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Vibration & Waves Simulator
Drumhead Vibration Simulator — Bessel Modes of a Circular Membrane
Computes vibration modes of a circular membrane (radius R, tension T, surface density ρ_s) from the Bessel-function zeros α_mn and displays the eigenfrequency f_mn together with the 2D displacement pattern. Tune the drum to feel why percussion partials are inharmonic.
Parameters
Radius R
m
Tension T
N/m
Surface density ρ_s
kg/m²
Mode index
Mode index 1=(0,1), 2=(1,1), 3=(2,1), 4=(0,2), 5=(3,1), 6=(1,2). The yellow box marks the currently selected mode.
Results
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Frequency f
—
α_mn
—
Period T_p
—
Wavenumber k
Membrane displacement u(r,θ)
Top view of the membrane. Blue = downward displacement / red = upward / white = nodes (|u|<ε). Each mode shows its node circles and diameter nodes.
Six lowest modes
Side-by-side view of the six lowest (m,n) modes with α_mn and relative frequency f/f_01. The current mode is highlighted in yellow; click a cell to switch.
Theory & Key Formulas
For a circular membrane of radius $R$, tension $T$ [N/m] and surface density $\rho_s$ [kg/m²], separating variables in the wave equation gives the eigenfunction
The wavenumber is $k_{mn}=\alpha_{mn}/R$ and the period is $T_p = 1/f_{mn}$. Low-order zeros are (0,1)=2.4048, (1,1)=3.8317, (2,1)=5.1356, (0,2)=5.5201, (3,1)=6.3802, (1,2)=7.0156 — not in integer ratios, so drum partials are inharmonic.
What is the Drumhead Vibration Simulator
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A drum does not have a clear pitch like a guitar. Why is that?
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Good observation. The reason is that the partials are not integer multiples. A guitar string has f, 2f, 3f, 4f..., so your ear locks onto f as 'the pitch'. A circular membrane has 1×, 1.594×, 2.135×, 2.296×, 2.653×, 2.917×... relative to f_01 — irrational ratios. You hear a 'thud' rather than a note, which is the percussion timbre.
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Why is α_mn = 2.4048 such an awkward number?
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It is a zero of the Bessel function $J_m(x)$ — a transcendental number with no closed form. The clamped boundary u(R)=0 forces $J_m(kR)=0$, so kR must equal one of those zeros, written $\alpha_{mn}$. The first zero of $J_0$ is 2.4048..., the next is 5.5201..., and so on. It is the same idea as $\sin(kL)=0$ giving $kL=n\pi$ for a string — Bessel just replaces sine for circular geometry.
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If I double the tension the frequency goes up by about √2. Does that match the formula?
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Exactly. $f_{mn} = (\alpha_{mn}/2\pi R)\sqrt{T/\rho_s}$, so doubling T gives a factor of $\sqrt{2}≈1.414$. This is the same scaling as a string $f=(n/2L)\sqrt{T/\mu}$, and is the physical basis for tightening drum lugs to change pitch. Timpani players use a foot pedal to vary T smoothly, sweeping the pitch continuously without changing R.
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There are both straight white lines and concentric circles in the picture. What is the difference?
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Both are nodal lines (where the displacement is zero), but they come from different factors. The straight lines are zeros of the angular factor $\cos(m\theta)$ — 2m lines for index m. The circles are zeros of the radial factor $J_m(\alpha_{mn} r/R)$ — more circles for larger n. Sprinkling sand on a real membrane driven at a resonance produces exactly this pattern, the 2D analogue of a Chladni figure.
FAQ
A string obeys the 1D Helmholtz equation $u_{tt}=c^2 u_{xx}$, whose boundary condition $\sin(kL)=0$ produces $k_n=n\pi/L$ — an integer-spaced spectrum. That is the basis of pitched tones. A circular membrane obeys the 2D equation in polar coordinates, which separates into a Bessel ODE radially. The zeros of $J_m(x)$ are not integer-spaced (they are transcendental numbers), so the membrane's spectrum is inharmonic. That is the fundamental physical reason drums are unpitched percussion — and it is exactly why timpani and tabla need extra tricks (mass loading, air coupling, head damping) to coax out near-integer ratios and sound 'pitched'.
