates lie in the rest surface, and positive displacement points up. The height envelope sets one magnification for the whole playback, shown in the readouts. The map shows the long-time RMS motion, and the trace uses physical units. Slow playback counts time in fundamental periods, not in real seconds.

Illustrative grains use a seeded random walk whose step variance rises with local calculated RMS displacement. The initial pattern has 850 preparation steps. Diffusion steps, grain size and vibration amplitude are rescaled independently; this model does not predict real sand lift-off, friction, impacts or air-driven inverse patterns. Clamped-mode indices label dominant trial functions and need not count exact nodal lines. High Ritz modes converge more slowly.

The model covers only small transverse motion and linear elasticity. Sand impacts, friction, sound synthesis, air loading, real material calibration, free plate edges and contact forces are not included. Keeping a finite number of modes limits the spatial detail.

The drum is an ideal tensioned membrane with a fixed rim. Its motion is an exact sum over a finite set of modes, slowed down and magnified for display. Sound radiation, material damping and the full impact waveform are not modeled.

Start with the essentials

Focus question: Why does moving the strike change the mode mixture without changing its natural frequencies?
One-sentence intuition: The membrane fixes the mode frequencies; where you strike decides how strongly each mode is excited.

Core mathematical model

Specified parameters

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

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

Wave equation for the membrane

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

The mass per unit area and the tension per unit length set the wave speed. The roots in the frequency formula are zeros of the Bessel function.

Cosine and sine partners of each mode

\begin{gathered}B_{mn}=J_m(j_{mn}r/R)\\\phi^c_{mn}=B_{mn}\cos(m\theta)\\\phi^s_{mn}=B_{mn}\sin(m\theta)\end{gathered}

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

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

A mode that is zero at the strike point gets no impulse. Energy is conserved within the finite set of retained modes.

Common difficulties

What the gold grains represent

Typical misconceptionThe simulated grains predict a real sand experiment's migration speed and exact pattern.

Better mental modelPlate grains illustrate a random walk with lower motility where vibration is weak. They use a separate dimensionless clock. Real lift-off, friction, impacts, air flow and support flexibility need additional models. The drum shows displacement directly.

Run the experiment

01
Find the quiet lines
Compare Two nodal diameters and Two internal nodal circles; inspect top and side views.
What to observe: The nodal pattern stays fixed, and neighboring lobes rise and fall in opposite directions.
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: Four times the tension doubles the frequency. The display gain leaves the physical displacement and energy unchanged.

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

Drum Modes & Strike Position

A strike chooses a superposition

Core mathematical model

Specified parameters

Wave equation for the membrane

Cosine and sine partners of each mode

A point impulse selects the modes

Common difficulties

What the gold grains represent

Run the experiment

Find the quiet lines

Move the strike

Separate physics from display

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