Formulario di Fisica 1

Formulario completo di Fisica 1: meccanica (cinematica, dinamica, urti, corpo rigido, oscillazioni, gravitazione), fluidi, termodinamica e onde. Tutte le formule in un unico PDF scaricabile.

Particle Kinematics
Physical Quantities and the International System (SI)
[v]=L⋅T−1=m/s[v] = L \cdot T^{-1} = \mathrm{m/s}
[a]=L⋅T−2=m/s2[a] = L \cdot T^{-2} = \mathrm{m/s^2}
[F]=M⋅L⋅T−2=N[F] = M \cdot L \cdot T^{-2} = \mathrm{N}
[W]=M⋅L2⋅T−2=J[W] = M \cdot L^2 \cdot T^{-2} = \mathrm{J}
Position, Displacement, and Velocity
v⃗m=Δr⃗Δt\vec{v}_m = \dfrac{\Delta\vec{r}}{\Delta t}
v⃗=dr⃗dt\vec{v} = \dfrac{d\vec{r}}{dt}
∣v⃗∣=dsdt|\vec{v}| = \dfrac{ds}{dt}
Δr⃗=r⃗2−r⃗1\Delta\vec{r} = \vec{r}_2 - \vec{r}_1
Acceleration — Tangential and Centripetal Components
a⃗=dv⃗dt=d2r⃗dt2\vec{a} = \dfrac{d\vec{v}}{dt} = \dfrac{d^2\vec{r}}{dt^2}
at=dvdta_t = \dfrac{dv}{dt}
an=v2Ra_n = \dfrac{v^2}{R}
∣a⃗∣2=at2+an2|\vec{a}|^2 = a_t^2 + a_n^2
Uniform Linear Motion (ULM)
x(t)=x0+v⋅tx(t) = x_0 + v \cdot t
v=const,a=0v = \mathrm{const},\quad a = 0
Δx=v⋅Δt\Delta x = v \cdot \Delta t
x(t) is a straight line of slope vx(t) \text{ is a straight line of slope } v
Uniformly Accelerated Motion (UAM)
v(t)=v0+a tv(t) = v_0 + a\,t
x(t)=x0+v0 t+12a t2x(t) = x_0 + v_0\,t + \tfrac{1}{2}a\,t^2
v2=v02+2a (x−x0)v^2 = v_0^2 + 2a\,(x - x_0)
Free fall: y(t)=y0+v0t−12g t2\text{Free fall: } y(t) = y_0 + v_0 t - \tfrac{1}{2}g\,t^2
Projectile Motion (Parabolic)
x(t)=v0cos⁡θ⋅tx(t) = v_0\cos\theta\cdot t
y(t)=v0sin⁡θ⋅t−12g t2y(t) = v_0\sin\theta\cdot t - \tfrac{1}{2}g\,t^2
R=v02sin⁡(2θ)gR = \dfrac{v_0^2\sin(2\theta)}{g}
hmax=v02sin⁡2θ2gh_{max} = \dfrac{v_0^2\sin^2\theta}{2g}
Uniform Circular Motion (UCM)
v=ω r,ac=v2/r=ω2rv = \omega\,r,\quad a_c = v^2/r = \omega^2 r
T=2π/ω,f=ω/(2π)T = 2\pi/\omega,\quad f = \omega/(2\pi)
Non-uniform: ω(t)=ω0+αt,  θ(t)=θ0+ω0t+12αt2\text{Non-uniform: }\omega(t)=\omega_0+\alpha t,\;\theta(t)=\theta_0+\omega_0 t+\tfrac{1}{2}\alpha t^2
Dynamics and Conservation Laws
First Law — Inertia and Inertial Frames
F⃗net=0  ⟺  v⃗=const\vec{F}_{net} = 0 \iff \vec{v} = \mathrm{const}
Second Law — F = ma
F⃗net=m a⃗\vec{F}_{net} = m\,\vec{a}
∑Fx=m ax,∑Fy=m ay\sum F_x = m\,a_x,\quad \sum F_y = m\,a_y
