Resolve boundary structure into harmonic modes

Consider the region 0<x<L and y>0, with zero potential on the side walls and a prescribed bottom potential f(x). Require decay as y tends to infinity. Assume a product of one function of x and another of y, substitute into Laplace’s equation, and divide by the product where it is nonzero. Since one resulting expression depends only on x and the other only on y, both must equal a separation constant.

Zero values at both side walls select sine functions with integer mode number n. Their second x derivatives are negative multiples of themselves, so the second y derivative must have the opposite sign. The vertical solutions are exponentials, not traveling sinusoids. Discard the growing exponential to meet the condition at infinity. The n=0 linear branch cannot satisfy both homogeneous side values unless it vanishes; it does not supply a missing constant mode here.

Linearity allows summing the separated modes. At the bottom boundary this becomes a Fourier sine expansion of f. Multiply by a chosen sine and integrate across the strip. Orthogonality removes every other mode and leaves its coefficient. This projection is why separation is practical: a coupled boundary condition becomes independent scalar integrals. The basis is chosen by geometry and boundary conditions, rather than by an arbitrary taste for trigonometric functions.

For a bottom electrode held at constant voltage between grounded side walls, the ideal corner values disagree. The sine series represents the constant on the open interval but is zero at the endpoints; the actual idealized corner is singular. That does not invalidate the interior harmonic solution. It does limit uniform convergence at the corner and can make field derivatives large there. A finite number of modes gives an approximation whose boundary error deserves attention, especially near discontinuities.

The amplitude of mode n falls by a factor e after moving a distance L/(nπ) into the strip. A pattern with twice the mode number has half this penetration length. The total potential at a particular x can nevertheless rise or change sign with y because different modes may initially cancel; monotonic decay of every coefficient does not imply monotonic pointwise decay of the complete sum. Distinguish modal statements from statements about every observation point.

In the spectral lab, Reset and set mode count to 3 and coefficient decay to 0; inspect the initial sum. Reset and set mode count to 3 and coefficient decay to 2; predict suppression of higher harmonics. This illustrates how coefficients control spatial detail. The instrument uses sin(nx−t)/nᵖ, with a common temporal frequency, so playback does not implement the exponential y decay of a Laplace solution or a physical electrostatic evolution.