Formula sheet · PU-103

Vibrations & Waves — formula sheet

Every boxed result from the 18 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.

D-057

Simple Harmonic Motion from a Linear Restoring Force

\[ m\ddot{x} = -k x, \qquad x(t)=x_0+A\cos(\omega t+\varphi), \qquad \boxed{\;\omega=\sqrt{\dfrac{k}{m}}\;},\quad T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}} \]

Near any stable minimum the force is a spring: it pulls back in proportion to displacement. The response is a pure cosine whose angular frequency is set entirely by the ratio of the potential's curvature \(k\) to the inertia \(m\). Crucially, \(\omega\) contains neither \(A\) nor \(\varphi\) — the oscillation is isochronous, ticking at the same rate whether the swing is large or vanishingly small (within the small-displacement regime). Stiffer wells and lighter masses oscillate faster.

D-058

Energy Exchange and Conservation in SHM

\[ E=T+U=\tfrac12 kA^2=\tfrac12 m\omega_0^2 A^2=\text{const},\qquad \langle T\rangle=\langle U\rangle=\tfrac14 kA^2,\qquad f_{\text{exch}}=2f_0 \]

The total energy is set once and for all by the amplitude and stiffness; it does not depend on where in the cycle you look. Kinetic and potential energy each swing between \(0\) and the full \(\tfrac12 kA^2\), perfectly anti-correlated, so that whenever one is maximal the other is zero. Averaged over a cycle they are equal (the virial result for a quadratic potential), each carrying half the energy. Because \(\sin^2\) and \(\cos^2\) complete a full cycle in half the period of \(\sin\) and \(\cos\), the trading happens at \(2\omega_0\) — twice the oscillation frequency.

D-059

The Damped Harmonic Oscillator

\[ m\lambda^{2}+c\lambda+k=0,\qquad \lambda_{\pm}=-\gamma\pm\sqrt{\gamma^{2}-\omega_0^{2}},\qquad \Delta=c^{2}-4mk \]

D-060

Quality Factor and Exponential Energy Decay

\[ \langle E\rangle(t)=E_0\,e^{-\gamma t}, \qquad Q=\frac{\omega_0}{\gamma}=2\pi\,\frac{E_{\text{stored}}}{E_{\text{lost per cycle}}} \]

The stored energy falls by a factor \( e \) every time \( t=1/\gamma \), i.e. after \( Q/(2\pi) \) full oscillations, or equivalently after the phase \( \omega_0 t \) advances by \( Q \) radians. A high-\( Q \) resonator therefore rings for many cycles: \( Q \) literally counts radians of oscillation per energy e-fold. Amplitude decays half as fast as energy (rate \( \gamma/2 \)), so the amplitude e-folding time is \( 2/\gamma \) and amplitude falls by \( e \) in \( Q/\pi \) cycles.

D-061

Driven Oscillator: Amplitude, Phase and Resonance

\[ x(t)=A(\omega)\cos(\omega t-\phi),\quad A(\omega)=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+(2\zeta\omega_0\omega)^2}},\quad \phi(\omega)=\arctan\frac{2\zeta\omega_0\omega}{\omega_0^2-\omega^2} \]

The mass settles into oscillation at the driving frequency, not its own natural frequency. The amplitude is large only when the drive is near \(\omega_0\), the sharpness of that peak set by the damping. The motion lags the force: in phase at low frequency (the spring can keep up), a quarter cycle behind at \(\omega_0\) (velocity in phase with force, maximum power absorption), and half a cycle behind at high frequency (inertia-dominated, mass moves opposite to the push). The amplitude peak \(\omega_{\text{res}}=\omega_0\sqrt{1-2\zeta^2}\) sits slightly below \(\omega_0\), whereas the peak of absorbed power sits exactly at \(\omega_0\).

D-062

Power Absorption and the Lorentzian Lineshape

\[ \boxed{\;\langle P\rangle(\omega)=\frac{F_0^2}{2m\gamma}\,\frac{(\gamma/2)^2}{(\omega-\omega_0)^2+(\gamma/2)^2}\;,\qquad \Delta\omega_{\rm FWHM}=\gamma=\frac{\omega_0}{Q}\;} \]

The absorbed power traces a symmetric bell centred on \( \omega_0 \). Its peak \( F_0^2/2m\gamma \) grows as damping falls, while its width \( \gamma \) shrinks in step: a sharper resonance absorbs more strongly but over a narrower band. The half-power points sit at \( \omega=\omega_0\pm\gamma/2 \), so measuring the width and the centre delivers \( Q=\omega_0/\gamma \) directly.