Third Law — Action and Reaction
F⃗AB=−F⃗BA\vec{F}_{AB} = -\vec{F}_{BA}
∣F⃗AB∣=∣F⃗BA∣|\vec{F}_{AB}| = |\vec{F}_{BA}|
Fundamental Forces — Weight, Normal, Friction
P=mg,N=mgcos⁡θ,f=μNP = mg,\quad N = mg\cos\theta,\quad f = \mu N
P∥=mgsin⁡θ,  P⊥=mgcos⁡θP_\parallel = mg\sin\theta,\; P_\perp = mg\cos\theta
Work and the Work-Energy Theorem
W=F dcos⁡θ=∫F⃗⋅ds⃗W = F\,d\cos\theta = \int\vec{F}\cdot d\vec{s}
Wnet=ΔKE=12mv2−12mv02W_{net} = \Delta KE = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2
P=dW/dt=F⃗⋅v⃗P = dW/dt = \vec{F}\cdot\vec{v}
Potential Energy and Conservation
Ug=mgh,Ue=12k x2U_g = mgh,\quad U_e = \tfrac{1}{2}k\,x^2
Emech=12mv2+U=constE_{mech} = \tfrac{1}{2}mv^2 + U = \mathrm{const}
Fspring=−k xF_{spring} = -k\,x
Relative Motion
Two Reference Frames — Vector Geometry
r⃗A=r⃗O′+r⃗A/S′\vec{r}_A = \vec{r}_{O'} + \vec{r}_{A/S'}
r⃗A/S′=position of A measured in S’\vec{r}_{A/S'} = \text{position of A \emph{measured in} S'}
Velocity Composition — Pure Translation
v⃗A=v⃗A/S′+v⃗S′/S\vec{v}_A = \vec{v}_{A/S'} + \vec{v}_{S'/S}
v⃗abs=v⃗rel+v⃗transport\vec{v}_{abs} = \vec{v}_{rel} + \vec{v}_{transport}
v⃗A/S′=(dr⃗A/S′dt)S′\vec{v}_{A/S'} = \left(\frac{d\vec{r}_{A/S'}}{dt}\right)_{S'}
Acceleration Composition — Pure Translation
a⃗A=a⃗A/S′+a⃗S′/S\vec{a}_A = \vec{a}_{A/S'} + \vec{a}_{S'/S}
F⃗fictitious=−m a⃗S′/S\vec{F}_{fictitious} = -m\,\vec{a}_{S'/S}
S′ inertial   ⟺  a⃗S′/S=0  ⟹  a⃗A=a⃗A/S′S'\text{ inertial }\iff \vec{a}_{S'/S}=0 \implies \vec{a}_A = \vec{a}_{A/S'}
Rotating Frames — Relative Derivative Theorem
(dA⃗dt)S=(dA⃗dt)S′+ω⃗×A⃗\left(\frac{d\vec{A}}{dt}\right)_S = \left(\frac{d\vec{A}}{dt}\right)_{S'} + \vec{\omega}\times\vec{A}
e^˙i=ω⃗×e^i\dot{\hat{e}}_i = \vec{\omega}\times\hat{e}_i
Velocity and Acceleration in Rotating Frame — Fictitious Forces
v⃗A=v⃗A/S′+ω⃗×r⃗A/S′+v⃗O′/S\vec{v}_A = \vec{v}_{A/S'} + \vec{\omega}\times\vec{r}_{A/S'} + \vec{v}_{O'/S}
F⃗Cor=−2m ω⃗×v⃗A/S′\vec{F}_{Cor} = -2m\,\vec{\omega}\times\vec{v}_{A/S'}
F⃗cf=mω2r⊥r^⊥\vec{F}_{cf} = m\omega^2 r_\perp \hat{r}_\perp
a⃗A=a⃗A/S′+2ω⃗×v⃗A/S′−ω2r⊥r^⊥\vec{a}_A = \vec{a}_{A/S'} + 2\vec{\omega}\times\vec{v}_{A/S'} - \omega^2 r_\perp\hat{r}_\perp
System Dynamics and Collision Theory
Momentum and Impulse
