Superposition of Waves Study Guide

INTRODUCTION TO WAVES

An oscillatory disturbance traveling through a medium without a change of form is called a wave. Waves serve as a mechanism to carry or transfer energy from one point to another without the permanent transfer of matter.

Key Characteristics:

• Waves are doubly periodic, meaning they repeat in both space and time.

• Common types include water waves, light waves, sound waves, mechanical waves, and electromagnetic waves.

PROGRESSIVE WAVES

A wave that travels through a medium continuously in a forward direction is known as a progressive wave.

Properties of Progressive Waves:

1. All particles of the medium vibrate about their mean positions, performing Simple Harmonic Motion (S.H.M.).

2. All vibrating particles have the same amplitude, period, and frequency.

3. The phase changes from one particle to another.

4. No particle remains permanently at rest; each comes to rest momentarily at the extreme positions of vibration.

5. Particles attain maximum velocity when passing through their mean positions.

6. Energy is transferred along the wave, but there is no transfer of matter.

7. The wave propagates through the medium with a certain velocity.

Types of Progressive Waves:

• Transverse Waves: Vibrations of particles are perpendicular to the direction of propagation. They produce crests and troughs alternately and can only pass through solids.

• Longitudinal Waves: Vibrations of particles are parallel to the direction of propagation. They produce compressions and rarefactions alternately and can pass through solids, liquids, and gases.

REFLECTION OF WAVES

Reflection occurs when a wave traveling through a medium reaches a boundary and part of the wave energy comes back into the same medium, changing its direction.

Reflection of a Transverse Wave

• Case 1: Rare Medium to Denser Medium (Rigid Support): When a crest generated in a string reaches a rigid support, it exerts an upward force. Per Newton's 3rd Law, the support exerts an equal and opposite downward reaction. This causes the crest to be reflected as a trough. There is a phase change of $\pi$ .

• Case 2: Denser Medium to Rarer Medium (Free End): If a string is attached to a frictionless ring on a rod, the ring moves upward when the crest arrives. The wave is reflected back as a crest, and there is no phase change.

• Case 3: Heavy vs. Light Strings:

  • A crest traveling from a heavy (denser) string to a light (rarer) string is reflected as a crest.

  • A crest traveling from a light (rarer) string to a heavy (denser) string is reflected as a trough.

Reflection of a Longitudinal Wave

• From a Denser Medium (Rigid Wall): When a longitudinal wave hits a rigid wall, it exerts force. The wall exerts an equal and opposite force. Consequently, a compression is reflected as a compression and a rarefaction as a rarefaction. There is a phase change of $\pi$.

• From a Rarer Medium: The particles get displaced in the direction of the wave. There is no phase change; however, a compression is reflected as a rarefaction and a rarefaction is reflected as a compression.

SUPERPOSITION OF WAVES

When two or more waves pass through a common point in a medium, each wave produces its own displacement independently of the others. The resultant displacement is the vector sum of the individual displacements. This principle applies to sound, light, and waves on strings without changing the individual shapes of the waves.

Interference Types

• Constructive Interference: Occurs when two wave pulses of equal amplitude and same phase meet. The resultant displacement is the sum of the individual displacements.

• Destructive Interference: Occurs when two wave pulses of equal amplitude and opposite phases meet (phase difference of $180^{\circ}$). The resultant displacement is zero.

Analytical Treatment of Resultant Amplitude

Consider two waves with same frequency but different amplitudes $A_1$ and $A_2$, differing in phase by $\phi$: $y_1 = A_1 \sin(\omega t)$ $y_2 = A_2 \sin(\omega t + \phi)$

Resultant displacement $y = y_1 + y_2$: $y = (A_1 + A_2 \cos(\phi)) \sin(\omega t) + A_2 \cos(\omega t) \sin(\phi)$

Let $A_1 + A_2 \cos(\phi) = A \cos(\theta)$ and $A_2 \sin(\phi) = A \sin(\theta)$. The resultant wave is $y = A \sin(\omega t + \theta)$.

The resultant amplitude $A$ is given by: $A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)}$

Special Cases:

• Case I ($\phi = 0$): Waves are in phase. $A_{max} = A_1 + A_2$. Intensity $I_{max} \propto (A_1 + A_2)^2$.

• Case II ($\phi = \pi$): Waves are out of phase. $A_{min} = A_1 - A_2$. Intensity $I_{min} \propto (A_1 - A_2)^2$.

STATIONARY WAVES

Stationary waves are produced by the superposition of two identical progressive waves (same amplitude, frequency, and speed) traveling along the same line in opposite directions.

Equation of a Stationary Wave

$y_1 = A \sin\left[ 2\pi \left(nt - \frac{x}{\lambda} \right) \right]$ $y_2 = A \sin\left[ 2\pi \left(nt + \frac{x}{\lambda} \right) \right]$

Resultant displacement: $y = R \sin(2\pi n t)$ Where resultant amplitude $R = 2A \cos \left( \frac{2\pi x}{\lambda} \right)$.

Nodes and Antinodes

• Nodes: Points of minimum displacement ($R = 0$).   Conditions: $\cos \left( \frac{2\pi x}{\lambda} \right) = 0$.   Positions: $x = \frac{\lambda}{4}, \frac{3\lambda}{4}, \dots$   Distance between successive nodes: $\frac{\lambda}{2}$.