D-033

Coupled Oscillators and Normal Modes

\[ \mathbf{K}\,\mathbf{a}^{(r)}=\omega_r^2\,\mathbf{M}\,\mathbf{a}^{(r)},\qquad \mathbf{q}(t)=\sum_{r=1}^{n}\mathbf{a}^{(r)}\big(A_r\cos\omega_r t+B_r\sin\omega_r t\big),\qquad \big(\mathbf{a}^{(r)}\big)^{\mathsf T}\mathbf{M}\,\mathbf{a}^{(s)}=\delta_{rs} \]

The eigenvalues of the pair \( (\mathbf K,\mathbf M) \) are the squared normal frequencies and the eigenvectors are the normal modes — spatial patterns in which every coordinate moves at the same frequency, passing through equilibrium together. Any free motion is a weighted sum of these modes; the weights (normal coordinates) each obey independent simple harmonic motion, so energy sloshes within a mode but never leaks between modes. The modes form an \( \mathbf M \)-orthonormal basis, guaranteeing the decomposition is unique.

D-063

Normal Coordinates for N Coupled Oscillators

\[ \boxed{\;\mathbf q(t)=\sum_{r=1}^{N}\mathbf a_r\,\eta_r(t),\quad \eta_r(t)=C_r\cos(\omega_r t+\phi_r),\quad (\mathbf K-\omega_r^2\mathbf M)\mathbf a_r=\mathbf 0,\quad \mathbf a_r^{\mathsf T}\mathbf M\,\mathbf a_s=\delta_{rs}\;}\]

Any small motion of the system is a superposition of \(N\) independent normal modes. Mode \(r\) has a fixed spatial pattern \(\mathbf a_r\) (all coordinates in phase or antiphase) oscillating at its own frequency \(\omega_r\); the amplitude \(C_r\) and phase \(\phi_r\) are set by the initial \(\mathbf q\) and \(\dot{\mathbf q}\). The mode shapes are orthonormal not in the ordinary sense but under the mass matrix as metric: the physically meaningful inner product is \(\langle\mathbf x,\mathbf y\rangle_{\mathbf M}=\mathbf x^{\mathsf T}\mathbf M\mathbf y\).

D-064

The Wave Equation as a Continuum Limit

\[ \boxed{\ \partial_t^2 y = c^2\,\partial_x^2 y\,,\qquad c=\sqrt{\frac{T}{\mu}}\ } \]

Every transverse disturbance of a light, uniformly tensioned string travels without change of shape at one speed \(c\), set only by the tension pulling it back and the mass it must accelerate. The general solution is d'Alembert's \(y=f(x-ct)+g(x+ct)\): a right-mover plus a left-mover, each rigid. Tighten the string and waves speed up (\(\propto\sqrt{T}\)); load it more heavily and they slow down (\(\propto 1/\sqrt{\mu}\)). No dispersion means every wavelength travels alike, so a pulse keeps its profile.

D-065

d'Alembert's General Solution

\[ u(x,t)=f(x-ct)+g(x+ct)=\frac{\phi(x-ct)+\phi(x+ct)}{2}+\frac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)\,ds \]

Any 1D wave is exactly two independent, shape-preserving pulses: one sliding right at speed \(c\), one sliding left at speed \(c\). Given a snapshot of shape \(\phi\) and velocity \(\psi\), the displacement at \((x,t)\) is the average of the initial displacement carried in along the two characteristics through \((x,t)\), plus the mean initial velocity over the interval \([x-ct,\,x+ct]\) they enclose — that interval is the domain of dependence.

D-066

Standing Waves and Boundary Quantization

\[ \boxed{\,k_n = \frac{n\pi}{L}, \qquad \omega_n = \frac{n\pi c}{L}, \qquad u_n(x,t)=\sin\!\left(\frac{n\pi x}{L}\right)\bigl[a_n\cos\omega_n t + b_n\sin\omega_n t\bigr],\quad n=1,2,3,\dots\,} \]

Only wavelengths \( \lambda_n = 2\pi/k_n = 2L/n \) fit between the two fixed ends — an integer number of half-wavelengths spans \( L \). Each such spatial pattern oscillates in time at its own frequency \( \omega_n \), an integer multiple of the fundamental \( \omega_1 = \pi c/L \). The general motion is a superposition \( u=\sum_n u_n \), a Fourier sine series whose coefficients are fixed by the initial shape and velocity.

D-067

Fourier Decomposition into Normal Modes

\[ y(x)=\sum_{n=1}^{\infty} b_n\,\sin\!\left(\frac{n\pi x}{L}\right),\qquad b_n=\frac{2}{L}\int_0^L y(x)\,\sin\!\left(\frac{n\pi x}{L}\right)dx \]

The string's shape is a weighted sum of standing modes, and the weight of each mode is its overlap with the shape, measured by the inner product and normalised by the mode's own norm \(L/2\). A mode "hears" only the part of \(y(x)\) that shares its spatial rhythm; everything orthogonal to it is invisible to the projection. Nothing about the shape is lost — the coefficients carry complete information.

