Oscillations and Harmonic Motion

A system undergoes simple harmonic motion when the restoring force is proportional to the displacement and directed toward the equilibrium position (Hooke's law, $F = -kx$). This is the most important model in oscillatory physics because it describes the behavior of every stable system near an equilibrium point.

The equation of motion $m\ddot{x} = -kx$ can be written in canonical form as: $$\ddot{x} + \omega_0^2 x = 0, \qquad \omega_0 = \sqrt{\frac{k}{m}}$$ This is a linear homogeneous second-order ODE with constant coefficients. The characteristic equation $\lambda^2 + \omega_0^2 = 0$ has complex conjugate roots $\lambda = \pm i\omega_0$, which lead to sinusoidal solutions. The general solution is a linear combination of sine and cosine, or equivalently: $$x(t) = A\cos(\omega_0 t + \phi)$$ where $A$ is the amplitude (maximum displacement) and $\phi$ the initial phase (angular position at $t=0$ on the phasor diagram).

Initial conditions: the constants $A$ and $\phi$ are determined from the initial position and velocity: $$A = \sqrt{x_0^2 + (v_0/\omega_0)^2}, \quad \tan\phi = -\frac{v_0}{\omega_0 x_0}$$ The phase is chosen in the interval $[0, 2\pi)$ based on the quadrant determined by the signs of $x_0$ and $v_0$.

Fundamental properties:

• Isochronism: the period $T = 2\pi/\omega_0 = 2\pi\sqrt{m/k}$ does not depend on amplitude. Large and small oscillations share the same period — a property discovered by Galileo while observing a swinging lamp in the Pisa cathedral.
• Natural angular frequency: $\omega_0 = \sqrt{k/m}$ increases with spring stiffness and decreases with mass. A stiffer spring → faster oscillations; a larger mass → slower oscillations.
• Frequency: $f = 1/T = \omega_0/(2\pi)$ in hertz (Hz), the number of complete oscillations per second.

Real-world examples of SHM:

• Mass attached to an ideal spring (the prototypical model).
• Motion of a diatomic molecule near equilibrium (Morse potential approximated by a parabola).
• LC circuit (inductor-capacitor): the equation $L\ddot{q} + q/C = 0$ is formally identical, with $q$ (charge) ↔ $x$, $L$ ↔ $m$, $1/C$ ↔ $k$.
• Simple pendulum for small angles (see §3).

Dimensional analysis: $\omega_0 = \sqrt{k/m}$ has dimensions $\sqrt{[k]/[m]} = \sqrt{(M T^{-2})/M} = T^{-1}$, i.e. rad/s. Check that $T = 2\pi\sqrt{m/k}$ has dimension $T$ (seconds), as required for a period.

Differentiating $x(t) = A\cos(\omega_0 t + \phi)$ with respect to time yields the instantaneous velocity and acceleration:

• $v(t) = \dot{x}(t) = -A\omega_0\sin(\omega_0 t + \phi)$
• $a(t) = \ddot{x}(t) = -A\omega_0^2\cos(\omega_0 t + \phi) = -\omega_0^2\,x(t)$

The acceleration is always proportional and opposite to the displacement: $a = -\omega_0^2 x$. At the endpoints ($x = \pm A$) the acceleration is maximal in magnitude and velocity is zero; at the center ($x = 0$) acceleration is zero and velocity is maximal. There is a phase shift of $\pi/2$ between position and velocity, and $\pi$ between position and acceleration.

Mechanical energy: the elastic force is conservative (depends only on position), so the total mechanical energy $E = K + U$ is conserved. The kinetic energy and potential energy are: $$K = \tfrac{1}{2}mv^2 = \tfrac{1}{2}m\omega_0^2 A^2\sin^2(\omega_0 t + \phi)$$ $$U = \tfrac{1}{2}kx^2 = \tfrac{1}{2}kA^2\cos^2(\omega_0 t + \phi)$$ Adding them, and recalling $\omega_0^2 = k/m$: $E = \tfrac{1}{2}kA^2[\sin^2(\cdot) + \cos^2(\cdot)] = \tfrac{1}{2}kA^2 = \text{constant}$. The total energy is therefore proportional to the square of the amplitude.

Cyclic energy exchange:

• When $x = \pm A$ (endpoints): $v = 0$, $U = E$, $K = 0$ — all energy is potential.
• When $x = 0$ (equilibrium): $v = \pm v_{max}$, $K = E$, $U = 0$ — all energy is kinetic.
• At any intermediate position: $E = K + U = \tfrac{1}{2}mv^2 + \tfrac{1}{2}kx^2 = \tfrac{1}{2}kA^2$.

Maximum velocity: $v_{max} = \omega_0 A = A\sqrt{k/m}$. Obtained from energy: $\tfrac{1}{2}mv_{max}^2 = E = \tfrac{1}{2}kA^2$.

