Oscillation and Waves

A motion that repeats itself at regular intervals of time is called periodic motion. If a particle in periodic motion moves to and fro about a fixed mean position, it is called an oscillatory or vibratory motion. Every oscillatory motion is periodic, but every periodic motion (e.g., planetary orbits) is not oscillatory.

Simple Harmonic Motion (SHM): SHM is the simplest and most fundamental form of oscillatory motion, defined as motion in which the restoring force (and acceleration) acting on the particle is directly proportional to its displacement from the mean position and is always directed toward that mean position:

where $k$ is the force constant (spring factor) and $\omega = \sqrt{k/m}$ is the angular frequency.

The instantaneous displacement $x(t)$ of a particle executing linear SHM is represented as a sinusoidal function of time:

• Amplitude ($A$): The maximum displacement of the oscillating particle on either side of the mean position.
• Time Period ($T$): The least time interval after which the periodic motion repeats: $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$.
• Frequency ($\nu$ or $f$): The number of complete oscillations executed per second: $\nu = \frac{1}{T} = \frac{\omega}{2\pi}$. Unit: $\text{Hertz (Hz)} = \text{s}^{-1}$.
• Phase ($(\omega t + \phi)$): The argument of the trigonometric function, representing the complete dynamical state (position and direction of motion) at time $t$.
• Initial Phase / Epoch ($\phi$): The phase of the particle at initial time $t = 0$.

Differentiating displacement with respect to time yields the instantaneous velocity ($v$):

• At the mean position ($x = 0$): Velocity is maximum: $v_{\max} = \omega A$.
• At extreme positions ($x = \pm A$): Velocity is zero: $v = 0$.

Differentiating velocity yields the instantaneous acceleration ($a$):

• At the mean position ($x = 0$): Acceleration is zero: $a = 0$.
• At extreme positions ($x = \pm A$): Acceleration is maximum: $a_{\max} = \omega^2 A$.

Applying Newton's second law $F = m \frac{d^2x}{dt^2} = -kx$ yields the canonical second-order linear homogeneous differential equation:

In an ideal undamped SHM, the total mechanical energy is conserved and oscillates continuously between kinetic energy ($K$) and potential energy ($U$):

• Kinetic Energy ($K$): Associated with particle velocity:$$K = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2) = \frac{1}{2} k (A^2 - x^2) = \frac{1}{2} k A^2 \sin^2(\omega t + \phi)$$
• Potential Energy ($U$): Work done against the linear restoring force in displacing the particle from $0$ to $x$:$$U = \int_0^x kx\,dx = \frac{1}{2} k x^2 = \frac{1}{2} m \omega^2 x^2 = \frac{1}{2} k A^2 \cos^2(\omega t + \phi)$$
• Total Mechanical Energy ($E$):$$E = K + U = \frac{1}{2} k (A^2 - x^2) + \frac{1}{2} k x^2 = \frac{1}{2} k A^2 = \frac{1}{2} m \omega^2 A^2 = \text{constant}$$

• Horizontal and Vertical Spring Oscillators: A mass $m$ attached to a spring of force constant $k$ executes SHM with period:$$T = 2\pi \sqrt{\frac{m}{k}}$$ In a vertical spring, gravity merely shifts the equilibrium position downwards by $y_0 = \frac{mg}{k}$; the oscillation frequency and time period remain completely unchanged.
• Springs in Series: Two springs of constants $k_1$ and $k_2$ connected end-to-end share the same restoring force ($F_1 = F_2 = F$) while total extension is $x = x_1 + x_2$:$$\frac{1}{k_s} = \frac{1}{k_1} + \frac{1}{k_2} \implies k_s = \frac{k_1 k_2}{k_1 + k_2} \implies T_s = 2\pi \sqrt{\frac{m(k_1+k_2)}{k_1 k_2}} = \sqrt{T_1^2 + T_2^2}$$
• Springs in Parallel: Two springs connected side-by-side share the same extension ($x_1 = x_2 = x$) while total force is $F = F_1 + F_2$:$$k_p = k_1 + k_2 \implies T_p = 2\pi \sqrt{\frac{m}{k_1 + k_2}} \implies \frac{1}{T_p^2} = \frac{1}{T_1^2} + \frac{1}{T_2^2}$$
• Cutting a Spring: The spring constant is inversely proportional to its natural length ($k \propto 1/L$). If a spring of constant $k$ is cut into two segments in the length ratio $m:n$, the spring constants of the parts become:$$k_1 = \frac{m+n}{m}k \quad \text{and} \quad k_2 = \frac{m+n}{n}k$$

