Why are the solutions to the Laplacian called harmonic functions?
The solutions to the LaPlacian (Del^2 V = 0 where Del^2 is the sum of 2nd partial derivative of V in terms of x,y,z) are called harmonic because they are the sums of infinite series of functions which are related “harmonically”. Each term is related to the others by being the same function like a sine, Bessel and Legendre functions, with each term of the form: f(x), f(2x), f(3x), … etc. The coefficient of x in each function is called the frequency of the term. Having integer frequency ratios like this is called a harmonic relationship.
The functione f(x) is sine(x) for systems with rectangular symmetry, a Bessel function, J(x), for systems with cylindrical symmetry, and a Legendre polynomial, P(x), for systems with spherical symmetry.
Each term is called a “harmonic” where x is the 0th harmonic or the fundamental, 2x is the 2nd harmonic of the fundamental, etc. You may also recognize the solution and the terminology from Fourier series, which these solutions happen to be, and generically from Fourier Transforms.
This terminology comes from music, dating back to ancient Greece when Pythagoras defined musical notes in terms of ratios lengths of vibrating strings. The root note is called the fundamental. The note of the same pitch but one higher octave is the 2nd harmonic. The ratio of the sting’s vibration frequency is the ratio of its length and is the same as the harmonic number.
Does this musical history relate back to Laplacians? Actually yes. The Laplacian differential equation describes the behavior of many musical instruments, neglecting nonlinear effects. The string vibration is a sine function, just as Pythagora’s analysis works out to be, because it can be described by a Laplacian and has a rectangular (one-dimensional) symmetry. A drumhead vibration is a Bessel function (it’s symmetry is cylindrical). Certain loudspeakers acoustic patterns have Legendre solutions because their symmetries are spherical.
Very impressive first and only post JeffG