Velocity-position relation: from energy conservation we obtain a useful time-independent relation: $$v(x) = \pm\omega_0\sqrt{A^2 - x^2}$$ The sign depends on the direction of motion. This formula shows that velocity is maximum at the center ($x=0$, $v = \pm\omega_0 A$) and zero at the endpoints ($x = \pm A$).

Energy diagram: the plot $U(x) = \tfrac{1}{2}kx^2$ is a parabola. The total energy $E$ is a horizontal line. The intersection points ($U(x) = E$) define the turning points $x = \pm A$, where $K = 0$. The particle oscillates between these two points like a ball rolling in a parabolic bowl, continuously converting kinetic energy into potential energy and back.

The simple pendulum consists of a point mass $m$ suspended from an inextensible massless string of length $L$. The restoring force is the tangential component of the weight: $F = -mg\sin\theta$. Writing Newton's second law along the tangent to the circular path: $$mL\ddot{\theta} = -mg\sin\theta \quad \longrightarrow \quad \ddot{\theta} + \frac{g}{L}\sin\theta = 0$$ This is a nonlinear equation (due to $\sin\theta$). To solve it analytically, we use the small-angle approximation: for $\theta \ll 1\,\mathrm{rad}$ (i.e., $\theta \lesssim 10°$), the Taylor series expansion gives $\sin\theta = \theta - \theta^3/6 + \ldots \approx \theta$. Keeping only the first term: $$\ddot{\theta} + \frac{g}{L}\theta = 0$$ which is formally identical to the SHM equation ($\ddot{x} + \omega_0^2 x = 0$) with: $$\omega_0 = \sqrt{\frac{g}{L}}, \qquad T = 2\pi\sqrt{\frac{L}{g}}$$
Fundamental properties:

• Mass independence: the period of a simple pendulum does not depend on mass $m$. Galileo deduced this experimentally by observing that pendulums of different masses but equal length oscillate with the same period (for equal amplitude). The mass cancels out in the equation of motion because it appears in both the restoring force ($mg$) and the inertia ($mL$).
• Approximate isochronism: for small oscillations, $T$ is independent of amplitude. For larger amplitudes, the actual period is longer than the approximation (corrected by $T \approx 2\pi\sqrt{L/g}\,(1 + \theta_0^2/16 + \ldots)$).
• Dependence on $g$: by measuring $T$ for a known $L$, one can determine $g$ precisely. This is the principle behind pendulum gravimeters used in geophysics.

Physical pendulum (compound pendulum): any rigid body oscillating about a horizontal pivot. Let $d$ be the distance between the pivot and the center of mass, and $I$ the moment of inertia about the pivot. The equation of motion (from Newton's second law for rotations, $\tau = I\alpha$) gives: $$I\ddot{\theta} = -Mgd\sin\theta \quad \longrightarrow \quad \ddot{\theta} + \frac{Mgd}{I}\theta = 0$$ hence $\omega_0 = \sqrt{Mgd/I}$ and $T = 2\pi\sqrt{I/(Mgd)}$. The equivalent length of a simple pendulum with the same period is $L_{eq} = I/(Md)$.

Historical applications:

• Galileo (1583): discovers pendulum isochronism by observing a lamp in the Pisa cathedral.
• Huygens (1656): invents the pendulum clock, improving time-measurement precision by two orders of magnitude.
• The meter was originally defined as the length of a pendulum with $T = 2\,\mathrm{s}$ (the definition has since changed).

Note on the approximation: the difference between $\sin\theta$ and $\theta$ is about 0.5% at $\theta = 10°$ and 4% at $\theta = 30°$. The approximate formula is therefore reliable for $\theta \lesssim 15°$.

In the presence of a viscous damping force $F_{damp} = -b\dot{x}$ (proportional and opposite to velocity, like the force exerted by a fluid at low speed), the equation of motion becomes: $$m\ddot{x} + b\dot{x} + kx = 0$$ Dividing by $m$ and introducing standard parameters: $$\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = 0, \qquad \gamma = \frac{b}{2m}, \quad \omega_0 = \sqrt{\frac{k}{m}}$$ $\gamma$ is the damping coefficient (in $\mathrm{s}^{-1}$). The characteristic equation $\lambda^2 + 2\gamma\lambda + \omega_0^2 = 0$ has roots $\lambda = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}$, whose discriminant determines the damping regime.

