Physics Theory & Explanations (Waves & Oscillations)

Based on KUET Physics PYQ (Subtopics A-E)

Subtopic A: Simple Harmonic Motion (SHM) Fundamentals

1. Define Simple Harmonic Motion (SHM).

Simple Harmonic Motion (SHM) is a special type of periodic motion where a particle moves to and fro about a fixed equilibrium position. It is defined by the condition that the restoring force (and thus acceleration) acting on the particle is:

Directly proportional to its displacement from the equilibrium position.
Always directed towards the equilibrium position.

Mathematically: $F \propto - x$ or $a \propto - x$ .

2. Explain SHM and discuss its oscillatory behavior.

SHM is the simplest form of oscillatory motion. When a particle is displaced from its mean position, a restoring force acts on it, trying to bring it back. Due to inertia, it overshoots the mean position, creating an oscillation.

Oscillatory Behavior: The particle oscillates between two extreme points (amplitude boundaries) with a constant time period. The velocity is maximum at the equilibrium position and zero at the extreme positions, while acceleration is maximum at the extremes and zero at the equilibrium.

3. Discuss some basic characteristics of Simple Harmonic Motion.

Linearity: The restoring force is linear ( $F = - k x$ ).
Isochronism: The time period ( $T$ ) is independent of the amplitude (for small oscillations).
Amplitude ( $A$ ): The maximum displacement from the mean position.
Phase: The state of the particle (position and direction of motion) at any instant.
Energy Conservation: The total mechanical energy (Kinetic + Potential) remains constant throughout the motion (in ideal conditions without friction).

Subtopic B: SHM Energy, Superposition, and Calculations

1. What is the Principle of Superposition?

The Principle of Superposition states that when two or more waves (or oscillations) overlap in space, the resultant displacement of the particle at any instant is the vector sum of the individual displacements produced by each wave acting independently.

Note: This principle is valid only for linear homogeneous differential equations (i.e., when wave amplitudes are not excessively large).

2. What are Lissajous figures?

Lissajous figures are the complex curve patterns traced by a particle when it is subjected to two mutually perpendicular Simple Harmonic Motions simultaneously.

The shape of the figure depends on:

The ratio of the frequencies (time periods) of the two component vibrations.
The phase difference ( $ϕ$ ) between them.
The ratio of their amplitudes.

3. Discuss Lissajous figures for different phase differences (assuming equal time periods).

If two SHMs of equal frequency act at right angles ( $x = A sin (ω t)$ and $y = B sin (ω t + ϕ)$ ):

$ϕ = 0$ or $2 π$ : The resultant is a straight line passing through the origin with a positive slope.
$ϕ = π$ : The resultant is a straight line with a negative slope.
$ϕ = π /2$ (90°): The resultant is an ellipse (or a circle if amplitudes $A = B$ ).
$ϕ = π /4$ : The resultant is an tilted ellipse.

Subtopic C: Damped and Forced Oscillations / Resonance

4. Discuss the sharpness of resonance.

Sharpness of resonance refers to how rapidly the amplitude of oscillation falls off as the driving frequency moves away from the resonant frequency.

Sharp Resonance: Occurs when damping is low. The peak amplitude is very high, and the resonance curve is narrow.
Flat Resonance: Occurs when damping is high. The peak amplitude is lower, and the resonance curve is broad.
It is quantitatively measured by the Quality Factor (Q).

5. Define Bandwidth of Resonance.

Bandwidth is the range of frequencies around the resonant frequency for which the power dissipated in the circuit (or mechanical system) is at least half of the maximum power. It is the difference between the two “half-power frequencies” ( $f_{2} - f_{1}$ ).

6. Describe the term “Quality Factor” (Q-Factor).

The Quality Factor is a dimensionless parameter that characterizes the damping of an oscillator. It indicates the efficiency of energy storage relative to energy loss.

Definition: $Q = 2 π \times \frac{Energy Stored}{Energy Dissipated per Cycle}$ .
Higher Q means lower damping and sharper resonance.