p⃗=mv⃗\vec{p} = m\vec{v}
J⃗=Δp⃗\vec{J} = \Delta\vec{p}
F⃗avg=Δp⃗/Δt\vec{F}_{avg} = \Delta\vec{p}/\Delta t
Center of Mass and Cardinal Equations
r⃗CM=∑mir⃗i/M\vec{r}_{CM} = \sum m_i\vec{r}_i/M
F⃗ext=Ma⃗CM\vec{F}_{ext} = M\vec{a}_{CM}
τ⃗ext=dL⃗/dt\vec{\tau}_{ext} = d\vec{L}/dt
Elastic Collision (1D)
v1′=m1−m2m1+m2 v1v_1' = \dfrac{m_1-m_2}{m_1+m_2}\,v_1
v2′=2m1m1+m2 v1v_2' = \dfrac{2m_1}{m_1+m_2}\,v_1
KEtot,before=KEtot,afterKE_{tot,before} = KE_{tot,after}
Inelastic Collision
(m1+m2)v′=m1v1+m2v2(m_1+m_2)v' = m_1 v_1 + m_2 v_2
ΔKE=−m1m2(v1−v2)2/[2(m1+m2)]\Delta KE = -m_1 m_2(v_1-v_2)^2/[2(m_1+m_2)]
e=∣v2′−v1′∣/∣v1−v2∣e = |v_2'-v_1'|/|v_1-v_2|
Moments of Inertia and Rigid Body
I=∫r2 dmI = \int r^2\,dm
I=ICM+Md2I = I_{CM} + Md^2
τ=Iα,KErot=12Iω2\tau = I\alpha,\quad KE_{rot} = \tfrac{1}{2}I\omega^2
Cylinder: I=12MR2;  Sphere: I=25MR2\text{Cylinder: }I=\tfrac{1}{2}MR^2;\;\text{Sphere: }I=\tfrac{2}{5}MR^2
Rigid Body Dynamics
Moment of Inertia — Resistance to Rotation
I=∫r2 dmI = \int r^2\,dm
Idisk=12mR2,Irod center=112mL2I_{disk} = \tfrac{1}{2}mR^2, \quad I_{rod\,center} = \tfrac{1}{12}mL^2
Isolid sphere=25mR2,Ihollow sphere=23mR2I_{solid\,sphere} = \tfrac{2}{5}mR^2, \quad I_{hollow\,sphere} = \tfrac{2}{3}mR^2
Iring=mR2,Iplate=112m(a2+b2)I_{ring} = mR^2, \quad I_{plate} = \tfrac{1}{12}m(a^2+b^2)
Parallel Axis Theorem (Steiner's Theorem)
I=ICM+Md2I = I_{CM} + Md^2
Irod, end=13mL2I_{rod,\,end} = \tfrac{1}{3}mL^2
Idisk, rim=32mR2I_{disk,\,rim} = \tfrac{3}{2}mR^2
Newton's Second Law for Rotation
τ=Iα(analogue of F=ma)\tau = I\alpha \quad (\text{analogue of } F=ma)
τ=rFsin⁡θ=r⊥F\tau = r F \sin\theta = r_\perp F
Krot=12Iω2K_{rot} = \tfrac{1}{2}I\omega^2
W=τΔθ,P=τωW = \tau\Delta\theta, \quad P = \tau\omega
Pure Rolling — Without Slipping
vCM=ωR,aCM=αRv_{CM} = \omega R, \quad a_{CM} = \alpha R
Ktot=12mvCM2+12ICMω2K_{tot} = \tfrac{1}{2}mv_{CM}^2 + \tfrac{1}{2}I_{CM}\omega^2
Ktot=12mvCM2 ⁣(1+ICMmR2)K_{tot} = \tfrac{1}{2}mv_{CM}^2\!\left(1 + \dfrac{I_{CM}}{mR^2}\right)
aCM=gsin⁡θ1+ICM/(mR2)a_{CM} = \dfrac{g\sin\theta}{1 + I_{CM}/(mR^2)}
Conservation of Angular Momentum
L=Iω(for a rigid body)L = I\omega \quad (\text{for a rigid body})
τ=dLdt(analogue of F=dp/dt)\tau = \dfrac{dL}{dt} \quad (\text{analogue of } F = dp/dt)