The three damping regimes:

• Underdamped ($\gamma < \omega_0$): the discriminant is negative, giving complex conjugate roots $\lambda = -\gamma \pm i\omega_1$ with $\omega_1 = \sqrt{\omega_0^2 - \gamma^2}$. The solution is a sinusoidal oscillation with exponentially decaying amplitude: $$x(t) = A_0 e^{-\gamma t}\cos(\omega_1 t + \phi)$$ The exponential envelope $e^{-\gamma t}$ describes energy dissipation. The time constant $\tau = 1/\gamma$ is the time for the amplitude to drop by a factor $e$.
• Critically damped ($\gamma = \omega_0$): the roots are real and equal $\lambda = -\gamma$. The solution is $x(t) = (C_1 + C_2 t)e^{-\gamma t}$, which returns to equilibrium in the shortest possible time without oscillating. This is the ideal regime for car shock absorbers, door dampers, and spring-based measuring instruments.
• Overdamped ($\gamma > \omega_0$): real and distinct roots, both negative. The solution is $x(t) = C_1 e^{-\lambda_1 t} + C_2 e^{-\lambda_2 t}$, with a slower return to equilibrium compared to the critical case.

Energy and damping: with viscous friction, mechanical energy is no longer conserved. The dissipated power is: $$P_{diss} = F_{damp}\cdot v = -bv^2$$ always negative (energy lost to friction). The total energy decays as $E(t) \approx E_0 e^{-2\gamma t}$ for weak damping.

Physical examples:

• Car shock absorbers: tuned near critical damping for a rapid return without oscillations after a bump.
• Door closers: viscous fluid mechanism preventing the door from slamming and oscillating.
• Galvanometer: the needle is critically damped to reach the measurement value quickly without oscillating.
• Seismometers: often overdamped to avoid spurious oscillations.

Quality factor: $Q = \omega_0/(2\gamma)$ measures damping: high $Q$ ↔ weak damping ↔ long-lasting oscillations. In energy terms, $Q = 2\pi \times (\text{energy stored})/(\text{energy lost per cycle})$.

When a damped oscillator is driven by a periodic external force $F(t) = F_0\cos(\omega t)$, the equation of motion becomes: $$m\ddot{x} + b\dot{x} + kx = F_0\cos(\omega t)$$ The general solution is the sum of two contributions:

• Transient: the homogeneous solution (damped oscillation) that decays after a time $\sim 1/\gamma$.
• Steady state (particular solution): an oscillation at the driving frequency $\omega$, with amplitude $A(\omega)$ and phase $\delta$.

The steady-state solution is $x(t) = A(\omega)\cos(\omega t - \delta)$, with amplitude: $$A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2}}$$ and phase shift $\delta = \arctan\left(\frac{2\gamma\omega}{\omega_0^2 - \omega^2}\right)$.

Resonance: the amplitude $A(\omega)$ reaches its maximum when the denominator is minimized, i.e. at: $$\omega_{res} = \sqrt{\omega_0^2 - 2\gamma^2} \approx \omega_0 \quad (\gamma \ll \omega_0)$$ At resonance, the amplitude is $A_{max} = \dfrac{F_0}{2m\gamma\omega_0} = \dfrac{F_0}{b\omega_0}$. As $\gamma \to 0$, $A_{max} \to \infty$ — the oscillator "absorbs" energy from the driving force with maximum efficiency.

Phase at resonance: when $\omega = \omega_0$, we have $\delta = \pi/2$ (velocity is in phase with the driving force, so the power $P = F\cdot v$ is maximal). For $\omega \ll \omega_0$, $\delta \approx 0$ (motion in phase with the driving force). For $\omega \gg \omega_0$, $\delta \approx \pi$ (motion opposite in phase).

Quality factor and bandwidth: $$Q = \frac{\omega_0}{2\gamma} = \frac{\omega_0 m}{b}$$ The bandwidth $\Delta\omega$ (difference between the frequencies at which $A = A_{max}/\sqrt{2}$) is related to $Q$ by: $$\Delta\omega = \frac{\omega_0}{Q} = 2\gamma$$ A high $Q$ means a narrow, tall resonance peak; a low $Q$ means a broad, low peak.

Examples and applications:

• Tacoma Narrows Bridge (1940): the bridge entered resonance with periodic wind gusts, producing increasingly large torsional oscillations until collapse. Classic example of the dangers of resonance in civil engineering.
• RLC circuit: resonance in an $LCR$ circuit is analogous: $\omega_0 = 1/\sqrt{LC}$, $Q = \frac{1}{R}\sqrt{L/C}$. Resonant circuits are the basis of radio tuning.
• Musical instruments: the soundboard of a violin or guitar selectively amplifies certain frequencies (acoustic resonance).
• Crystal glass: an opera singer can shatter it if the voice matches the glass's resonance frequency with sufficient amplitude.
• Magnetic resonance imaging (MRI): atomic nuclei in a magnetic field absorb radio waves at the Larmor frequency ($\omega = \gamma B_0$), the principle behind nuclear magnetic resonance and medical imaging.

Response curve: the plot $A(\omega)$ is called the resonance curve. As $\gamma$ increases, the peak lowers and widens, shifting slightly to the left ($\omega_{res} < \omega_0$). For $\gamma \ge \omega_0/\sqrt{2}$, there is no longer a pronounced maximum (overdamped system).