OpenStax University Physics/V1/Equations (master)

Equations (master) | Formulas (master) | Equations | Formulas | College Physics

Equations inspired by the Chapter Summaries of OpenStax University Physics Volume 1. Instructors who wish to base their course notes on Wikiversity should not use this version, but instead copy this much more user-friendly that contains easily understood transclusions to this "master". A four-page summary suitable for use during in-class exams is available in two different versions: "master" (online viewing) and "compact". The "compact" version is also available in this pdf form.

 Introduction

 

Units_and_Measurement
The base SI units are mass: kg (kilogram); length: m (meter); time: s (second). Percent error is $$(\delta A/A)\times 100%$$  

Vectors
Vector $$\vec A = A_x \, \hat i+ A_y \, \hat j + A_z \, \hat k$$ involves components (Ax,Ay,Az) and three orthonormal unit vectors. &#9645; If $$\vec A + \vec B =\vec C$$, then Ax+Bx=Cx, etc, and vector subtraction is defined by $$\vec B =\vec C - \vec A$$.

&#9645; The two-dimensional displacement from the origin is $$\vec r = x\hat i + y\hat j$$. The magnitude is $$ A \equiv |\vec A|= \sqrt{A_x^2 + A_y^2}$$. The angle (phase) is $$\theta=\tan^{-1}{(y/x)} $$.

&#9645; Scalar multiplication $$\alpha\vec A = \alpha A_x \hat i     +  \alpha A_y \hat j+...\quad$$

&#9645; Any vector divided by its magnitude is a unit vector and has unit magnitude: $$|\hat V| = 1$$ where $$\hat V \equiv \vec V/V$$

&#9645; Dot product $$\vec A\cdot\vec B = AB\cos\theta = A_xB_x+A_yB_y+...\quad$$ and $$\vec A\cdot\vec A = A^2$$

&#9645; Cross product $$\vec A = \vec B \times \vec C\Rightarrow$$ $$A_\alpha = B_\beta C_\gamma -C_\gamma A_\beta$$ where $$(\alpha,\beta,\gamma)$$ is any cyclic permutation of $$(x,y,z)$$, i.e., (&alpha;,&beta;,&gamma;) represents either (x,y,z) or (y,z,x) or (z,x,y).

&#9645; Cross-product magnitudes obey $$A=BC\sin\theta$$  where $$\theta$$ is the angle between $$\vec B$$ and $$\vec C$$,  and $$\vec A\perp\{\vec B,\vec C\}$$ by the right hand rule.

&#9645; Vector identities $$ \;c (\mathbf{A}+\mathbf{B})=c\mathbf{A}+c\mathbf{B}\quad$$

&#9645; $$ \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}\quad$$

&#9645; $$ \mathbf{A}+(\mathbf{B}+\mathbf{C})= (\mathbf{A}+\mathbf{B})+\mathbf{C}\quad$$

&#9645; $$ \mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A}\quad$$

&#9645; $$ \mathbf{A}\times\mathbf{B}=\mathbf{-B}\times\mathbf{A}\quad$$

&#9645; $$ \left(\mathbf{A}+\mathbf{B}\right)\cdot\mathbf{C}=\mathbf{A}\cdot\mathbf{C} +\mathbf{B}\cdot\mathbf{C} \quad$$

&#9645; $$ \left(\mathbf{A}+\mathbf{B}\right)\times\mathbf{C}=\mathbf{A}\times\mathbf{C} +\mathbf{B}\times\mathbf{C} \quad$$

&#9645; $$\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)= \mathbf{B}\cdot\left(\mathbf{C}\times\mathbf{A}\right)=\left(\mathbf{A}\times\mathbf{B}\right)\cdot\mathbf{C}\quad$$

&#9645; $$ \mathbf{A\times} \left(\mathbf{B}\times\mathbf{C}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right) \mathbf{B}-\left(\mathbf{A}\cdot\mathbf{B}\right)\mathbf{C}\quad$$

&#9645; $$ \mathbf{\left(A\times B\right)\cdot}\left(\mathbf{C} \times\mathbf{D}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right) \left(\mathbf{B}\cdot\mathbf{D}\right)- \left(\mathbf{B}\cdot\mathbf{C}\right)\left(\mathbf{A}\cdot\mathbf{D}\right)$$  

Motion_Along_a_Straight_Line
Delta as difference $$\Delta x =x_f-x_i \rightarrow dx\rightarrow 0$$ in limit of differential calculus.

