---
title: "Drum Modes & Strike Position Interactive Lab | Sandbox Physics"
description: "Move the strike. Change what rings. How does strike position choose the vibrations?"
canonical_url: "https://sandboxphysics.com/labs/drum-modes/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/drum-modes/"
last_modified: "2026-10-18"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## A strike chooses a superposition

**Background** A stretched membrane resists transverse motion through tension. Every natural pattern obeys the same fixed circular boundary.

**Why it matters** Connect the moving surface, quiet lines and a quantitative frequency account.

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 misconception** Every pale line predicts a measured sand pattern.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

Circular drum overtones are not an integer harmonic ladder. The center suppresses every nonaxisymmetric mode.

Primary sources: [Penn State Acoustics · Circular membrane modes](https://www.acs.psu.edu/drussell/Demos/MembraneCircle/Circle.html) [NIST DLMF · Bessel zeros](https://dlmf.nist.gov/10.21)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Measure gravity, split motion into components, add contact forces and collisions, then carry the same laws into orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)