6.4 Oscillations, Simple Harmonic Motion, Waves & Sound Mechanics

6.4 Oscillations, Simple Harmonic Motion, Waves & Sound Mechanics

Oscillatory systems, wave propagation, standing waves, and acoustic mechanics constitute a critical domain in Physics. This section provides quantitative theory, mathematical derivations, wave equations, sound speed corrections, and Doppler effect analysis required for the AMC Initial Test.

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1. Simple Harmonic Motion (SHM) Mechanics

Definition & Governing Differential Equation

Simple Harmonic Motion (SHM) occurs when the restoring force $F$ acting on an oscillating body is directly proportional to its displacement $x$ from the mean position and directed toward that mean position: $F = -k x \implies a = -\left(\frac{k}{m}\right) x = -\omega^2 x$ where $\omega = \sqrt{\frac{k}{m}}$ is angular frequency. Differential equation of SHM: $\frac{d^2 x}{dt^2} + \omega^2 x = 0$

Displacement, Velocity & Acceleration Equations

• Displacement: $x(t) = x_0 \sin(\omega t + \phi)$
• Velocity: $v(t) = \frac{dx}{dt} = x_0 \omega \cos(\omega t + \phi) = \pm \omega \sqrt{x_0^2 - x^2}$
  • Maximum velocity at mean position ($x = 0$): $v_{max} = \omega x_0$
  • Zero velocity at extreme positions ($x = \pm x_0$)
• Acceleration: $a(t) = -\omega^2 x$
  • Maximum acceleration at extreme positions ($x = \pm x_0$): $a_{max} = \omega^2 x_0$
  • Zero acceleration at mean position ($x = 0$)

Oscillatory Systems

• Mass-Spring System: Period $T = 2\pi \sqrt{\frac{m}{k}}$, Frequency $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
• Simple Pendulum: Period $T = 2\pi \sqrt{\frac{L}{g}}$ (valid for small angular displacements $\theta < 10^\circ$)
  • Seconds Pendulum: Pendulum with period $T = 2.0\text{ s}$ ($L \approx 0.992\text{ m} \approx 1\text{ m}$ on Earth).

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2. Energy Conservation in SHM & Damped/Forced Oscillations

Energy Formulas

• Kinetic Energy: $K = \frac{1}{2} m v^2 = \frac{1}{2} k (x_0^2 - x^2)$
• Potential Energy: $U = \frac{1}{2} k x^2$
• Total Mechanical Energy: $E_{total} = K + U = \frac{1}{2} k x_0^2 = \text{constant}$

          Energy Distribution in SHM
    Energy ^
           |       U (Potential)    K (Kinetic)
   E_total +------*---------------*------ (Constant)
           |     * \             / *
           |    *   \           /   *
           |   *     \         /     *
           |  *       \  E/2  /       *
           | *         \  |  /         *
         0 +------------+-+-+-------------> Displacement (x)
             -x_0       0   x_0/\sqrt{2}  +x_0

Equal Energy Condition: At displacement $x = \frac{x_0}{\sqrt{2}}$, $K = U = \frac{1}{2} E_{total}$.

Damped & Forced Oscillations, Resonance

• Damped Oscillations: Mechanical energy progressively dissipates due to friction/viscosity ($x(t) = x_0 e^{-b t / 2m} \cos(\omega' t)$).
• Resonance: Occurs when driving frequency matches natural frequency ($\omega_{driven} = \omega_0$), yielding maximum displacement amplitude.

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3. Wave Mechanics & Speed of Sound

Wave Fundamentals

• Wave equation: $v = f \lambda = \frac{\lambda}{T}$
• Phase difference: $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$
• Transverse (perpendicular vibration) vs Longitudinal (parallel vibration, e.g. sound waves).

Newton's Formula & Laplace's Correction for Sound Speed

Newton assumed sound compression/rarefaction in gas was isothermal ($E = P$): $v_{Newton} = \sqrt{\frac{P}{\rho}} \approx 281\text{ m/s } (\text{at STP for air})$ This underestimated experimental sound speed ($332\text{ m/s}$) by $\approx 16%$.