A simple pendulum consists of a point mass (bob) of mass $m$ suspended from a rigid, frictionless support by a light, inextensible string of length $L$.

When displaced through a small angular displacement $\theta$ from the vertical, the restoring torque about the suspension point is:

Using rotational dynamics $\tau = I \alpha = (m L^2) \frac{d^2\theta}{dt^2}$:$$m L^2 \frac{d^2\theta}{dt^2} + mg L \theta = 0 \implies \frac{d^2\theta}{dt^2} + \frac{g}{L} \theta = 0$$

Comparing with the angular SHM equation $\frac{d^2\theta}{dt^2} + \omega^2 \theta = 0$ yields:

• Seconds Pendulum: A pendulum whose time period is exactly 2 seconds ($T = 2.0\,\text{s}$). Its length at sea level ($g = 9.8\,\text{m/s}^2$) is: $L = \frac{g T^2}{4\pi^2} = \frac{9.8 \times 4}{4\pi^2} \approx 0.993\,\text{m} \approx 1\,\text{meter}$.
• Pendulum in an Accelerating Elevator:
  • Accelerating upwards with acceleration $a$: $g_{\text{eff}} = g + a \implies T = 2\pi \sqrt{\frac{L}{g+a}}$ (Period decreases, clock runs fast).
  • Accelerating downwards with acceleration $a$: $g_{\text{eff}} = g - a \implies T = 2\pi \sqrt{\frac{L}{g-a}}$ (Period increases, clock runs slow).
  • Free fall ($a = g$): $g_{\text{eff}} = 0 \implies T \to \infty$ (Pendulum ceases to oscillate).

In real physical systems, energy is dissipated due to viscous drag or friction. If the damping force is proportional to velocity ($F_d = -b v = -b \frac{dx}{dt}$, where $b$ is the damping coefficient), Newton's equation becomes:

where $\gamma = \frac{b}{2m}$ and $\omega_0 = \sqrt{\frac{k}{m}}$. For weak damping ($\gamma < \omega_0$), the solution is an amplitude-decaying oscillation:

The total mechanical energy decays exponentially with time: $E(t) = E_0 e^{-\frac{b t}{m}}$.

When an oscillator is subjected to an external periodic driving force $F(t) = F_0 \cos(\omega_d t)$, the steady-state amplitude is given by:

Resonance: When the driving frequency matches the natural frequency of the oscillator ($\omega_d \approx \omega_0$), the amplitude of oscillation becomes exceptionally large. If damping $b \to 0$, the resonance amplitude tends to infinity. In practical engineering, damping controls the sharpness of resonance (Quality factor $Q = \frac{\omega_0 m}{b}$).

A wave is a disturbance that travels through an elastic medium, transferring energy and momentum without transporting matter.

• Transverse Waves: The particles of the medium oscillate perpendicular to the direction of wave propagation. Characterized by crests and troughs. They require shear elasticity and can propagate only in solids and on liquid surfaces (e.g., waves on a guitar string, electromagnetic waves).
• Longitudinal Waves: The particles of the medium oscillate parallel to the direction of wave propagation. Characterized by alternating compressions (high density/pressure) and rarefactions (low density/pressure). They require volume elasticity (bulk modulus) and can propagate through solids, liquids, and gases (e.g., sound waves, ultrasound).

