Equations and Concepts for Mechanical Waves, Sound, Fluid Mechanics, and Thermodynamics (Chapters 12-15)

Mechanical Waves & Sound

Wave Properties

Mechanical waves transfer energy through a medium without transporting matter. The fundamental properties of waves include speed, frequency, wavelength, and amplitude.

• Wave Speed: The speed of a wave is given by $v = f\lambda$, where f is frequency and \lambda is wavelength.

• Wave Speed on a String: $v = \sqrt{\frac{F_T}{\mu}}$, where F_T is the tension in the string and \mu is the mass per unit length.

• Wave Function: $y(x, t) = A \sin\left(2\pi f \left(t - \frac{x}{v}\right)\right)$ or $y(x, t) = A \sin\left(2\pi \left(\frac{t}{T} - \frac{x}{\lambda}\right)\right)$, where A is amplitude, T is period.

Interference and Standing Waves

Interference occurs when two or more waves overlap, resulting in constructive or destructive patterns.

• Constructive Interference: Path difference $= n\lambda$, $n = 0, \pm1, \pm2, ...$

• Destructive Interference: Path difference $= \left(n + \frac{1}{2}\right)\lambda$, $n = 0, \pm1, \pm2, ...$

• Standing Waves (String): $f_n = n\left(\frac{v}{2L}\right)$, $n = 1, 2, 3, ...$

• Standing Waves (Open/Closed Tube): $f_n = n\left(\frac{v}{4L}\right)$, $n = 1, 3, 5, ...$

Sound Intensity and Decibels

Sound intensity measures the power per unit area, and decibels quantify loudness.

• Intensity: $I = \frac{P}{4\pi r^2}$, where P is power, r is distance.

• Decibel Level: $\beta = (10\ \text{dB}) \log\left(\frac{I}{I_0}\right)$, $I_0 = 10^{-12}\ \text{W/m}^2$

• Beat Frequency: $f_{\text{beat}} = |f_1 - f_2|$

• Doppler Effect: $f_L = \left(\frac{v + v_L}{v + v_S}\right)f_S$, where v_L is listener speed, v_S is source speed.

Fluid Mechanics

Density and Pressure

Fluids are characterized by their density and the pressure they exert.

• Density: $\rho = \frac{m}{V}$, where m is mass, V is volume.

• Pressure: $p = \frac{F_{\perp}}{A}$, where F_{\perp} is force perpendicular to area A.

• Hydrostatic Pressure: $p = p_0 + \rho gh$, where p_0 is atmospheric pressure, g is gravity, h is depth.

Buoyancy

Buoyant force is the upward force exerted by a fluid on a submerged object.

• Buoyant Force: $F_B = \rho_{\text{fluid}} V_{\text{disp}} g$, where V_{\text{disp}} is displaced volume.

Temperature & Heat

Temperature Scales

Temperature can be measured in Celsius, Fahrenheit, or Kelvin.

• Fahrenheit to Celsius: $T_F = \frac{9}{5}T_C + 32^\circ$

• Celsius to Fahrenheit: $T_C = \frac{5}{9}(T_F - 32^\circ)$

• Celsius to Kelvin: $T_K = T_C + 273.15$

Thermal Expansion

Materials expand when heated.

• Linear Expansion: $\Delta L = \alpha L_0 \Delta T$

• Volume Expansion: $\Delta V = V_0 \beta \Delta T$

Heat and Phase Changes

Heat is energy transferred due to temperature difference. Phase changes require latent heat.

• Specific Heat: $Q = mc\Delta T$

• Latent Heat: $Q = \pm mL$

• Heat Transfer (Conduction): $H = kA\frac{T_H - T_C}{L}$

• Heat Transfer (Radiation): $H = Ae\sigma T^4$

Common Values

• Specific heat of ice: $c_{\text{ice}} = 2010\ \text{J/(kg·K)}$

• Specific heat of water: $c_{\text{water}} = 4190\ \text{J/(kg·K)}$

• Latent heat of fusion (water): $L_f = 3.34 \times 10^5\ \text{J/kg}$

• Latent heat of vaporization (water): $L_v = 2.256 \times 10^6\ \text{J/kg}$

• Stefan-Boltzmann constant: $\sigma = 5.67 \times 10^{-8}\ \text{W/(m}^2\ \text{K}^4)$

Thermal Properties of Matter & The Second Law of Thermodynamics

Ideal Gas Law and Related Quantities

Gases obey relationships between pressure, volume, temperature, and number of molecules.

• Avogadro's Number: $N = 6.022 \times 10^{23}\ \text{molecules/mol}$

• Molar Mass: $m_{\text{total}} = nM$

• Ideal Gas Law: $pV = nRT$

• Density of Gas: $\rho = \frac{pM}{RT}$

• Gas Constant: $R = 8.314\ \text{J/(mol·K)}$

• Boltzmann Constant: $k = 1.381 \times 10^{-23}\ \text{J/K}$

Kinetic Theory of Gases

The kinetic theory relates the motion of molecules to macroscopic properties.

• Translational Kinetic Energy: $K_{\text{tr.}} = \frac{3}{2}nRT$

• Average Kinetic Energy: $K_{\text{av.}} = \frac{1}{2}m\langle v^2 \rangle_{\text{av.}} = \frac{3}{2}kT$

• Ideal Gas Law (Molecules): $pV = NkT$

• Root Mean Square Speed: $v_{\text{rms}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

Heat, Work, and Internal Energy

Energy transfer in thermodynamic systems is governed by the first law of thermodynamics.

• Heat Capacity: $Q = nC\Delta T$

• Work Done by Gas: $W = p\Delta V$

• Change in Internal Energy: $\Delta U = Q - W$

• Relationship of Heat Capacities: $C_p = C_V + R$

Adiabatic Processes

Adiabatic processes occur without heat exchange.

• Adiabatic Condition: $p_1V_1^{\gamma} = p_2V_2^{\gamma}$

• Adiabatic Temperature Relation: $T_1V_1^{\gamma-1} = T_2V_2^{\gamma-1}$

• Adiabatic Index: $\gamma = \frac{C_P}{C_V}$

• For Monatomic Ideal Gas: $C_V = \frac{3R}{2}$, $C_p = \frac{5R}{2}$

Summary Table: Key Constants and Properties

Quantity	Symbol	Value
Avogadro's Number	N	6.022 × 1023 molecules/mol
Gas Constant	R	8.314 J/(mol·K)
Boltzmann Constant	k	1.381 × 10-23 J/K
Specific Heat of Ice	cice	2010 J/(kg·K)
Specific Heat of Water	cwater	4190 J/(kg·K)
Latent Heat of Fusion (water)	Lf	3.34 × 105 J/kg
Latent Heat of Vaporization (water)	Lv	2.256 × 106 J/kg
Stefan-Boltzmann Constant	σ	5.67 × 10-8 W/(m2·K4)

Example: To calculate the heat required to melt 2 kg of ice at 0°C, use $Q = mL_f = 2 \times 3.34 \times 10^5 = 6.68 \times 10^5$ J.

Additional info: Academic context and explanations have been added to clarify the meaning and application of each equation and concept.