Chladni Plate
z = (cos(n · π · x) · cos(m · π · y) − cos(m · π · x) · cos(n · π · y)) · cos(t)
Open in the app The dials and keys named below are the app's.
What it draws
cos(n·π·x)·cos(m·π·y) is one standing mode of a square. The π inside it puts a full wave every 2/n along x, so with Span of x, y at 1 there are n waves across and m the other way. cos(m·π·x)·cos(n·π·y) is the very same mode with the two directions exchanged, which on a square costs nothing: it vibrates at the same rate. The height is their difference, and cos t swings the whole sheet up and back every 2π ≈ 6.28 seconds at Speed 1.
The nodal lines
Wherever the two terms are equal the difference is zero, and it stays zero for every t: the sheet never moves there. Those lines are the figure. Exchanging x and y swaps the two terms, so the difference vanishes along the diagonal y = x. Cosine is even, so it vanishes along y = −x as well. Both diagonals are still at every setting of the letters, and the rest of the nodal set curves between them.
How many pieces those lines cut the square into jumps with each integer step. It is 8 at n = 1 and m = 2, 12 at 2 and 3, 16 at 3 and 4, and 48 at the opening pair, 3 and 5. When n = m the two terms are the same mode, the difference is identically zero, and the sheet lies flat for all time.
History
Ernst Chladni drew a violin bow along the edge of a metal plate strewn with sand. The sand was thrown clear of the parts that moved and settled along the parts that did not, so the nodal set drew itself. He published the method in 1787 in Entdeckungen über die Theorie des Klanges, and the patterns have carried his name since. What is on screen is the usual textbook stand-in for a square plate, with the right symmetry and the right pair of modes. A real plate obeys a fourth order equation whose free edges do not separate into products like these.
Try
- Drag n to 5, where it meets m: the two modes coincide, cancel exactly and the sheet goes flat.
- Drag m to 4: a coarser figure, 16 pieces instead of 48, with the two diagonals unchanged.
- Drag n to 10 and raise Mesh to follow it: the grid needs several cells per wave or the fine pattern is lost.
- Change the minus to a plus, (cos(n·π·x)·cos(m·π·y) + cos(m·π·x)·cos(n·π·y))·cos t: the symmetric partner of the same pair, whose diagonals are no longer nodal and which does not cancel at n = m.
- Clamp the edges with (sin(n·π·x)·sin(m·π·y) − sin(m·π·x)·sin(n·π·y))·cos t: every sine is zero at x = ±1, so the boundary of the square is nodal too.