Vibrating membrane normal modes
A stretched membrane can vibrate in a set of special patterns called normal modes. In each one, every point moves at the same frequency, and some lines on the surface never move at all.
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What you will see
A square drum skin, lit and clamped in a frame, vibrates in one mode at a time. Nodal lines, where the membrane stays still, divide it into patches that move in opposite directions. Step through the modes from (1,1) to (3,3) to see the patterns grow more intricate.
Try this
- Compare mode (1,2) with (2,1): same frequency, rotated pattern.
- Count the nodal lines in mode (3,2).
- Note how the frequency grows with √(m² + n²).
Common misconception
A drum does not vibrate in just one mode. A real strike excites many modes at once, and their frequencies are not whole-number multiples, which is why drums sound less tonal than strings.
Exam-style question
For a square membrane, f ∝ √(m² + n²). What is the ratio of the (2,1) mode’s frequency to the (1,1) mode’s?
√(2² + 1²) / √(1² + 1²) = √5 / √2 ≈ 1.58. Unlike a string’s harmonics, this is not a whole number.
Key equations
f_mn ∝ √(m² + n²)∂²u/∂t² = c²∇²u
Curriculum
AQA A-level physics 3.3.1–3.3.2, IB Physics C.2–C.3.