The propagation speed of a transverse mechanical wave on a flexible string under tension $T$ and linear mass density $\mu = \frac{m}{L}$ is given by:

In general, the speed of a longitudinal wave in an elastic fluid medium is governed by the bulk modulus $B$ and density $\rho$:

1. Newton's Formula (1687): Newton assumed that the compressions and rarefactions of sound waves occur very slowly, allowing heat to flow freely so that temperature remains constant (an isothermal process). For an isothermal process ($PV = \text{const}$), the isothermal bulk modulus is $B_{\text{iso}} = P$.

At standard temperature and pressure ($P = 1.013 \times 10^5\,\text{Pa}, \rho = 1.293\,\text{kg/m}^3$):$$v_{\text{Newton}} = \sqrt{\frac{1.013 \times 10^5}{1.293}} \approx 280\,\text{m/s}$$

The experimental speed of sound in air at STP is $332\,\text{m/s}$. Newton's formula yielded a discrepancy of approximately $15.7\%$.

2. Laplace's Correction (1816): Pierre-Simon Laplace resolved the discrepancy by recognizing that sound compressions and rarefactions occur with high speed, and air is a poor conductor of heat. Therefore, no heat exchange occurs with the surrounding air, making the process adiabatic ($P V^\gamma = \text{const}$).

Differentiating $P V^\gamma = \text{const}$: $\gamma P V^{\gamma - 1} dV + V^\gamma dP = 0 \implies -V \frac{dP}{dV} = \gamma P$. Hence, the adiabatic bulk modulus is $B_{\text{ad}} = \gamma P$, where $\gamma = C_p/C_v$ is the adiabatic index.

For diatomic air ($\gamma = 1.40$, $M = 0.0289\,\text{kg/mol}$):$$v_{\text{Laplace}} = \sqrt{\frac{1.40 \times 1.013 \times 10^5}{1.293}} = 280 \times \sqrt{1.40} \approx 331.3\,\text{m/s}$$

This matches experimental measurements perfectly.

• Effect of Pressure: At constant temperature, $P/\rho = \frac{RT}{M} = \text{constant}$. Thus, sound speed is independent of atmospheric pressure at constant temperature.
• Effect of Temperature: $v = \sqrt{\frac{\gamma RT}{M}} \implies v \propto \sqrt{T}$. Speed increases with absolute temperature. For small Celsius temperatures $t$, $v_t \approx v_0 + 0.61\,t\,\text{m/s}$.
• Effect of Humidity: Molar mass of water vapor ($18\,\text{g/mol}$) is less than dry air ($28.9\,\text{g/mol}$). Moist air has lower density than dry air ($\rho_{\text{moist}} < \rho_{\text{dry}}$), so sound travels faster in humid air.
• Effect of Wind: If wind blows with speed $w$ at an angle $\theta$ to the sound direction, effective speed is $v_{\text{eff}} = v \pm w \cos\theta$.

A harmonic progressive wave propagating in the positive $x$-direction is expressed as:

• Propagation constant / Angular wave number: $k = \frac{2\pi}{\lambda}$ (unit: $\text{rad/m}$).
• Wave velocity: $v = \frac{\omega}{k} = \nu \lambda$.
• Particle velocity: $v_p = \frac{\partial y}{\partial t} = -\omega A \cos(kx - \omega t)$. Relation to wave speed: $v_p = -v \left(\frac{\partial y}{\partial x}\right) = -v \times (\text{wave slope})$.

When two or more waves traverse the same elastic medium simultaneously, the net displacement of any particle is the algebraic sum of the individual displacements: $y = y_1 + y_2 + \dots$.

• Reflection at a Rigid / Fixed Boundary: The boundary exerts an equal and opposite reaction, introducing a phase change of $\pi$ radians ($180^\circ$): $y_r = -A \sin(kx + \omega t)$. Crest reflects as trough.
• Reflection at a Free / Open Boundary: No phase change occurs: $y_r = A \sin(kx + \omega t)$. Crest reflects as crest.

