# Harmonic oscillator

• Many of the mechanical elements constituting a musical instrument behave approximately as linear systems. By this we mean that the acoustic output is a linear function of the mechanical input, so that the output obtained from two inputs applied simultaneousl is just the sum of the outputs that would be obtained if they were applied separately. For this statement to be true for the instrument as a whole, it must also be true for all of its parts, so that deflections must be proportional to applied forces, flows to applied pressures, and so on. Mathematically, this property is reflected in the requirement that the differential equations describing the behavior of the system are also linear, in the sense that the dependent variable occurs only to the first power. An example is the equation for the displacement ${\displaystyle y}$ of a simple harmonic oscillator under the action of an applied force ${\displaystyle F(t)}$:   ${\displaystyle m{\frac {\mathrm {d} ^{2}y}{\mathrm {d} t^{2}}}+R{\frac {\mathrm {d} y}{\mathrm {d} t}}+Ky=F(t)}$, ... where ${\displaystyle m}$, ${\displaystyle R}$, and ${\displaystyle K}$ are, respectively, the mass, damping coefficeint, and spring coefficent, all of which are taken to be constants. ... A little consideration shows, of course, that this description must be an over-simplification ...