&#9645; Average velocity $$\bar v = \Delta x /\Delta t  \rightarrow v=dx/dt$$ (instantaneous velocity)

&#9645; Acceleration $$\bar a = \Delta v/\Delta t \rightarrow a=dv/dt$$.

&#9645; WLOG set $$\Delta t=t\;$$ and $$\Delta x = x-x_0\;$$ if $$t_i=0$$. Then $$\Delta v = v-v_0$$, and $$\; v(t) = \int_0^t a(t')dt' + v_0 $$, $$\;x(t) = \int_0^t v(t')dt' + x_0 = x_0 + \bar v t $$, where $$\bar v = \frac 1 t \int_0^t v(t')dt'$$ is the average velocity.

&#9645; At constant acceleration: $$\bar v = \tfrac{v_0+v}{2},\quad v=v_0+at,\quad x=x_0+ v_0t+\tfrac 1 2 at^2,\,$$ $$ v^2=v_0^2+2a\Delta x$$.

&#9645; For free fall, replace $$x\rightarrow y$$ (positive up) and $$a\rightarrow -g$$, where $$g$$ = 9.81 m/s2 at Earth's surface).  

Motion_in_Two_and_Three_Dimensions
Instantaneous velocity: $$\vec v(t)=v_x(t)\hat i + v_y(t)\hat j + v_z(t)\hat k\; = \frac{dx}{dt}\hat i + \frac{dy}{dt}\hat j + \frac{dz}{dt}\hat k\;$$ $$= \lim_{\Delta t\rightarrow 0}{\tfrac{\Delta \vec r}{\Delta t}} =\lim_{\Delta t\rightarrow 0}{\tfrac{\vec r(t+\Delta t)-\vec r(t)}{\Delta t}}$$, where $$\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k$$ &#9645; Acceleration $$\vec a = a_x\hat i + a_y\hat j + a_z\hat k$$, where $$a_x(t)=dv_x/dt=d^2x/dt^2$$.

&#9645; Average values: $$\vec v_{ave} = \tfrac{\Delta\vec r}{\Delta t}=\tfrac{\vec r(t_2)-\vec r(t_2)}{t_2-t_1} $$, and $$\;\vec a_{ave} = \tfrac{\Delta\vec v}{\Delta t}= \tfrac{\vec v(t_2)-\vec v(t_2)}{t_2-t_1}$$

&#9645; Free fall time of flight $$\,T_{of}= \tfrac{2(v_0\sin\theta_0)}{g}\,,\,$$ &#9645; Trajectory $$ y=(\tan\theta_0)x - \left[\tfrac{g}{2(v_0\cos\theta_0)^2} \right]x^2\,,\,$$ &#9645; Range $$ R=\tfrac{v_0^2\sin2\theta_0}{g}$$

&#9645; Uniform circular motion: $$|\vec a|=a_C=\omega^2r = v^2/r$$ where $$v\equiv|\vec v|=\omega r$$
 * $$\vec r = A\cos\omega t\hat i + A\sin\omega t\hat j \,,\,$$ $$

\vec v = -A\omega\sin\omega t\hat i + A\omega\cos\omega t\hat j \,,\,$$ $$ \vec a = -A\omega^2\cos\omega t\hat i - A\omega^2\sin\omega t\hat j\,. $$

&#9645; Tangential and centripetal acceleration $$\vec a = \vec a_c + \vec a_T$$ where $$a_T=d|\vec v|/dt$$.

&#9645; Relative motion: $$\,\vec r_{PS}=\vec r_{PS'}+\vec r_{S'S}$$, $$\,\vec v_{PS}=\vec v_{PS'}+\vec v_{S'S}$$, $$\,\vec v_{PC}=\vec v_{PA}+\vec v_{AB}+\vec v_{BC}$$, $$\,\vec a_{PS}=\vec a_{PS'}+\vec a_{S'S}$$  

Newton's_Laws_of_Motion
Newton's 2nd Law $$\;m\vec a = d\vec p/dt =\sum \vec F_j$$, where $$\vec p =m\vec v$$ is momentum, $$m$$ is mass, and  $$\sum \vec F_j$$ is the sum of all forces  This sum  needs only include external forces because all internal forces cancel by the 3rd law  $$\vec F_{AB}=-\vec F_{BA}$$. The 1st law is that velocity is constant if the net force is zero. &#9645; Weight$$=\vec w=m\vec g$$.