I1ω1=I2ω2(if τext=0)I_1\omega_1 = I_2\omega_2 \quad (\text{if } \tau_{ext}=0)
Ω=τ/L(gyroscope precession)\Omega = \tau/L \quad (\text{gyroscope precession})
Static Equilibrium of a Rigid Body
∑Fx=0,∑Fy=0\sum F_x = 0, \quad \sum F_y = 0
∑τO=0(∀ O)\sum \tau_O = 0 \quad (\forall\, O)
Ustable≈12kθ2(for small oscillations)U_{stable} \approx \tfrac{1}{2}k\theta^2 \quad (\text{for small oscillations})
Oscillations and Harmonic Motion
Simple Harmonic Motion (SHM) — Differential Equation and Solution
mx¨+kx=0(equation of motion)m\ddot{x} + kx = 0 \quad (\text{equation of motion})
x(t)=Acos⁡(ω0t+ϕ)x(t) = A\cos(\omega_0 t + \phi)
ω0=k/m[rad/s]\omega_0 = \sqrt{k/m} \quad [\mathrm{rad/s}]
T=2πω0=2πmk,f=1T=ω02πT = \dfrac{2\pi}{\omega_0} = 2\pi\sqrt{\dfrac{m}{k}}, \quad f = \dfrac{1}{T} = \dfrac{\omega_0}{2\pi}
A=x02+(v0/ω0)2,tan⁡ϕ=−v0ω0x0A = \sqrt{x_0^2 + (v_0/\omega_0)^2}, \quad \tan\phi = -\dfrac{v_0}{\omega_0 x_0}
Velocity, Acceleration, and Energy in SHM
v(t)=−Aω0sin⁡(ω0t+ϕ)v(t) = -A\omega_0\sin(\omega_0 t+\phi)
a(t)=−Aω02cos⁡(ω0t+ϕ)=−ω02 xa(t) = -A\omega_0^2\cos(\omega_0 t+\phi) = -\omega_0^2\,x
E=12mv2+12kx2=12kA2=constE = \tfrac{1}{2}mv^2 + \tfrac{1}{2}kx^2 = \tfrac{1}{2}kA^2 = \mathrm{const}
vmax=ω0A=Ak/mv_{max} = \omega_0 A = A\sqrt{k/m}
v(x)=±ω0A2−x2v(x) = \pm\omega_0\sqrt{A^2 - x^2}
Simple Pendulum — Small Angle Approximation
ω0=g/L(simple pendulum)\omega_0 = \sqrt{g/L} \quad (\text{simple pendulum})
T=2πL/g(independent of m and A for small angles)T = 2\pi\sqrt{L/g} \quad (\text{independent of } m \text{ and } A \text{ for small angles})
T=2πI/(Mgd)(physical pendulum)T = 2\pi\sqrt{I/(Mgd)} \quad (\text{physical pendulum})
Leq=I/(Md)(equivalent length)L_{eq} = I/(Md) \quad (\text{equivalent length})
Damped Harmonic Oscillator
mx¨+bx˙+kx=0m\ddot{x} + b\dot{x} + kx = 0
x(t)=A0e−γtcos⁡(ω1t+ϕ)(underdamped)x(t) = A_0 e^{-\gamma t}\cos(\omega_1 t + \phi) \quad (\text{underdamped})
ω1=ω02−γ2,γ=b2m\omega_1 = \sqrt{\omega_0^2 - \gamma^2}, \quad \gamma = \dfrac{b}{2m}
τ=1/γ(characteristic decay time)\tau = 1/\gamma \quad (\text{characteristic decay time})
Q=ω02γ=ω0mbQ = \dfrac{\omega_0}{2\gamma} = \dfrac{\omega_0 m}{b}
Forced Oscillations and Resonance