Laplace's Correction: Compressions/rarefactions occur so rapidly that no heat exchange occurs (adiabatic process, $E = \gamma P$): $v_{Laplace} = \sqrt{\frac{\gamma P}{\rho}}$ For diatomic air ($\gamma = 1.40$), $v = \sqrt{1.40 \times \frac{1.013 \times 10^5}{1.293}} \approx 332\text{ m/s}$ at $0^\circ\text{C}$.

Environmental Dependencies of Sound Speed

• Temperature: $v \propto \sqrt{T(\text{Kelvin})}$. Linear approximation near room temperature: $v_T = v_0 + 0.61 T(^\circ\text{C})$ Speed increases by $0.61\text{ m/s}$ per $1^\circ\text{C}$ temperature rise.
• Humidity: Moist air is less dense than dry air ($\rho_{moist} < \rho_{dry}$), so sound travels faster in humid air.
• Pressure: Sound speed is independent of pressure changes at constant temperature.

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4. Superposition, Standing Waves & Doppler Effect

Beats

Superposition of two waves of slightly different frequencies $f_1, f_2$ yields periodic amplitude variations (beats): $f_{beat} = |f_1 - f_2|$

Standing Waves in Pipes & Strings

• Stretched String (fixed ends) & Open Organ Pipe: $f_n = n \left(\frac{v}{2L}\right) = n f_1 \quad (n = 1, 2, 3, \dots)$ Contains all integer harmonic multiples.
• Closed Organ Pipe (closed at one end): $f_n = (2n - 1) \left(\frac{v}{4L}\right) \quad (n = 1, 2, 3, \dots)$ Contains only odd harmonics ($f_1, 3f_1, 5f_1, \dots$). Fundamental frequency: $f_1 = \frac{v}{4L}$.

Doppler Effect

Apparent frequency shift due to relative motion between source ($s$) and observer ($o$): $f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$

• Observer moving towards stationary source ($+v_o$): $f' = f \left(\frac{v + v_o}{v}\right)$ (pitch increases)
• Observer moving away from stationary source ($-v_o$): $f' = f \left(\frac{v - v_o}{v}\right)$ (pitch decreases)
• Source moving towards stationary observer ($-v_s$): $f' = f \left(\frac{v}{v - v_s}\right)$ (pitch increases)
• Source moving away from stationary observer ($+v_s$): $f' = f \left(\frac{v}{v + v_s}\right)$ (pitch decreases)

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5. Worked AMC Numerical Examples

Example 1: Closed Organ Pipe Fundamental Frequency

Question: Calculate fundamental frequency $f_1$ of an organ pipe closed at one end with length $L = 0.85\text{ m}$ given sound speed $v = 340\text{ m/s}$.

Solution:

1. Formula for closed pipe: $f_1 = \frac{v}{4L}$.
2. Substitute values: $f_1 = \frac{340}{4 \times 0.85} = \frac{340}{3.4} = 100\text{ Hz}$.

Example 2: Temperature Effect on Sound Speed

Question: Find the speed of sound in air at $25^\circ\text{C}$ taking speed at $0^\circ\text{C}$ as $v_0 = 332\text{ m/s}$.

Solution:

1. Use formula: $v_T = v_0 + 0.61 T$.
2. Calculate: $v_{25} = 332 + 0.61(25) = 332 + 15.25 = 347.25\text{ m/s}$.

Example 3: Doppler Frequency Shift

Question: An ambulance siren emits $f = 500\text{ Hz}$ sound while moving at $v_s = 30\text{ m/s}$ towards a stationary bystander. Given sound speed $v = 330\text{ m/s}$, calculate apparent frequency $f'$.

Solution:

1. Use source moving towards observer formula: $f' = f \left(\frac{v}{v - v_s}\right)$.
2. Substitute values: $f' = 500 \left(\frac{330}{330 - 30}\right) = 500 \left(\frac{330}{300}\right) = 500 \times 1.1 = 550\text{ Hz}$.