&#9645; normal force is a component of the contact force by the surface. If the only forces are contact and weight, $$|\vec N|=N=mg\cos\theta$$ where $$\theta$$ is the angle of incline.

&#9645; Hooke's law $$F=-kx$$ where $$k$$ is the spring constant. 



Applications_of_Newton's_Laws
$$f_s\le\mu_sN \text{ and } f_k=\mu_kN$$: $$\,f=$$ friction, $$\mu_{s,k}=$$ coefficient of (static,kinetic) friction, $$N=$$ normal force. &#9645; Centripetal force$$F_c=m v^2/r=mr\omega^2$$ for uniform circular motion. Angular velocity $$\omega$$ is measured in radians per second.

&#9645; Ideal angle of banked curve: $$\tan\theta=v^2/(rg)$$ for curve of radius $$r$$ banked at angle $$\theta$$.

&#9645; Drag equation $$F_D=\tfrac 1 2 C \rho A v^2$$ where $$C=$$ Drag coefficient, $$\rho=$$ mass density, $$A=$$ area, $$v=$$ speed. Holds approximately for large Reynold's number $$=\mathrm{Re} = \rho v L/ \eta$$, where $$\eta =$$dynamic viscosity; $$L=$$ characteristic length.

&#9645; Stokes's law models a sphere of radius $$r$$ at small Reynold's number: $$F_s=6\pi r\eta v$$.  

Work_and_Kinetic_Energy
Infinitesimal work done by force: $$ dW=\vec F\cdot d\vec r = |\vec F|\,|d\vec r|\cos\theta$$ leads to the path integral $$W_{AB}=\int_A^B\vec F\cdot d\vec r$$ &#9645; Work done from A&rarr;B by friction $$-f_k|\ell_{AB}|,\;$$gravity $$-mg(y_B-y_A),\;$$ and spring $$-\tfrac 1 2 k\left(x_B^2-x_A^2\right)$$

&#9645; Work-energy theorem: The work done on a particle is $$W_{net}=K_B-K_A$$ where kinetic energy $$=K=\tfrac 1 2 mv^2 = \frac{p^2}{2m}$$.

&#9645; Power$$=P=dW/dt=\vec F\cdot \vec v$$.  <section begin=Potential_Energy_and_Conservation_of_Energy/>

Potential_Energy_and_Conservation_of_Energy
Potential Energy: $$\Delta U_{AB} = U_B-U_A=-W_{AB}$$; PE at $$\vec r$$ WRT  $$\vec r_0$$ is $$\Delta U = U(\vec r)-U(\vec r_0)$$ $$U=mgy+\mathcal{C}$$ (gravitational PE Earth's surface. $$U=\tfrac 1 2 k x^2+\mathcal{C}$$ (ideal spring)

&#9645; Conservative force: $$\oint\vec F_{\text{cons}}\cdot d\vec r = 0$$. In 2D, $$\vec F(x,y)$$ is conservative if and only if $$ \vec F = -(\partial U/\partial x) \,\hat i -(\partial U/\partial y) \,\hat j \iff \partial F_x/\partial y = \partial F_y/\partial x$$

&#9645; Mechanical energy is conserved if no non-conservative forces are present: $$0=W_{nc,AB}=\Delta (K+U)_{AB} =\Delta E_{AB}$$ <section end=Potential_Energy_and_Conservation_of_Energy/>

<section begin=Linear_Momentum_and_Collisions/>

Linear_Momentum_and_Collisions
$$\vec F(t) = d\vec p/dt \text{, where } \vec p=m\vec v$$ is momentum. &#9645; Impulse-momentum theorem $$\vec J = F_{ave}\Delta t = \int_{t_i}^{t_f} \vec F dt = \Delta\vec p$$.

&#9645; For 2 particles in 2D $$\text{If } \vec F_{ext}=0 \text{ then } \sum_{j=1}^N\vec p_j =0 \Rightarrow p_{f,\alpha}=p_{1,i,\alpha}+p_{2,i,\alpha}$$ where (&alpha;,&beta;)=(x,y)

&#9645; Center of mass: $$\vec r_{CM}= \tfrac 1 M \sum_{j=1}^N m_j\vec r_j \rightarrow \tfrac 1 M \int\vec r dm, $$ $$\vec v_{CM}= \tfrac{\,d}{dt}\vec r_{CM}$$, and $$\vec p_{CM}=\sum_{j=1}^N m_j\vec v_j =M\vec v_{CM}.$$