mx¨+bx˙+kx=F0cos⁡(ωt)m\ddot{x} + b\dot{x} + kx = F_0\cos(\omega t)
A(ω)=F0/m(ω02−ω2)2+4γ2ω2A(\omega) = \dfrac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2 + 4\gamma^2\omega^2}}
ωres=ω02−2γ2≈ω0(γ≪ω0)\omega_{res} = \sqrt{\omega_0^2 - 2\gamma^2} \approx \omega_0 \quad (\gamma \ll \omega_0)
Δω=ω0Q=2γ\Delta\omega = \dfrac{\omega_0}{Q} = 2\gamma
Gravitation — Kepler and Newton
Law of Universal Gravitation
F=Gm1m2/r2F = G m_1 m_2/r^2
G=6.674×10−11 N⋅m2/kg2G = 6.674\times10^{-11}\,\mathrm{N\cdot m^2/kg^2}
g=GMT/RT2g = GM_T/R_T^2
Gravitational Field and Potential
g⃗(r)=−GMr^/r2\vec{g}(r) = -GM\hat{r}/r^2
V(r)=−GM/r,U=mVV(r) = -GM/r,\quad U = mV
g⃗=−∇V\vec{g} = -\nabla V
Kepler's Three Laws
r=p/(1+ecos⁡θ)r = p/(1+e\cos\theta)
dA/dt=L/(2m)=constdA/dt = L/(2m) = \mathrm{const}
T2=(4π2/GM) a3T^2 = (4\pi^2/GM)\,a^3
Escape Velocity and Orbits
vesc=2GM/Rv_{esc} = \sqrt{2GM/R}
vorb=GM/rv_{orb} = \sqrt{GM/r}
Eorb=−GMm/(2a)E_{orb} = -GMm/(2a)
vesc=2 vorbv_{esc} = \sqrt{2}\,v_{orb}
Hydrostatics
Definition of Fluid and Pressure
P=F/A[Pa]P = F/A \quad [\mathrm{Pa}]
1 atm=101325 Pa1\,\mathrm{atm} = 101325\,\mathrm{Pa}
Pascal's Law — Hydraulic Press
F2/F1=A2/A1F_2/F_1 = A_2/A_1
F1d1=F2d2F_1 d_1 = F_2 d_2
Stevin's Law — Hydrostatic Pressure
P(h)=P0+ρghP(h) = P_0 + \rho g h
ΔP=ρg Δh\Delta P = \rho g\,\Delta h
ρ1h1=ρ2h2(communicating vessels)\rho_1 h_1 = \rho_2 h_2 \quad (\text{communicating vessels})
Archimedes' Principle
FA=ρfluid g VsubmF_A = \rho_{fluid}\,g\,V_{subm}
ρbody≤ρfluid⇒floats\rho_{body} \leq \rho_{fluid} \Rightarrow \text{floats}
Vsubm/Vtot=ρbody/ρfluidV_{subm}/V_{tot} = \rho_{body}/\rho_{fluid}
Ideal Fluid Dynamics
Ideal Fluid and Steady Flow
ρ=const,η=0\rho = \mathrm{const},\quad \eta = 0
∂(⋅)/∂t=0\partial(\cdot)/\partial t = 0
Continuity Equation — Flow Rate
Q=Av=const  [m3/s]Q = Av = \mathrm{const}\;[\mathrm{m^3/s}]
A1v1=A2v2A_1 v_1 = A_2 v_2
Bernoulli's Theorem
P+12ρv2+ρgh=constP + \tfrac{1}{2}\rho v^2 + \rho g h = \mathrm{const}
Applications: Venturi, Torricelli, Lift
v=2gh(Torricelli)v = \sqrt{2gh} \quad (\text{Torricelli})
Q=Ahole2ghQ = A_{hole}\sqrt{2gh}
Ideal Gases, Real Gases, and Kinetic Theory
Zeroth Law — Temperature and Thermometric Scales
T(K)=T(°C)+273.15T(K) = T(°C) + 273.15
