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A chirp is an oscillation whose instantaneous frequency changes with time. For $s(t)=A\cos \varphi (t)$, with a smoothly increasing wave phase $\varphi$, the instantaneous frequency is $f(t)={\varphi }^{\prime }(t)/(2\pi )$. A linear chirp hasThe chirp rate $\alpha$ has units of hertz per second; positive and negative values give increasing and decreasing frequency over intervals where the stated interpretation applies.

A swept tone can probe the response of a coupled oscillator over a band, although a rapid sweep need not reproduce the steady response at each frequency. In radar or sonar, comparing a returned chirp with the transmitted signal supports pulse compression: a long transmitted signal can be processed into a narrow delay peak. A finite-duration chirp occupies a range of frequencies, even though its local wave phase supplies a well-defined instantaneous rate.

Coupled oscillators influence one another's motion through shared forces or other interactions. Two pendulums connected by a light spring provide a visible example: moving either bob changes the force on the other. For equal masses and lengths, the symmetric motion leaves the connecting spring unstretched, while the antisymmetric motion stretches it and has a higher frequency.

For a conservative linear system, the equations of motion take the form $M\ddot{\mathbf{q}}+K\mathbf{q}=0$, where $M$ is the mass matrix and $K$ the elastic stiffness matrix. A normal mode has a fixed displacement pattern $\mathbf{a}$ satisfying $K\mathbf{a}={\omega }^{2}M\mathbf{a}$. Expanding the initial motion in normal modes separates the evolution. Displacing just one pendulum generally excites several normal modes, producing beats and visible transfer of energy. A driving force excites only the modes onto which its spatial pattern has a nonzero projection.

An oscillator is a system capable of repeated motion or variation about a central state. A mass $m$ on an ideal spring with spring constant $k$ satisfiesThe restoring force accelerates the mass toward zero displacement; inertia carries it through that point, producing simple harmonic motion. The kinetic energy is largest at zero displacement, and the elastic potential energy is largest at the turning points.

Real oscillators can include damping, external driving and anharmonicity. A damped harmonic oscillator distinguishes the free transient from the forced periodic response, and a coupled oscillator adds exchange of energy with another dynamical coordinate. This makes an oscillator useful both as a physical device and as an approximation to small disturbances of much more complicated systems.

A normal mode of a stable undamped linear coupled oscillator has a single angular frequency and a fixed ratio of component displacements. All components share a sinusoidal time factor, with relative oscillation phases of zero or $\pi$ for real mode shapes. A general free motion is a superposition of the normal modes. For this system the symmetric stiffness matrix gives real eigenvalues and orthogonal eigenvectors; positive spring constants make the squared angular frequencies positive.

Write $m=m_1=m_2>0$ and $\kappa =k_{12}$. Assume ideal massless linear springs, negligible damping and positive $k_1,k_2,\kappa$. By Hooke's law and Newton's second law,The coupling spring extends by $x_2-x_1$, giving equal and opposite coupling forces.

For a normal mode, take $\mathbf{x}=\mathrm{Re}(\mathbf{a}{e}^{i\omega t})$. Nonzero eigenvectors requireSolving this quadratic equation in $m{\omega }^2$ givesA corresponding mode shape has ratioand the general real solution is $\mathbf{x}=C_{-}{\mathbf{a}}_{-}\cos ({\omega }_{-}t+{\varphi }_{-})+C_{+}{\mathbf{a}}_{+}\cos ({\omega }_{+}t+{\varphi }_{+})$. The arbitrary amplitudes and oscillation phases account for four independent initial data.

Weak coupling alone does not guarantee large beats. A useful representative sketch assumes nearly equal uncoupled angular frequencies. In the resonant case $k_1=k_2=k$ with $\kappa \ll k$, the mode shapes are $(1,1)$ and $(1,-1)$, andFor $x_1(0)=A$, $x_2(0)={\dot{x}}_1(0)={\dot{x}}_2(0)=0$, both normal modes have equal coefficients. With $\bar{\omega }=({\omega }_{+}+{\omega }_{-})/2$ and $\Delta \omega ={\omega }_{+}-{\omega }_{-}$,These rapid oscillations have slowly varying complementary wave envelopes. The initial displacement shifts almost entirely to the other mass at $t=\pi /\Delta \omega$, and returns after $2\pi /\Delta \omega$. Arbitrary initial data give a different superposition; if $|k_1-k_2|\gg \kappa$, the normal modes are localized mainly on individual masses, and strong exchange is suppressed.

Holding one mass fixed leaves the coupling spring attached and stretched by the other mass. Therefore the two constrained angular frequencies areThese differ from the angular frequencies obtained by physically removing the coupling spring. Ordinary frequencies in hertz are each corresponding $\omega /(2\pi )$.

Finally, for $k_1>k_2$ and $\kappa >0$, put $D=\sqrt{(k_1-k_2{)}^2+4{\kappa }^2}>k_1-k_2$. ThenSince ${\omega }_1>{\omega }_2$, taking positive square roots provesAt $\kappa =0$, the outer inequalities become equalities.

Coupled oscillators have more than one collective pattern of motion. For two equal masses $m$, each attached to a fixed support by a spring of stiffness $k$ and linked to each other by stiffness $\kappa$, small displacements obeyHere equal linear damping $b$ is included to make finite steady-state responses at resonance. The normal mode coordinates $q_{+}=(x_1+x_2)/\sqrt{2}$ and $q_{-}=(x_1-x_2)/\sqrt{2}$ diagonalize these equations. Their natural angular frequencies areThe symmetric normal mode leaves the coupling spring unextended; the antisymmetric normal mode stretches it and therefore has the higher frequency. With $F_{\pm }=(F_1\pm F_2)/\sqrt{2}$, their complex steady-state responses areThus the drive's spatial pattern matters as well as its frequency. Equal in-phase forces drive only the symmetric normal mode; opposite forces drive only the antisymmetric one; driving a single mass generally excites both. Finite damping broadens the resonance peaks and produces a phase lag. In the ideal undamped system, a resonant drive with nonzero projection onto a normal mode produces secular growth rather than a bounded resonant steady state.

The displacement of a particular mass adds contributions from both normal modes. They can cancel at an antiresonance. For $F_2=0$ and no damping, $x_1$ vanishes at ${\Omega }^2=(k+\kappa )/m$, while the partner still moves. This differs from the two nearby resonances, where a mode responds strongly; a small measured response need not mean that no motion or stored energy exists elsewhere.