&#9645; $$\vec F = \tfrac{\,d}{dt}\vec p_{CM}=m\vec a_{CM}=\sum_{j=1}^Nm_j\vec a_j$$

&#9645; Rocket equation $$mdv= -udm\Rightarrow \Delta v = u\ln(m_f/m_i) $$ where u is the gas speed WRT the rocket. <section end=Linear_Momentum_and_Collisions/> <section begin=Fixed-Axis_Rotation/>

Fixed-Axis_Rotation
$$\theta=s/r\,$$ is angle in radians,$$\,\omega=d\theta/dt\,$$ is angular velocity;

&#9645; $$\,v_t=\omega r=ds/dt\,$$ is tangential speed. Angular acceleration is $$\alpha=d\omega/dt=d^2\theta/dt^2\,$$. $$a_t=\alpha r= d^2s/dt^2\,$$ is the tangential acceleration.

&#9645; Constant angular acceleration $$\bar\omega=\tfrac 1 2 (\omega_0+\omega_f)\,$$ is average angular velocity.

&#9645; $$\theta_f=\theta_0+\bar\omega t = \theta_0+\omega_0t+\tfrac 1 2 \alpha t^2\,.$$

&#9645; $$\omega_f=\omega_0+\alpha t.\,$$ $$\omega_f^2=\omega_0^2 + 2\alpha\Delta\theta\,.$$

&#9645; Total acceleration is centripetal plus tangential: $$\vec a = \vec a_c + \vec a_t.\,$$

&#9645; Rotational kinetic energy is $$K=\tfrac 1 2 I\omega^2,\,$$ where $$I =\sum_j m_j r_j^2\rightarrow\int r^2 dm$$ is the Moment of inertia.

&#9645; parallel axis theorem $$I_{parallel-axis}=I_{center\,of\,mass}+md^2$$

&#9645; Restricting ourselves to fixed axis rotation, $$r$$ is the distance from a fixed axis; the sum of torques, $$\vec\tau=\vec r\times\vec F$$ requires only one component, summed as $$\tau_{net}=\sum\tau_j=\sum r_{\perp_j}F_j=I\alpha$$.

&#9645; Work done by a torque is $$dW=\left(\sum\tau_j\right)d\theta$$. The Work-energy theorem is $$K_B-K_A=W_{AB}=\int_{\theta_A}^{\theta_B}\left(\sum_j\tau_j\right)d\theta$$.

&#9645; Rotational power $$=P=\tau\omega$$. <section end=Fixed-Axis_Rotation/> <section begin=Angular_Momentum/>

Angular_Momentum
Center of mass (rolling without slip) $$d_{CM}= r\theta,\;$$ $$v_{CM}= r\omega,\;$$$$a_{MC}=R\alpha= \tfrac{mg\sin\theta/}{m+\left(I_{cm}/r^2\right)} $$ &#9645; Total angular momentum and net torque: $$d\vec L/dt = \sum \vec\tau $$ $$= \vec l_1+\vec l_2 + ...;$$ $$\vec l = \vec r\times\vec p\,$$ for a single particle. $$L_{total}=I\omega.$$

&#9645; Precession of a top $$\omega_P=mrg/(I\omega).$$ <section end=Angular_Momentum/> <section begin=Static_Equilibrium_and_Elasticity/>

Static_Equilibrium_and_Elasticity
Equilibrium $$\sum \vec F_j=0=\sum\vec \tau_j.\,$$ Stress = elastic modulus &middot; strain (analogous to Force = k &middot; &Delta; x ) &#9645; (Young's, Bulk , Shear) modulus: $$\left( \tfrac{F_\perp}{A}=Y\cdot\tfrac{\Delta L}{L_0}\,,\; \Delta p          =B\cdot\tfrac{-\Delta V}{V_0}\,,\; \tfrac{F_\parallel}{A}=S\cdot\tfrac{\Delta x}{L_0}\right) $$ <section end=Static_Equilibrium_and_Elasticity/> <section begin=Gravitation/>

Gravitation
Newton's law of gravity $$\vec F_{12}=G\tfrac{m_1m_2}{r^2}\hat r_{12}$$ &#9645; Earth's gravity $$g=G\tfrac{M_E}{r^2}$$