• Antinodes: Points of maximum displacement ($R = \pm 2A$).   Conditions: $\cos \left( \frac{2\pi x}{\lambda} \right) = \pm 1$.   Positions: $x = 0, \frac{\lambda}{2}, \lambda, \dots$   Distance between successive antinodes:$\frac{\lambda}{2}$

• Node to Adjacent Antinode: The distance is $\frac{\lambda}{4}$.

Properties of Stationary Waves

• Energy is not transferred through the medium.

• All particles perform S.H.M with the same period, but amplitudes vary (zero at nodes, maximum at antinodes).

• It is doubly periodic.

• Particles within one loop are in the same phase; particles in adjacent loops are out of phase.

HARMONICS AND OVERTONES

• Harmonics: Integral multiples of the fundamental frequency ($n$). The fundamental is the first harmonic.

• Overtones: Higher frequencies actually present in addition to the fundamental. The first frequency higher than the fundamental is the first overtone.

End Correction

In vibrating air columns, the antinode forms slightly beyond the open end of a pipe. This distance is the end correction ($e$): $e = 0.3d$ (where $d$ is the inner diameter).

• Pipe closed at one end: Correct length $L = l + e$.

• Pipe open at both ends: Correct length $L = l + 2e$.

Vibrations of Air Column

1. Pipe Closed at One End:

• Boundary conditions: Node at closed end, antinode at open end.

• Fundamental Frequency: $n = \frac{v}{4L} = \frac{v}{4(l+e)}$.

• Harmonics present: Only odd harmonics ($n, 3n, 5n, \dots$). Even harmonics are absent.

• $p^{th}$ overtone frequency: $n_p = (2p + 1)n$.

2. Pipe Open at Both Ends:

• Boundary conditions: Antinodes at both open ends.

• Fundamental Frequency: $n = \frac{v}{2L} = \frac{v}{2(l+2e)}$.

• Harmonics present: All harmonics ($n, 2n, 3n, \dots$).

• $p^{th}$ overtone frequency: $n_p = (p + 1)n$.

VIBRATIONS OF A STRETCHED STRING

The velocity of a transverse wave on a string is $v = \sqrt{\frac{T}{m}}$, where $T$ is tension and $m$ is mass per unit length (linear density).

Fundamental Frequency: $n = \frac{1}{2l} \sqrt{\frac{T}{m}}$ All harmonics are present in a stretched string ($n, 2n, 3n, \dots$).

Laws of Vibrating Strings

1. Law of Length: $n \propto \frac{1}{l}$ (if $T, m$ constant).

2. Law of Tension: $n \propto \sqrt{T}$ (if $l, m$ constant).

3. Law of Linear Density: $n \propto \frac{1}{\sqrt{m}}$ (if $T, l$ constant).

4. Law of Radius: $n \propto \frac{1}{r}$ (if $\rho, l, T$ constant).

5. Law of Density: $n \propto \frac{1}{\sqrt{\rho}}$ (if $l, T, r$ constant).

SONOMETER AND BEATS

Sonometer

Consists of a sound box, two bridges ($P, Q$), and a wire. Resonance occurs when the tuning fork frequency matches the frequency of the vibrating string length, causing a paper rider to fall.

• Verification: Used to verify the laws of length, tension, and linear density.

Beats

Beats are produced by the superposition of two sound waves of the same amplitude but slightly different frequencies ($n_1$ and $n_2$) traveling in the same direction.

• Waxing: Maximum sound intensity.

• Waning: Minimum sound intensity.

• Beat Frequency ($N$): $N = n_1 - n_2$.

• Beat Period ($T$): $T = \frac{1}{n_1 - n_2}$.

CHARACTERISTICS OF SOUND AND INSTRUMENTS

Characteristics

1. Loudness: Related to intensity ($I$). Sound level $\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)$, measured in decibels (dB), where $I_0 = 10^{-12}\,W/m^2$.

2. Pitch: Perception of frequency; higher frequency corresponds to higher pitch.

3. Quality (Timbre): Depends on the number of overtones; allows distinction between different sources of the same pitch and loudness.

Musical Instruments

• Stringed: Plucked (Sitar), Bowed (Violin), Struck (Piano).

• Wind: Free wind (Harmonium), Edge (Flute), Reed pipes (Saxophone).

• Percussion: Stretched membrane (Tabla, Drum), Metal (Cymbals).

QUESTIONS & DISCUSSION

• Q: State any two characteristics of progressive waves.

• A: (i) All particles perform S.H.M with same amplitude and frequency. (ii) Energy is transferred through the medium without matter transfer.

• Q: Distinguish between free and forced vibrations.

• A: Free vibrations occur at natural frequency after an initial disturbance; forced vibrations occur under an external periodic force at the forcing frequency.

• Q: Derive an expression for practical determination of end correction for a pipe closed at one end.

• A: Using two lengths $l_1$ and $l_2$ for the same frequency: $e = \frac{n_2 l_2 - n_1 l_1}{n_1 - n_2}$. (Based on the transcript derivation: $e = \frac{l_2 - 3l_1}{2}$ for overtones, or simplified comparison methods as shown in equations).