D-068

Energy Density and Power Transport

\[ \langle u\rangle=\frac12\,\mu\,\omega^{2}A^{2},\qquad \langle P\rangle=\frac12\,\mu v\,\omega^{2}A^{2}=\frac12\sqrt{\mu T}\,\omega^{2}A^{2}=v\,\langle u\rangle. \]

A travelling wave stores, per unit length, an average energy \(\tfrac12\mu\omega^{2}A^{2}\) split equally between kinetic and potential form, and carries that energy forward at the wave speed \(v\). Both the stored energy and the transported power grow as the square of the amplitude and as the square of the frequency, so frequency is the stronger lever. The transported power is the energy density times the speed at which it moves, the universal density-times-velocity flux law.

D-069

Reflection, Transmission and Impedance Matching

\[ r=\frac{Z_1-Z_2}{Z_1+Z_2},\qquad \tau=\frac{2Z_1}{Z_1+Z_2},\qquad R=r^2,\qquad \mathcal{T}=\frac{4Z_1Z_2}{(Z_1+Z_2)^2},\qquad R+\mathcal{T}=1 \]

The amplitude split depends only on the impedance mismatch \(Z_1-Z_2\) measured against the total \(Z_1+Z_2\). A wave entering a heavier (higher-\(Z\)) medium reflects with a sign flip (\(r<0\), a half-cycle phase shift); entering a lighter medium it reflects in phase (\(r>0\)). The transmitted amplitude can be as large as twice the incident (\(\tau\to2\) as \(Z_2\to0\)) yet no energy paradox arises, because the power that flows is weighted by \(Z\), and \(\mathcal{T}\) never exceeds one.

D-070

Longitudinal Sound Waves in a Fluid

\[ \boxed{\,c=\sqrt{\dfrac{\gamma P_0}{\rho_0}}=\sqrt{\dfrac{\gamma R T}{M}}\,} \]

The speed of sound is set by the fluid's stiffness against compression (the adiabatic bulk modulus \(\gamma P_0\)) divided by its inertia (density \(\rho_0\)). A stiffer or lighter gas carries sound faster. Using the ideal-gas law \(P_0=\rho_0 RT/M\), the pressure and density dependence cancels: at fixed temperature \(c\) depends only on \(\gamma\) and the molar mass \(M\), not on pressure — which is why the sound speed in air is essentially altitude-independent at fixed temperature.

D-071

Dispersion, Phase Velocity and Group Velocity

\[ v_p=\frac{\omega}{k},\qquad v_g=\frac{d\omega}{dk}=v_p+k\,\frac{dv_p}{dk} \]

The phase velocity is the speed of an individual crest and is set pointwise by the ratio \(\omega/k\). The group velocity is the speed of the packet's envelope and is set by the local slope of the dispersion curve. They agree only when that curve is a straight line through the origin (\(\omega=ck\)); every departure from proportionality — every real medium — makes them differ, and \(v_g\) is the one that transports amplitude, energy, and information to leading order.

D-072

Wave Packets, Group Velocity and Spreading

\[ \boxed{\ v_g=\left.\frac{d\omega}{dk}\right|_{k_0}\ ,\qquad \Delta x(t)=\Delta x_0\sqrt{1+\left(\frac{\beta\,t}{2\,\Delta x_0^{\,2}}\right)^{2}}\ ,\qquad \beta=\left.\frac{d^2\omega}{dk^2}\right|_{k_0}\ } \]

The centre of the packet moves at the group velocity \(v_g\), the slope of the dispersion curve — not the phase velocity \(\omega/k\). The curvature \(\beta\) of that same curve makes the packet spread: high- and low-\(k\) components travel at slightly different group speeds, so an initially compact group unavoidably widens, symmetrically and quadratically in time at late times. A narrower initial packet (small \(\Delta x_0\)) spreads faster, because it needs a broader band of \(k\) to build and those components disperse more.

D-073

The Doppler Effect and Shock Fronts

\[ f_o = f_s\,\frac{c-\vec{v}_o\cdot\hat{n}}{c-\vec{v}_s\cdot\hat{n}}\,,\qquad\qquad \sin\mu=\frac{c}{v_s}=\frac{1}{M}\ \ (M>1) \]

The observed frequency is the emitted frequency scaled by the closing speed of the crests over their spacing in the medium. The numerator raises the pitch when the observer moves into the waves; the denominator raises it when the source chases its own sound. Only line-of-sight components count. Source and observer enter asymmetrically — only the source term can drive the wavelength to zero — so equal speeds of the two do not cancel. Once \(v_s\) reaches \(c\) (\(M=1\)) the forward formula fails and the fronts organize into a cone whose half-angle \(\mu\) narrows as the source goes faster; measuring \(\mu\) alone returns \(M\).

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