Oscillations and Waves: Study Notes for Physics

Chapter 15: Oscillations

Restoring Force and Simple Harmonic Motion (SHM)

Oscillatory motion occurs when a restoring force acts to return a system to its equilibrium position. This force is typically proportional to the displacement from equilibrium.

• Restoring Force: For a spring, $F = -kx$, where k is the spring constant and x is the displacement.

• Equilibrium Position: The system returns to equilibrium at $x = 0$.

• Physical Pendulum: The angular frequency is $oxed{ ext{ω} = rac{ ext{Mg}l}{I}}$, and the period is $T = 2 ext{π} rac{ ext{Mg}l}{I}$, where M is mass, g is gravity, l is the distance to the center of mass, and I is the moment of inertia.

Energy in Oscillatory Systems

If there is no friction or dissipation, kinetic and potential energy are alternately transformed into each other, but the total mechanical energy is conserved.

• Total Mechanical Energy: $E = K + U$

• For a spring-mass system: $E = rac{1}{2}mv^2 + rac{1}{2}kx^2 = rac{1}{2}m(v_{max})^2 = rac{1}{2}kA^2$

• Damped Oscillator: The energy of a lightly damped oscillator decays exponentially: $E = E_0 e^{-t/ au}$, where $au$ is the time constant.

Simple Harmonic Motion (SHM)

SHM is a sinusoidal oscillation described by a cosine or sine function. It is the projection onto the axis of uniform circular motion.

• General Solution: $x(t) = A ext{cos}( ext{ω}t + ext{φ}_0)$

• Phase Constant: $ext{φ}_0$ is determined by initial conditions.

• Maximum Speed: $v_{max} = ext{ω}A$

• Velocity: $v(t) = - ext{ω}A ext{sin}( ext{ω}t + ext{φ}_0)$

Damping

Damping occurs when a drag force opposes motion, reducing amplitude over time.

• Damping Force: $F_{damp} = -b v$, where b is the damping constant.

• Damped SHM Solution: $x(t) = Ae^{-bt/2m} ext{cos}( ext{ω}t + ext{φ}_0)$

• Time Constant for Energy Loss: $au = m/b$

Chapter 16: Waves

Traveling Waves

A traveling wave is an organized disturbance that moves with a definite speed v. The displacement of particles is parallel (longitudinal) or perpendicular (transverse) to the direction of wave propagation.

• Mechanical Waves: Require a medium (e.g., sound, waves on strings).

• Electromagnetic Waves: Can travel through a vacuum; speed depends on the medium's index of refraction.

Wave Functions and Sinusoidal Waves

Sinusoidal waves are periodic in both time and space. The wave function describes the displacement as a function of position and time.

• General Wave Function: $D(x, t) = A ext{sin}[2 ext{π}(x/ ext{λ} - t/T)] = A ext{sin}(kx - ext{ω}t + ext{φ}_0)$

• Amplitude (A): Maximum displacement from equilibrium.

• Wavelength (λ): Distance between successive crests.

• Angular Frequency (ω): $ext{ω} = 2 ext{π}/T$

• Wave Number (k): $k = 2 ext{π}/ ext{λ}$

Wave Speed and Intensity

• Wave Speed on a String: $v = ext{√}(T/ ext{μ})$, where T is tension and μ is mass per unit length.

• Speed of Sound in Air: $v = ext{√}(B/ ho) = 343$ m/s at 20°C

• Speed of Light in Vacuum: $c = 3.00 imes 10^8$ m/s

• Wave Intensity: Power per unit area carried by the wave.

• Sound Intensity Level: $β = (10 ext{ dB}) ext{log}(I/I_0)$

Doppler Effect

The Doppler effect describes the change in frequency observed when the source or observer is moving.

Situation	Observed Frequency $f_o$
Stationary observer, stationary source	$f_o = f_s$
Observer moving towards source	$f_o = f_s rac{v + v_o}{v}$
Observer moving away from source	$f_o = f_s rac{v - v_o}{v}$
Source moving towards observer	$f_o = f_s rac{v}{v - v_s}$
Source moving away from observer	$f_o = f_s rac{v}{v + v_s}$

Chapter 17: Superposition and Boundary Conditions

Superposition Principle

When more than one wave is present in a medium, the displacement at each point is the sum of the displacements due to each individual wave.

• Constructive Interference: Occurs when waves are in phase, leading to increased amplitude.

• Destructive Interference: Occurs when waves are out of phase, leading to reduced amplitude.

Boundary Conditions for Standing Waves

Standing waves form under specific boundary conditions, such as fixed or free ends.

• Closed-Closed Tube/String: Nodes at both ends. Wavelengths: $λ_m = rac{2L}{m}$ Frequencies: $f_m = rac{m v}{2L} = m f_1$ $m = 1, 2, 3, ext{...}$

• Open-Open Tube: Antinodes at both ends. Same as closed-closed.

• Open-Closed Tube: Node at closed end, antinode at open end. Wavelengths: $λ_m = rac{4L}{m}$ Frequencies: $f_m = rac{m v}{4L} = m f_1$ $m = 1, 3, 5, 7, ext{...}$

Applications and Examples

• Musical Instruments: Standing waves explain the resonant frequencies of strings and air columns.

• Sound Waves: The boundary conditions determine the harmonics produced in tubes and strings.

Additional info: These notes expand on the provided slides by including definitions, formulas, and examples for each topic, ensuring completeness and academic clarity.