Cheat sheet

Every key quantitative relation in the book, in canonical order, at a glance. Each formula links to the lesson that develops it.

2 — The body before the ear

\Delta L \approx r(\theta + \sin\theta)

\text{ITD} = \Delta L / c

f_\text{res} = \dfrac{c}{4L}

f_n = (2n-1)\,\dfrac{c}{4L}

f_n = n\,\dfrac{c}{2L}

Z = p/v

R = \dfrac{Z_2 - Z_1}{Z_1 + Z_2}

T = 1 - \lvert R\rvert^2

Z = \rho c

\dfrac{p_\text{OW}}{p_\text{TM}} = \dfrac{A_\text{TM}}{A_\text{footplate}} \approx 17

G_p = \dfrac{A_\text{TM}}{A_\text{footplate}}\cdot\dfrac{L_\text{malleus}}{L_\text{incus}} \approx 22

20\log_{10}(22) \approx 27\,\text{dB}

m\ddot\eta + b\dot\eta + k\eta = P(x,t)

\omega_0(x) = \sqrt{k(x)/m(x)}

\lvert H(\omega)\rvert = \dfrac{1}{m\sqrt{(\omega_0^2-\omega^2)^2 + \gamma^2\omega^2}}

\phi(\omega) = -\arctan\dfrac{\gamma\omega}{\omega_0^2-\omega^2}

Q = \omega_0/\gamma, width \approx \omega_0/Q

Z_\text{BM} = b + i\!\left(\omega m - \dfrac{k}{\omega}\right)

\dfrac{d^2P}{dx^2} + \kappa^2 P = 0

\kappa^2 = \dfrac{2i\omega\rho}{A\,Z_\text{BM}}

P(x) \approx \dfrac{P_0}{\sqrt{\kappa}}\exp\!\left(i\!\int_0^x \kappa\,dx'\right)

f(x) = A\bigl(10^{\alpha(1-x/L)} - K\bigr)

A = 165.4\,\text{Hz},\ \alpha = 2.1,\ K = 0.88

m\ddot\eta + (b-\beta)\dot\eta + k\eta = P,\ \ b_\text{eff} = b - \beta

P_\text{open}(x) = \dfrac{1}{1 + \exp[-(x - x_0)/\kappa]}

\text{VS} = \left\lvert \dfrac{1}{N}\sum_{i=1}^{N} e^{i\phi_i} \right\rvert

P(M\mid S) = \dfrac{P(S\mid M)\,P(M)}{P(S)}

\Delta\theta_i \propto \pi_i\bigl(\theta_{i-1} - g(\theta_i)\bigr) - \pi_{i+1}\bigl(\partial g/\partial\theta_i\bigr)\bigl(\theta_i - g(\theta_{i+1})\bigr)

F = \mathbb{E}_q[\log q(\theta)] - \mathbb{E}_q[\log p(\theta,S)] = \mathrm{KL}\bigl(q(\theta)\,\Vert\,p(\theta\mid S)\bigr) - \log p(S)

V_\text{Nernst} = \dfrac{RT}{zF}\ln\dfrac{c^\text{out}}{c^\text{in}}

V_\text{rest} = \dfrac{RT}{F}\ln\dfrac{P_K c_K^\text{out} + P_\text{Na} c_\text{Na}^\text{out} + P_\text{Cl} c_\text{Cl}^\text{in}}{P_K c_K^\text{in} + P_\text{Na} c_\text{Na}^\text{in} + P_\text{Cl} c_\text{Cl}^\text{out}}

\Delta V_\text{MET} = V_\text{endolymph} - V_\text{cytoplasm}

P_\text{open}(x) = \dfrac{1}{1 + e^{(\Delta G_0 - K_\text{TL}\,d\,x)/k_B T}}