ΔL=α L0 ΔT\Delta L = \alpha\,L_0\,\Delta T
ΔV=β V0 ΔT\Delta V = \beta\,V_0\,\Delta T
Ideal Gas Equation of State
PV=nRTPV = nRT
R=8.314 J/(mol⋅K)R = 8.314\,\mathrm{J/(mol\cdot K)}
PV=NkBT,kB=1.381×10−23 J/KPV = Nk_BT,\quad k_B = 1.381\times10^{-23}\,\mathrm{J/K}
Kinetic Theory of Gases
⟨KE⟩=32kBT\langle KE\rangle = \tfrac{3}{2}k_B T
vrms=3RT/Mv_{rms} = \sqrt{3RT/M}
Cv=fR/2,Cp=(f+2)R/2C_v = fR/2,\quad C_p = (f+2)R/2
Real Gas — Van der Waals Equation
(P+a/V2)(V−b)=RT(P + a/V^2)(V-b) = RT
Tc=8a/(27Rb),Pc=a/(27b2)T_c = 8a/(27Rb),\quad P_c = a/(27b^2)
Thermodynamic Processes and First Law
Work in Thermodynamic Processes
W=∫P dVW = \int P\,dV
Wisobar=PΔV=nRΔTW_{isobar} = P\Delta V = nR\Delta T
Wisoth=nRTln⁡(V2/V1)W_{isoth} = nRT\ln(V_2/V_1)
Wad=−ΔU=−nCvΔTW_{ad} = -\Delta U = -nC_v\Delta T
First Law of Thermodynamics — Conservation of Energy
ΔU=Q−W\Delta U = Q - W
ΔU=nCvΔT\Delta U = nC_v\Delta T
Cycle: ΔU=0\text{Cycle: }\Delta U = 0
The Four Fundamental Gas Processes
γ=Cp/Cv\gamma = C_p/C_v
γmono=5/3,  γdi=1.4\gamma_{mono}=5/3,\; \gamma_{di}=1.4
PVγ=const,  TVγ−1=constPV^\gamma=\mathrm{const},\; TV^{\gamma-1}=\mathrm{const}
Cp−Cv=R  (Mayer)C_p-C_v=R \;(\text{Mayer})
Specific Heat and Latent Heat — Heat Transfer
Q=mc ΔTQ = mc\,\Delta T
Q=mLQ = mL
Prad=εσAT4P_{rad} = \varepsilon\sigma A T^4
Heat Engines — Carnot and Otto Cycles
Heat Engines — Layout and Efficiency
W=QH−QCW = Q_H - Q_C
η=W/QH=1−QC/QH<1\eta = W/Q_H = 1 - Q_C/Q_H < 1
COPref=QC/WCOP_{ref} = Q_C/W
Carnot Cycle — Maximum Theoretical Efficiency
ηCarnot=1−TC/TH\eta_{Carnot} = 1 - T_C/T_H
QH/TH=QC/TCQ_H/T_H = Q_C/T_C
ηreal≤ηCarnot\eta_{real} \leq \eta_{Carnot}
Otto Cycle — The Petrol Engine
ηOtto=1−1/rγ−1\eta_{Otto} = 1 - 1/r^{\gamma-1}
r=Vmax/Vminr = V_{max}/V_{min}
r=8÷12⇒η≈40÷55%r=8\div12 \Rightarrow \eta\approx40\div55\%
Kelvin and Clausius Postulates — The Second Law Statements
η<1\eta < 1
COPref,max=TC/(TH−TC)COP_{ref,max} = T_C/(T_H-T_C)
COPhp,max=TH/(TH−TC)COP_{hp,max} = T_H/(T_H-T_C)
Entropy and the Second Law of Thermodynamics
Definition of Entropy according to Clausius
dS=δQrev/TdS = \delta Q_{rev}/T
ΔS=∫12δQrev/T\Delta S = \int_1^2 \delta Q_{rev}/T
∮δQ/T≤0\oint \delta Q/T \leq 0
