Normal modes and harmonics

What is a normal mode?

A normal mode is a natural pattern of vibration in which every part of a system oscillates at the same frequency with a fixed shape. For a string fixed at both ends, each normal mode has a particular number of nodes and antinodes.

The mode shape for a fixed string is

ight)$$ with $$n=1,2,3,ldots$$ Each mode has its own frequency: $$f_n=rac{nv}{2L}$$ ## Fundamental and harmonics The lowest allowed frequency is the fundamental frequency: $$f_1=rac{v}{2L}$$ Higher frequencies are harmonics: $$f_n=nf_1$$ For a string fixed at both ends, all integer harmonics are allowed. The second harmonic has twice the fundamental frequency. The third harmonic has three times the fundamental frequency. ## Mode number and shape The mode number $n$ tells how many half-wavelengths fit on the string: $$L=nrac{lambda_n}{2}$$ For $n=1$, the string has one antinode and nodes at the ends. For $n=2$, there is an additional node at the center. For $n=3$, there are three loops, and so on. The higher the mode number, the shorter the wavelength and the higher the frequency. ## Orthogonality of modes Normal modes are independent in a mathematical sense. For the string, different sine modes are orthogonal over the interval from 0 to $L$: $$int_0^L sinleft(rac{mpi x}{L} ight)sinleft(rac{npi x}{L} ight)dx=0$$ when $m eq n$. Orthogonality means one mode can be excited without directly being the same as another. It is similar to perpendicular vectors in ordinary geometry. ## Superposition of modes A general vibration of the string can be written as a sum of normal modes: $$y(x,t)=sum_{n=1}^{infty} A_nsinleft(rac{npi x}{L} ight)cos(omega_n t+phi_n)$$ This is the connection between normal modes and Fourier analysis. The initial shape of the string determines how much of each mode is present. Plucking a guitar string near the center excites a different mixture of harmonics than plucking near the bridge. That affects timbre. ## Normal modes beyond strings Normal modes occur in many systems: air columns, drumheads, bridges, molecules, buildings, electrical circuits, and coupled pendulums. A molecule's vibrational spectrum is a set of normal modes. A building has structural modes that engineers analyze to avoid dangerous resonance. In two-dimensional systems, such as drumheads, mode shapes can be more complicated, with nodal lines instead of simple nodal points. ## Degeneracy Sometimes two different mode shapes have the same frequency. This is called degeneracy. Degeneracy often results from symmetry. For example, circular membranes can have different orientations of a mode with the same frequency. Symmetry and normal modes are deeply connected in advanced physics. ## The big idea Normal modes are the natural vibration patterns of a system. Harmonics are mode frequencies related to a fundamental frequency. Because modes can be superposed, complicated vibrations can be decomposed into simpler independent patterns. This idea unifies musical instruments, structures, molecules, waves, and quantum systems.