PHY222 Midterm #2 Study Guide: Waves, Sound, and Thermodynamics

Wave Optics: Diffraction and Interference

Double-Slit Interference

The double-slit experiment demonstrates the wave nature of light through the creation of interference patterns. When monochromatic light passes through two closely spaced slits, it produces bright and dark fringes on a screen due to constructive and destructive interference.

• Constructive Interference: Occurs when the path difference between the two slits is an integer multiple of the wavelength ($d \sin\theta = m\lambda$, where $m = 0, \pm1, \pm2, ...$).

• Destructive Interference: Occurs when the path difference is a half-integer multiple of the wavelength ($d \sin\theta = (m+\frac{1}{2})\lambda$).

• Fringe Spacing: The distance between adjacent bright fringes can be calculated using $y = \frac{m\lambda L}{d}$, where $L$ is the distance to the screen.

• Example: If the first order fringe is 6.00 cm from the center, with slit separation $d = 0.05$ mm and screen distance $L = 4.00$ m, the wavelength can be found using $\lambda = \frac{y d}{L}$.

Thin Film Interference

Thin films, such as oil or soap, produce colorful patterns due to interference between light reflected from the top and bottom surfaces. The conditions for constructive and destructive interference depend on the thickness of the film and the wavelength of light.

• Constructive Interference (for thin film): $2 n t = m \lambda$ (with phase shift considerations).

• Destructive Interference (for thin film): $2 n t = (m+\frac{1}{2}) \lambda$.

• Example: For a thin oil layer ($n=1.50$) on water ($n=1.33$), the thinnest layer for constructive interference at $\lambda = 700$ nm is $t = \frac{\lambda}{4n}$.

Wave Motion

Transverse Waves on a String

Transverse waves are described mathematically by sinusoidal functions. The displacement $D(x,t)$ of a wave can be written as $D(x,t) = A \sin(kx - \omega t)$, where $A$ is amplitude, $k$ is wavenumber, and $\omega$ is angular frequency.

• Amplitude ($A$): Maximum displacement from equilibrium.

• Wavenumber ($k$): $k = \frac{2\pi}{\lambda}$, where $\lambda$ is wavelength.

• Angular Frequency ($\omega$): $\omega = 2\pi f$, where $f$ is frequency.

• Wave Speed ($v$): $v = \frac{\omega}{k}$.

• Period ($T$): $T = \frac{1}{f}$.

• Direction of Propagation: Determined by the sign in the argument of the sine function.

• Example: For $D(x,t) = 4 \sin(5\pi x - 6\pi t)$, $A = 4$, $k = 5\pi$, $\omega = 6\pi$, $v = \frac{6\pi}{5\pi} = 1.2$ m/s.

Standing Waves

Standing waves are formed by the superposition of two waves traveling in opposite directions. They have nodes (points of zero displacement) and antinodes (points of maximum displacement).

• Wave Function: $D(x,t) = A \sin(kx) \cos(\omega t)$.

• Nodes: Points where $D(x,t) = 0$ for all $t$.

• Antinodes: Points where $D(x,t)$ reaches maximum amplitude.

• Example: In a 30 cm open tube, nodes occur at the ends and antinodes at the center.

Sound and the Doppler Effect

Standing Sound Waves in Pipes

Sound waves in pipes can form standing waves, with nodes and antinodes depending on whether the pipe is open or closed at the ends.

• Wavelength in Open Pipe: $\lambda = 2L$ for the fundamental mode.

• Frequency: $f = \frac{v}{\lambda}$, where $v$ is the speed of sound.

• Example: For a 30 cm pipe, $\lambda = 60$ cm for the fundamental, but the problem may show a higher mode.

Doppler Effect

The Doppler effect describes the change in frequency heard by an observer moving relative to a sound source.

• Observed Frequency: $f' = f \left(\frac{v + v_o}{v - v_s}\right)$, where $v_o$ is observer speed, $v_s$ is source speed.

• Example: If you run toward a stationary sound source, the frequency increases.

Beat Frequency

When two sound waves of slightly different frequencies interfere, they produce beats with a frequency equal to the difference between the two frequencies.

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

• Wavelength of Beat: $\lambda_{beat} = \frac{v}{f_{beat}}$.

Thermodynamics: Engine Cycles and Entropy

Ideal Gas Processes

Engine cycles often involve different thermodynamic processes: isobaric (constant pressure), isochoric (constant volume), and adiabatic (no heat exchange).

• Isobaric Process: $Q = n C_p \Delta T$

• Isochoric Process: $Q = n C_v \Delta T$

• Adiabatic Process: $PV^\gamma = \text{constant}$, where $\gamma = \frac{C_p}{C_v}$

• Example: For a monoatomic ideal gas, $C_v = \frac{3R}{2}$, $C_p = \frac{5R}{2}$

Heat and Efficiency

The efficiency of a heat engine is the ratio of work output to heat input. The heat absorbed ($Q_H$) and heat released ($Q_L$) are calculated for each process.

• Efficiency: $e = 1 - \frac{Q_L}{Q_H}$

• Example: In a three-step cycle, identify which step absorbs heat and which releases it.

Entropy Changes

Entropy measures the disorder or randomness in a system. For reversible processes, the change in entropy is given by $\Delta S = \frac{Q_{rev}}{T}$.

• Isobaric: $\Delta S = n C_p \ln\left(\frac{T_2}{T_1}\right)$

• Isochoric: $\Delta S = n C_v \ln\left(\frac{T_2}{T_1}\right)$

• Adiabatic: $\Delta S = 0$ (reversible adiabatic)

Equation Sheet Reference

Key Formulas for Exam

• Wave Equations: $D(x,t) = A \sin(kx - \omega t)$, $v = f \lambda$

• Sound: $f = \frac{v}{\lambda}$, $f' = f \left(\frac{v + v_o}{v - v_s}\right)$

• Thermodynamics: $Q = n C \Delta T$, $e = 1 - \frac{Q_L}{Q_H}$, $\Delta S = \frac{Q_{rev}}{T}$