Fix a positive normal from medium 1 toward medium 2, and define all velocity amplitudes in that coordinate. A right-going linear sound wave satisfies $p^{\prime }=Z{u}^{\prime }$ and a left-going wave satisfies $p^{\prime }=-Z{u}^{\prime }$, with acoustic impedance $Z=\rho c_s$. Continuity gives $u_i+u_r=u_t$ and $Z_1(u_i-u_r)=Z_2{u}_t$. Eliminating $u_t$ gives the displayed reflected-to-incident velocity ratio. The pressure reflection coefficient is its negative. A surface mass, surface restoring force or oblique incidence changes the boundary calculation.

The radiative efficiency is the escaping radiative power divided by an adopted accreted rest-mass energy rate. Its meaning depends on where the mass rate is measured and which emission is counted. For efficient Newtonian accretion onto a stellar surface it scales as $GM/(R_{\ast }{c}^2)$, whereas advection or outflows can reduce luminosity for the same supplied mass.

Integrating $L_{\mathrm{acc}}=GM\dot{M}/r_0$ at fixed radius gives this result. With negligible seed mass it supplies the radiative requirement of monatomic cloud collapse only if ${\Omega }_f=-G{M}^2/r_0$ under the adopted zero-initial-energy approximation. A different density-profile coefficient, retained accretion heat, a wind or a changing radius modifies the budget. Positive binding-energy magnitude and negative potential energy must not be interchanged silently.

A fossil acanthodian whose preserved braincase informs early jawed-vertebrate comparisons.

The 2-adic metric is $d_2(x,y)=|x-y{|}_2$, where $|x{|}_2=2^{-v_2(x)}$ for $x\ne 0$ and $|0{|}_2=0$. Closeness means divisibility of the difference by a large power of two.

For uniform coherent plane-wave illumination, an ideal binary amplitude zone plate transmits alternate Fresnel half-period zones. If $N_o$ complete odd zones are open, every one contributes approximately twice the unobstructed axial complex amplitude, in the same phase, and $I_f/I_0\simeq 4N_o^2$. This is a local light intensity gain relative to removing the plate and outer stop. It differs from comparison with a finite clear aperture: an aperture containing an odd integer number of half-period zones has axial intensity $4I_0$, giving plate-to-aperture ratio $N_o^2$. Neither gain is a total power efficiency.

For the separable square-lattice potential $V=-V_0(\cos Qx+\cos Qy)$, four free plane waves with $(k_x,k_y)=(\pm Q/2,\pm Q/2)$ meet at a Brillouin zone corner. Their common unperturbed energy is $E_Q/2$, where $E_Q={\hslash }^2{Q}^2/(2m)$.

Each cosine couples the opposite signs of its own momentum component with matrix element $-V_0/2$. In the four-state space the perturbation is $-(V_0/2)({\sigma }_x\otimes I+I\otimes {\sigma }_x)$, with eigenvalues $-V_0,0,0,V_0$. Thus the corner energies at leading order are $E_Q/2-V_0$, twice-degenerate $E_Q/2$, and $E_Q/2+V_0$. Additional Fourier components or coupling to remote waves can change this result. A local corner splitting alone does not establish a global insulating gap.

The physical Higgs boson interaction $\mathcal{L}\supset -(m_e/v)h\bar{e}e$ after electroweak symmetry breaking. It is distinguished from the unbroken-phase electron Yukawa coupling $y_e=\sqrt{2}{m}_e/v$. This small coefficient explains the electron Yukawa suppression of Higgs production in direct electron–positron annihilation.

A small arc of a rotating thin hoop needs inward force per length $\rho A{\omega }^{2}r$. Its circumferential tension supplies this through curvature, giving tension $\rho A{\omega }^2{r}^2$ and tensile stress $\rho {\omega }^2{r}^2$. Hooke's law converts that stress to circumferential strain, which equals fractional radius change. Small strain, a slender cross-section and uniaxial linear elasticity are required; the cross-sectional area cancels at leading order.

Combining reversible work of adhesion per area, $W={\gamma }_{SV}+{\gamma }_{LV}-{\gamma }_{SL}$, with Young's equation gives the displayed relation. It assumes the same smooth equilibrium interfaces; contact-angle hysteresis and dissipation are not included.

Subtracting Young's equations for two liquids on the same solid eliminates the unknown solid–vapour interfacial energy. The result assumes equilibrium angles and an unchanged solid surface. Swelling, contamination or contact-angle hysteresis can invalidate this cancellation as a comparison of equilibrium material properties.