7. Distinguish between Free, Forced, and Damped vibrations.

Free Vibration: Oscillates at its own natural frequency after being disturbed; no external force acts thereafter; amplitude is constant (ideally).
Damped Vibration: Oscillates at a frequency slightly lower than natural frequency; experiences resistive forces; amplitude decays exponentially with time.
Forced Vibration: Oscillates at the frequency of the external driver; amplitude is constant but depends on the difference between driving and natural frequencies.

Subtopic D: Wave Motion

1. What is meant by the intensity of a plane progressive wave?

Intensity is defined as the average amount of energy flowing per unit time per unit area held perpendicular to the direction of propagation of the wave.

Unit: Watt per square meter ( $W / m^{2}$ ).
It is directly proportional to the square of the amplitude ( $A^{2}$ ) and the square of the frequency ( $f^{2}$ ).

2. What is Wave Velocity?

Wave velocity (or Phase Velocity) is the speed at which the disturbance (or a specific phase, like a crest) travels through the medium. It is given by $v = f λ$ .

3. What are Stationary (Standing) Waves?

Stationary waves are formed by the superposition of two identical progressive waves traveling in opposite directions along the same line.

Properties: The wave profile does not move through the medium. Energy is not transported from one point to another; it is confined between nodes.

6. Discuss the formation of stationary waves due to reflection at a free end boundary.

When a wave reflects from a “free” boundary (like a rope with a ring sliding on a pole), the reflected wave does not undergo a phase change (phase shift = 0). The incident crest reflects as a crest. The superposition of the incoming and outgoing waves creates an Anti-node at the free boundary (maximum displacement).

8. Mention the basic difference between phase velocity and group velocity.

Phase velocity is the speed of individual ripples, while group velocity is the speed of the wave packet as a whole. In a dispersive medium (where velocity depends on frequency), they are different ( $v_{g} < v_{p}$ usually). In a non-dispersive medium (like sound in air), they are equal.

Subtopic E: Acoustics and Sound

1. What is Doppler’s Effect in Sound?

Doppler’s effect is the phenomenon of the apparent change in the frequency (pitch) of a sound heard by an observer due to the relative motion between the source of the sound and the observer.

Approach: Frequency appears to increase.
Recession: Frequency appears to decrease.

2. What are Beats? How are they produced?

Definition: Beats are the periodic variations in the intensity (loudness) of sound heard when two sound waves of slightly different frequencies interfere with each other.
Production: They are produced by the superposition principle. The two waves alternately constructively interfere (waxing/loud sound) and destructively interfere (waning/soft sound).
Beat Frequency: The number of beats per second is equal to the difference in frequencies ( $f_{1} - f_{2}$ ).

4. Discuss the factors influencing loudness.

Loudness depends on:

Amplitude: Loudness $\propto$ Amplitude $^{2}$ .
Surface Area: Larger vibrating area increases loudness.
Distance: Loudness $\propto 1/ r^{2}$ .
Density of Medium: Higher density transmits louder sound.
Motion of Medium: Wind blowing towards the listener increases loudness.
Sensitivity of the Ear: Varies with frequency.

5. Explain Sabine’s Reverberation Formula.

Sabine’s formula relates the reverberation time ( $T$ ) of a room to its volume ( $V$ ) and total absorption ( $A$ ).

Formula: $T = \frac{0.161 V}{\sum a S}$
- $V$ = Volume of the hall (m³).
- $\sum a S$ = Total absorption (Sum of surface areas $\times$ absorption coefficients).
It states that reverberation time is directly proportional to volume and inversely proportional to total absorption.

6. Write down the requisites for a good auditorium.

For good acoustics, an auditorium must have:

Optimum Reverberation Time: Neither too long (echoes/blurring) nor too short (dead sound).
Adequate Loudness: Every seat should receive sufficient sound intensity.
Uniform Distribution: Sound should be spread evenly (using diffusers/curved surfaces).
Absence of Defects: No echoes, echelon effects (staircase echoes), or sound shadows.
Resonance Control: The hall should not resonate with specific frequencies.
Noise Insulation: Isolation from outside noise and internal machinery noise.

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Waves & Oscillations PYQ Theory