When two identical progressive waves of equal amplitude and frequency travel in opposite directions along the same line, their superposition produces a standing (stationary) wave:

• Position-dependent Amplitude: $A_{\text{res}}(x) = 2A |\sin kx|$.
• Nodes ($N$): Points of zero displacement ($A_{\text{res}} = 0$). Occur where $\sin kx = 0 \implies kx = n\pi \implies x = 0, \frac{\lambda}{2}, \lambda, \frac{3\lambda}{2}, \dots$. Distance between consecutive nodes is $\frac{\lambda}{2}$. Pressure and strain variations are maximum at nodes.
• Antinodes ($A$): Points of maximum displacement ($A_{\text{res}} = 2A$). Occur where $|\sin kx| = 1 \implies x = \frac{\lambda}{4}, \frac{3\lambda}{4}, \frac{5\lambda}{4}, \dots$. Distance between consecutive antinodes is $\frac{\lambda}{2}$. Distance between a node and adjacent antinode is $\frac{\lambda}{4}$. Pressure variation is zero at antinodes.
• Standing waves do NOT transport energy across the medium; energy remains trapped between nodes.

End Correction ($e$): Due to air momentum, the displacement antinode forms slightly outside an open pipe end by $e \approx 0.6\,r$ (where $r$ is pipe radius). Effective length is $L_{\text{eff}} = L + 0.6r$ for a closed pipe, and $L_{\text{eff}} = L + 1.2r$ for an open pipe.

When two sound waves of slightly different frequencies $\nu_1$ and $\nu_2$ (with $|\nu_1 - \nu_2| \le 10\,\text{Hz}$) travel simultaneously in the same direction, their superposition produces a periodic rise and fall in the intensity of sound called beats.

Let the two waves be $y_1 = A \sin(2\pi \nu_1 t)$ and $y_2 = A \sin(2\pi \nu_2 t)$. By superposition:

• Modulated Amplitude: $A_{\text{mod}} = 2A \left|\cos\left[\pi (\nu_1 - \nu_2) t\right]\right|$.
• Waxing (Maximum Intensity): Occurs when $\cos[\pi (\nu_1 - \nu_2) t] = \pm 1$, at intervals $\Delta t = \frac{1}{\nu_1 - \nu_2}$.
• Waning (Minimum Intensity): Occurs when $\cos[\pi (\nu_1 - \nu_2) t] = 0$, at intervals $\Delta t = \frac{1}{\nu_1 - \nu_2}$.
• Beat Frequency ($f_{\text{beat}}$): The number of beats heard per second:$$f_{\text{beat}} = |\nu_1 - \nu_2|$$
• Persistence of Hearing: Due to human persistence of hearing ($0.1\,\text{s}$), beats are distinctly audible to the human ear only when $f_{\text{beat}} \le 10\,\text{Hz}$.
• Tuning Fork Loading & Filing Rules:
  • Waxing / Loading a prong increases its mass, thereby decreasing its frequency ($\nu \downarrow$).
  • Filing a prong reduces its mass, thereby increasing its frequency ($\nu \uparrow$).

The apparent change in the observed frequency of sound due to the relative motion between the source of sound, the observer, and the medium is called the Doppler Effect.

General Doppler Equation:

where $v$ is the speed of sound, $v_o$ is observer speed, $v_s$ is source speed, and $\nu$ is source true frequency.

Sign Convention Guide:

• Upper signs ($+v_o$ in numerator, $-v_s$ in denominator) apply when the relative motion is APPROACH (apparent frequency increases: $\nu' > \nu$).
• Lower signs ($-v_o$ in numerator, $+v_s$ in denominator) apply when the relative motion is RECEDING (apparent frequency decreases: $\nu' < \nu$).