13  Formulae for Oscillations and Waves

The list of formulae below is the standard set of formulae made available for assessment.

13.1 Physical Constants

Constant	Value
Acceleration due to gravity	\(g = 9.81 \mathrm{\,m\,s}^{-2}\)
Speed of light	\(c = 2.99 \times 10^8 \mathrm{\,m\,s}^{-2}\)
Ideal gas constant	\(R = 8.31 \mathrm{\,J\,K}^{-1}\mathrm{\,mol}^{-1}\)
Boltzmann constant	\(k_B = 1.38 \times 10^{-23} \mathrm{\,J\,K}^{-1}\)
Avogadro constant	\(N_A = 6.02 \times10^{23} \mathrm{\,mol}^{-1}\)
Stefan-Boltzmann constant	\(\sigma = 5.67 \times10^{-8} \mathrm{\,W\,m}^{-2}\mathrm{\,K}^{-4}\)
Wien displacement constant	\(b = 2.897 \times 10^{-3} \mathrm{\,m\,K}\)
Reduced Planck constant	\(\hbar = 1.05 \times 10^{-34} \mathrm{\,m}^2 \mathrm{\,kg\,s}^{-1}\)
Planck constant	\(h = 6.63\times 10^{-34} \mathrm{\,m}^2 \mathrm{\,kg\,s}^{-1}\)
Electron volt	\(1\,eV = 1.60 \times 10^{-19} \mathrm{\,J}\)
Atomic mass unit (Dalton)	\(1\,u = 1.66\times 10^{-27} \mathrm{\,kg}\)

13.2 Trigonometric Relationships

\(\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B\)

\(\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1\mp \tan A \tan B}\)

\(\cos (A \pm B)=\cos A \cos B \mp \sin A \sin B\)

\(\cos^2 A + \sin^2 A = 1\)

\(\sin 2A = 2 \sin A \cos A\)

\(\cos A \cos B = \frac{1}{2} \left[ \cos(A+B) + \cos (A - B) \right]\)

\(\cos 2A = \cos^2 A - \sin^2 A\)

\(\sin A \cos B = \frac{1}{2} \left[ \sin(A+B) + \sin(A-B) \right]\)

\(\tan 2A = \dfrac{2 \tan A}{1 - \tan^2 A}\)

\(\sin A + \sin B = 2 \sin \dfrac{A+B}{2} \cos \dfrac{A-B}{2}\)

\(\cos A + \cos B = 2 \cos \dfrac{A+B}{2} \cos \dfrac{A-B}{2}\)

\(\sin A - \sin B = 2 \cos \dfrac{A+B}{2} \sin \dfrac{A-B}{2}\)

\(\cos A - \cos B = 2 \sin \dfrac{A+B}{2} \sin \dfrac{B-A}{2}\)

\(\cos^2 A = \frac{1}{2} (1+\cos 2A)\)

\(\sin^2 A = \frac{1}{2} (1-\cos 2A)\)

13.3 Oscillations

Undamped SHM: \(\dfrac{\textrm{d}^2 x}{\textrm{d} t^2} = -\omega^2 x\)

Solution: \(x = A_0 \cos (\omega t + \delta)\)

Energy: \(E_{\mathrm{total}} = \frac{1}{2} k A_0^2\)

Simple Pendulum \(T = 2\pi \sqrt{\frac{l}{g}}\)

Damped SHM: \(m \dfrac{\mathrm{d}^2 x}{\mathrm{d}t^2} + b \dfrac{\mathrm{d}x}{\mathrm{d}t}+ kx =0\)

Damped SHM Solution:

• Position \(x = A_0 e^{-\frac{t}{2\tau}} e^{i(\omega' t + \delta)}\)
• Characteristic decay constant \(\tau = \frac{m}{b}\)
• Damped frequency \(\omega' = \sqrt{\omega_0^2 - (b/2m)^2}\)

Critical damping: \(b = 2m\omega_0\)

Quality factor: \(Q = \omega_0 \tau = \dfrac{2\pi}{ \left( |\Delta E | / E \right)}\)

Forced SHM: \(m \dfrac{\mathrm{d}^2 x}{\mathrm{d}t^2} + b \dfrac{\mathrm{d}x}{\mathrm{d}t}+ kx = F_0 \mathrm{e}^{\mathrm{i}\omega t}\)

Solution:

• Position \(x = A_0 \mathrm{e}^{- \left(\frac{b}{2m}\right)t} \mathrm{e}^{\mathrm{i} \left( \omega^\prime t + \delta^\prime \right)} + A e^{i(\omega t - \delta)}\)

• where \(\delta = \arctan \left( \dfrac{b\omega}{m(\omega_0^2 - \omega^2)}\right)\)

Steady-state amplitude: \(A = \dfrac{F_0}{\sqrt{m^2 \left(\omega_0^2 - \omega^2 \right)^2 + b^2 \omega^2}}\)

Width (at half height) of resonance peak (\(\Delta \omega\)): \(\dfrac{\Delta \omega}{\omega_0} = \dfrac{1}{Q}\)

Coupled oscillator (\(n = 2\)):

\[ \begin{cases} \begin{array}{l} m_A \dfrac{\mathrm{d}^2 x_A}{\mathrm{d}t^2} = -(k_A + k_{AB}) x_A + k_{AB}x_B \\ m_B \dfrac{\mathrm{d}^2 x_B}{\mathrm{d}t^2} = -(k_B + k_{AB}) x_B + k_{AB}x_A \end{array} \end{cases} \]

Coupled Oscillator Solutions:

\[\begin{cases} x_1 = \frac{1}{2} B_1 \cos(\omega_1 t + \phi_1) + \frac{1}{2}B_2 \cos (\omega_2 t + \phi_2)\\ x_2 = \frac{1}{2} B_1 \cos(\omega_1 t + \phi_1) - \frac{1}{2}B_2 \cos (\omega_2 t + \phi_2)\\ \end{cases}\]

For \(m_A = m_B = m\) and \(k_A = k_B = k\):

\(\omega_1 = \sqrt{\dfrac{k}{m}}\) and \(\omega_2 = \sqrt{\dfrac{k + 2 k_{AB}}{m}}\)

Matrix representation:

\[ \left( \begin{array}{cc} m_A & 0 \\ 0 & m_B \end{array} \right) \left( \begin{array}{c} \frac{\mathrm{d}^2 x_A}{\mathrm{d}t^2} \\ \frac{\mathrm{d}^2 x_B}{\mathrm{d}t^2} \end{array} \right) = - \left( \begin{array}{cc} (k_A + k_{AB}) & -k_{AB} \\ -k_{AB} & (k_B + k_{AB}) \end{array} \right) \left( \begin{array}{c} x_A \\ x_B \end{array} \right) \]

13.4 Waves

Wave Equation: \(\dfrac{\partial^2 y}{\partial x^2} = \dfrac{\mu}{F} \dfrac{\partial^2 y}{\partial t^2}\)

Harmonic wave:

• Sinusoidal representation: \(y = A \sin (kx - \omega t)\)
• Complex representation \(y = Ae^{i(kx - \omega t)}\)
• De Moivres representation \(Ae^{i\omega t} = A(\cos \omega t + i \sin \omega t)\)

Phase Velocity: \(v_p = \dfrac{\omega}{k}\)

Group velocity: \(v_g = \dfrac{\partial \omega(k)}{\partial k}\)

For wave on a string:

• Phase velocity: \(v_p = \frac{\omega}{k} = \sqrt{F/\mu}\)
  
• Energy density: \(\epsilon = \frac{1}{2} \mu \omega^2 A^2\)
• Average power transmitted: \(P_{\mathrm{av}} = \frac{1}{2}\mu \omega^2 A^2 v\)
• Impedance: \(Z = \frac{Fk}{\omega}\)

Reflection and Transmission at Boundaries:

• \(\dfrac{\mathrm{Reflected\,power}}{\mathrm{Incident\,power}} = \left( \dfrac{Z_1 - Z_2}{Z_1 + Z_2} \right)^2\)

• \(\dfrac{\mathrm{Transmitted\,power}}{\mathrm{Incident\,power}} = \dfrac{4 Z_1 Z_2}{(Z_1 + Z_2)^2}\)

Intensity level: \(\beta = 10 \log_{10} \left( \dfrac{I}{I_0}\right)\)

• …where \(I_0 = 10^{-12} \mathrm{\,W\,m}^{-2}\)

Doppler shift: \(f^\prime = \left(\dfrac{v\pm u_r}{v \pm u_s} \right) f_0\)

Beats: \(y = 2y_0 \cos \left( \dfrac{\Delta kx - \Delta \omega t}{2} \right) \sin (k_{\mathrm{av}} x - \omega_{\mathrm{av}} t)\)