An ideal circular membrane is inharmonic, but timpani exploit three tricks: striking near 1/4 of the radius, air coupling through the kettle shell, and a slight mass concentration near the center. Together these suppress the axisymmetric (0,1) mode and leave the m≥1 modes (1,1), (2,1), (3,1), (4,1) with α = 3.832, 5.136, 6.380, 7.588. Normalizing by (1,1) gives 1.000, 1.340, 1.665, 1.981 — the last is almost a perfect octave, and 5.136/3.832 = 1.340 is close to a perfect fifth. Real timpani therefore sound much closer to harmonic, which is why a tuned kettle drum has a recognizable note.
The eigenfunction $u_{mn}(r,\theta) = J_m(\alpha_{mn} r/R)\cos(m\theta)$ is the exact analytical mode shape produced by separation of variables for a circular membrane. Running Ansys, Abaqus, or Nastran on a thin circular plate (Kirchhoff plate) yields exactly these patterns plus a bending-stiffness correction. Touch panels, speaker cone diaphragms, MEMS sensor membranes, and electronic-instrument transducers all rely on this analysis. The tool is a useful sanity check for FEM output: an analytical reference any CAE engineer can reproduce by hand.
Yes. For m≥1, the eigenfunctions $\cos(m\theta)$ and $\sin(m\theta)$ have the same frequency, so any linear combination is also an eigenfunction. This tool displays only the cosine variant, but in practice the two combine according to the strike location and small membrane asymmetries, making the nodal lines appear to rotate. A slightly elliptical drumhead breaks the degeneracy and the two modes split into two slightly different frequencies — that is sometimes audible as a slow beating. Mode m=0 is axisymmetric and never degenerate, so (0,1) and (0,2) sit on single frequencies.
Real-world applications
Percussion design (timpani, tabla, conga): Timpani pedals vary the tension T continuously, sweeping the fundamental over almost an octave as $\sqrt{T}$. Tabla (Indian percussion) glues a black mass-loaded patch (the syahi) at the center to suppress particular modes and produce a near-integer harmonic series — the reason a tabla can play a melody. Congas and djembes combine the membrane modes with a wooden shell air resonance to emphasize low-frequency modes. Varying tension, density, and radius in this tool reproduces those tuning principles.
Speaker-cone diaphragms: Dynamic-speaker cones and dome tweeters can be modelled as circular membranes (or conical extensions). Resonant modes inside the audible band colour the sound, so designers push (0,1), (1,1), (2,1) frequencies outside the operating range by tuning cone geometry, density, and surround stiffness. This simulator is accurate enough for first-pass analysis of dome tweeters.
MEMS pressure sensors and microphones: A capacitive MEMS pressure sensor reads displacement of a 100 µm–1 mm circular silicon diaphragm. Its operating bandwidth is limited by the (0,1) eigenfrequency, so designers tune radius, tension, and thickness to push that mode above 30 kHz. With an extra factor for plate bending stiffness, the formulas here are useful for back-of-the-envelope MEMS sizing.
Drum recording and PA engineering: A microphone placed above the head centre primarily picks up the (0,1) mode (maximum displacement); placed near the rim it captures higher m≥1 modes. The nodal map in this tool explains the timbre differences between centre- and edge-miking. Engineers use centre miking for kick drums and edge miking for snares precisely because the nodal layout selects which modes leak into the signal.
Common misconceptions and caveats
The most common error is to assume drum partials are integers like a string's, i.e. f, 2f, 3f, 4f. In reality the circular-membrane spectrum scales with the α_mn ratios — 1, 1.594, 2.135, 2.296, 2.653, 2.917, ... — which are irrational. That is the essential acoustic difference between strings and percussion and why a drum's pitch feels indefinite. The six-mode comparison in this tool shows the relative frequencies f/f_01 directly, so the inharmonicity is immediately visible. 'Pitched' drums (timpani, tabla) are special: their physical design hand-picks a near-integer subset of those partials.
Another frequent assumption is that α_mn can be computed from a simple closed-form formula. Bessel-function zeros have no closed form and must be computed numerically (Newton iteration or truncated series). This tool hard-codes the six lowest values as constants and evaluates J_m via a 60-term power series for the membrane shape. In engineering practice one looks them up in tables (Abramowitz–Stegun) or uses libraries such as Boost.Math or SciPy.
Finally, do not assume the rim is perfectly clamped in reality. The tool uses the idealized u(R)=0 boundary, but real drum heads are clamped to a hoop with finite stiffness, which lowers the eigenfrequencies slightly. Production CAE/FEM models add a finite spring constant at the boundary for realism, and the values produced here represent the 'ideal upper bound' that experimental measurements approach as the rim becomes stiffer.