&#9645; Gravitational PE beyond Earth $$U =-G\tfrac{M_Em}{r}$$

&#9645; Energy conservation $$\tfrac 1 2 mv_1^2-G\tfrac{Mm} {r_1}=\tfrac 1 2 mv_2^2-G\tfrac{Mm}{r_2}$$

&#9645; Escape velocity $$v_{esc} =\sqrt{\tfrac{2GM_E}{r}}$$

&#9645; Orbital speed $$v_{orbit}=\sqrt{\tfrac{GM_E}{r}}$$

&#9645; Orbital period $$T=2\pi\sqrt{\tfrac{r^3}{GM_E}}$$

&#9645; Energy in circular orbit $$E=K+U=-\tfrac{GmM_E}{2r}$$

&#9645; Conic section $$\tfrac{\alpha}{r}=1+e\,\!\cos\theta$$

&#9645; Kepler's third law $$T^2=\tfrac{4\pi^2}{GM}a^3$$

&#9645; Schwarzschild radius $$R_S=\tfrac{2GM}{c^2}$$ <section end=Gravitation/> <section begin=Fluid_Mechanics/>

Fluid_Mechanics
Mass density $$\rho=m/V\;$$ &#9645; Pressure $$P=F/A\;$$

&#9645; $$B=\rho_{flu}(A\Delta h )g$$ and &#9645; $$W = \rho_{obj}(A\Delta h )g = M_{obj}g$$

&#9645; Pressure vs depth/height (constant density)$$\,p=p_o+\rho gh \Leftarrow dp/dy = -\rho g$$

&#9645; Absolute vs gauge pressure $$\,p_{abs}=p_g+p_{atm} \;$$

&#9645; Pascal's principle: $$\,F/A\,$$ depends only on depth, not on orientation of A.

&#9645; Volume flow rate $$Q=dV/dt\;$$

&#9645; Continuity equation $$\rho_1A_1v_1=\rho_2A_2v_2$$$$\Rightarrow A_1v_1=A_2v_2\text{ if }\rho=const.\;$$

&#9645; Bernoulli's principle $$p_1+\tfrac 1 2 \rho v_1^2 + \rho gy_1=p_2+\tfrac 1 2 \rho v_2^2 + \rho gy_2$$

&#9645; Viscosity $$\eta = \tfrac{FL}{vA}$$ where F is the force applied by a fluid that is moving along a distance L from an area A.

&#9645; Poiseuille equation $$p_2-p_1 = QR$$ where $$R=\tfrac{8\eta\ell}{\pi r^4}$$ is "resistance" for a pipe of radius $$r$$ and length $$\ell$$.

<section end=Fluid_Mechanics/> <section begin=Oscillations/>

Oscillations
Frequency $$f$$, period $$T$$ and angular frequency $$ \omega\,:\;$$ $$fT=1\,,\quad\omega T=2\pi$$ &#9645; Simple harmonic motion $$x(t)=A\cos(\omega t+\phi),\, $$ $$v(t)=-A\omega\sin(\omega t+\phi),\,$$ $$ a(t)=-A\omega^2\cos(\omega t+\phi)$$ also models the x-component of uniform circular motion.

&#9645; For $$(A,\omega)$$ positive: $$\,x_{max}=A,\;v_{max}=A\omega,\;a_{max}=A\omega^2 $$

&#9645; Mass-spring $$ \omega=2\pi/T= 2\pi f=\sqrt{k/m};\,$$

&#9645; Energy $$E_{Tot}=\tfrac 1 2 kx^2 + \tfrac 1 2 mv^2 = \tfrac 1 2 mv_{max}^2 = \tfrac 1 2 kx_{max}^2 \Rightarrow $$$$v = \pm\sqrt{\tfrac k m \left(A^2-x^2\right)}$$

&#9645; Simple pendulum $$ \omega\approx\sqrt{g/L}$$

&#9645; Physical pendulum $$\tau=-MgL\sin\theta\approx -MgL\theta \Rightarrow \;$$$$\omega=\sqrt{mgL/I}$$ and $$L$$ measures from pivot to CM.