ΔSuniv≥0\Delta S_{univ} \geq 0
Reversible and Irreversible Processes
ΔSuniv=ΔSsys+ΔSsurr≥0\Delta S_{univ} = \Delta S_{sys}+\Delta S_{surr} \geq 0
ΔSgas=nCvln⁡(T2/T1)+nRln⁡(V2/V1)\Delta S_{gas} = nC_v\ln(T_2/T_1)+nR\ln(V_2/V_1)
Boltzmann's Statistical Interpretation — Entropy as Disorder
S=kBln⁡ΩS = k_B\ln\Omega
kB=1.381×10−23 J/Kk_B = 1.381\times10^{-23}\,\mathrm{J/K}
ΔS>0  ⟺  Ωfinal>Ωinitial\Delta S>0 \iff \Omega_{final}>\Omega_{initial}
Mechanical Waves
What is a Wave — Fundamental Concepts
v=λf=λ/Tv = \lambda f = \lambda/T
f=1/T,ω=2πf,k=2π/λf = 1/T, \quad \omega = 2\pi f, \quad k = 2\pi/\lambda
E∝A2 — energy carriedE \propto A^2 \text{ — energy carried}
Wave Equation — Mathematical Description
y(x,t)=Acos⁡(kx−ωt+ϕ0)y(x,t) = A\cos(kx - \omega t + \phi_0)
v=ω/k=λfv = \omega/k = \lambda f
∂2y/∂t2=v2 ∂2y/∂x2\partial^2 y/\partial t^2 = v^2\,\partial^2 y/\partial x^2
Wave Speed — Dependence on Medium
vstring=FT/μv_{string} = \sqrt{F_T/\mu}
vsolid=E/ρv_{solid} = \sqrt{E/\rho}
vsound=γRT/M≈331T/273  m/sv_{sound} = \sqrt{\gamma RT/M} \approx 331\sqrt{T/273}\;\mathrm{m/s}
Superposition Principle and Interference
y1+y2=2Acos⁡(Δϕ/2)cos⁡(kx−ωt+Δϕ/2)y_1+y_2 = 2A\cos(\Delta\phi/2)\cos(kx-\omega t + \Delta\phi/2)
Constr.: Δr=mλ\text{Constr.: } \Delta r = m\lambda
Destr.: Δr=(m+12)λ\text{Destr.: } \Delta r = (m+\tfrac{1}{2})\lambda
fbeat=∣f2−f1∣f_{beat} = |f_2 - f_1|
Standing Waves — Resonance
fn=n v/(2L)(n=1,2,3,…) — string / o-o pipef_n = n\,v/(2L) \quad (n=1,2,3,\ldots) \text{ — string / o-o pipe}
fn=(2n−1) v/(4L)(n=1,2,3,…) — o-c pipef_n = (2n-1)\,v/(4L) \quad (n=1,2,3,\ldots) \text{ — o-c pipe}
λn=2L/n    (string),λn=4L/(2n−1)    (o-c)\lambda_n = 2L/n \;\;\text{(string)}, \quad \lambda_n = 4L/(2n-1) \;\;\text{(o-c)}
Sound — Intensity and Sound Level
I=P/(4πr2)I = P/(4\pi r^2)
β=10log⁡10(I/I0)  [dB]\beta = 10\log_{10}(I/I_0) \;\mathrm{[dB]}
I0=10−12  W/m2I_0 = 10^{-12} \;\mathrm{W/m^2}
I∝1/r2I \propto 1/r^2
Doppler Effect
f′=f0 (v±vo)/(v∓vs)f' = f_0\,(v \pm v_o)/(v \mp v_s)
f′>f0 (approach),f′<f0 (recede)f' > f_0 \text{ (approach)}, \quad f' < f_0 \text{ (recede)}
Mach cone: sin⁡θ=v/vs  (vs>v)\text{Mach cone: } \sin\theta = v/v_s \;(v_s > v)
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