&#9645; Torsional pendulum $$\tau=-\kappa\theta$$$$ \Rightarrow \omega =\sqrt{I/\kappa}$$

&#9645; Damped harmonic oscillator $$ m\tfrac{d^2x}{dt^2}=-kx-b\tfrac{dx}{dt} $$$$  \Rightarrow x=A_0e^{\frac{b}{2m}t}\cos{(\omega t + \phi)}$$ where $$ \omega=\sqrt{\omega_0^2-\left(\tfrac{b}{2m}\right)^2}$$ and $$ \omega_0=\sqrt\tfrac k m.$$

&#9645; Forced harmonic oscillator (MIT wiki!) $$ m\tfrac{d^2x}{dt^2}=-kx-b\tfrac{dx}{dt}+F_0\sin\omega t $$$$ \Rightarrow x=Ae^{\frac{b}{2m}t}\cos{(\omega t + \phi)}$$ where $$ A= \tfrac{F_0}{\sqrt{ m^2(\omega-\omega_0)^2   +   b^2\omega^2}}$$. <section end=Oscillations/> <section begin=Waves/>

Waves
Wave speed (phase velocity) $$v=\lambda/T=\lambda f = \omega/k$$ where $$k=2\pi/\lambda$$ is wavenumber. &#9645; Wave and pulse speed of a stretched string $$=\sqrt{F_T/\mu}$$ where $$F_T$$ is tension and $$\mu$$ is linear mass density. &#9645; Speed of a compression wave in a fluid $$v=\sqrt{B/\rho}.$$

&#9645; Periodic travelling wave $$y(x,t)=A\sin(kx\mp\omega t)$$ travels in the positive/negative direction. The phase is $$kx\mp\omega t$$ and the amplitude is $$A$$.

&#9645; The resultant of two waves with identical amplitude and frequency $$ y_R(x,t) = \left[2A \cos\left( \tfrac\phi 2 \right) \right]\sin\left(kx-\omega t + \tfrac\phi 2\right) $$ where $$\phi$$ is the phase shift.

&#9645; This wave equation $$\partial^2 y/\partial t^2 = v_w^2\,\partial^2 y/\partial x^2$$ is linear in $$y=y(x,t)$$

&#9645; Power in a tranverse stretched string wave $$P_{ave}=\tfrac 1 2 \mu A^2\omega^2v$$.

&#9645; Intensity of a plane wave $$I= P/A \Rightarrow \tfrac{P}{4\pi r^2}$$ in a spherical wave.

&#9645; Standing wave $$y(x,t)=A\sin(kx)\cos(\omega t+\phi)$$ For symmetric boundary conditions $$\lambda_n=2\pi/k_n=\tfrac 2\pi L$$ $$n=1,2,3,...$$, or equivalently $$f=nf_1$$ where $$f_1=\tfrac{v}{2L} $$ is the fundamental frequency. <section end=Waves/> <section begin=Sound/>

Sound
Pressure and displacement fluctuations in a sound wave $$P=\Delta P_{max}\sin(kx\mp\omega t +\phi)$$ and $$s=s_{max}\cos(kx\mp\omega t +\phi)$$

&#9645; Speed of sound in a fluid $$v=f\lambda=\sqrt{\beta/\rho}$$, &#9645; in a solid $$\sqrt{Y/\rho}$$, &#9645; in an idal gas $$\sqrt{\gamma RT/M}$$, &#9645; in air $$331\tfrac m s\sqrt{\tfrac{T_K}{273\,K}}=331\tfrac m s\sqrt{1+\tfrac{T_C}{273^oC}}$$

&#9645; Decreasing intensity spherical wave $$I_2=I_1\left(\tfrac{r_1}{r_2}\right)^2$$

&#9645; Sound intensity $$I=\tfrac{\langle P\rangle}{A}=\tfrac{\left(\Delta P_{max}\right)^2}{2\rho v}$$ &#9645; ...level $$10\log_{10}{I/I_0}$$

&#9645; Resonance tube One end closed: $$\lambda_n=\tfrac 4 n L,$$ $$f_n=n\tfrac {v}{4L},$$ $$n=1,3,5,...$$ &#9645; Both ends open: $$\lambda_n=\tfrac 2 n L,$$ $$f_n=n\tfrac {v}{2L},$$ $$n=1,2,3,...$$

&#9645; Beat frequency $$f_{beat}=|f_2-f_1|$$

&#9645; (nonrelativistic) Doppler effect $$f_o=f_s\tfrac{v\pm v_o}{v\mp v_s}$$ where $$v$$ is the speed of sound, $$v_s$$ is the velocity of the source, and $$v_o$$ is the velocity of the observer. &#9645; Angle of shock wave   $$\sin\theta=v/v_s=1/M$$ where $$v$$ is the speed of sound, $$v_s$$ is the speed of the source, and $$M$$ is the Mach number. <section end=Sound/>