Class 11 Physics Revision Notes & Formula Sheet

Chapter: Waves (तरंगें)

1. Fundamental Concepts

Wave Motion (तरंग गति): A periodic disturbance that travels through a medium, transferring energy and momentum from one point to another without the actual transfer of matter.
Mechanical Waves (यांत्रिक तरंगें): Require a material medium for propagation (e.g., Sound waves, water waves, string waves).
Non-Mechanical / Electromagnetic Waves (अयांत्रिक तरंगें): Do not require any medium for propagation (e.g., Light waves, X-rays, Radio waves).
Transverse Waves (अनुप्रस्थ तरंगें): Particles of the medium vibrate perpendicular to the direction of wave propagation. (Consists of Crests and Troughs).
Longitudinal Waves (अनुदैर्ध्य तरंगें): Particles of the medium vibrate parallel to the direction of wave propagation. (Consists of Compressions and Rarefactions).

2. Basic Wave Parameters

Amplitude ($A$): Maximum displacement of a particle from its mean position.
Wavelength ($\lambda$): Distance between two consecutive crests/troughs or compressions/rarefactions.
Time Period ($T$): Time taken to complete one full vibration.
Frequency ($f$ or $\nu$): Number of vibrations per second ($f = 1 / T$).
Angular Frequency ($\omega$): $$\omega = 2\pi f = \frac{2\pi}{T}$$
Angular Wave Number / Propagation Constant ($k$): $$k = \frac{2\pi}{\lambda}$$
Wave Velocity ($v$): Speed at which the wave profile moves forward. $$v = f \lambda = \frac{\omega}{k}$$

3. Equation of a Progressive Wave (प्रगामी तरंग)

A simple harmonic wave travelling along the positive x-axis is represented as:

$$y(x, t) = A \sin(\omega t - kx + \phi)$$

or,

$$y(x, t) = A \sin 2\pi \left( \frac{t}{T} - \frac{x}{\lambda} \right)$$

If travelling along negative x-axis: $y(x, t) = A \sin(\omega t + kx + \phi)$
Particle Velocity ($v_p$): $$v_p = \frac{\partial y}{\partial t} = -v \times \text{Slope of wave curve} = -v \left(\frac{\partial y}{\partial x}\right)$$
Maximum Particle Velocity: $(v_p)_{\text{max}} = A \omega$

Phase Difference ($\Delta \phi$) Relations:

With Path Difference ($\Delta x$): $$\Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta x$$
With Time Difference ($\Delta t$): $$\Delta \phi = \frac{2\pi}{T} \cdot \Delta t$$

4. Speed of Sound and Mechanical Waves

A. Speed of Transverse Wave on a Stretched String:

$$v = \sqrt{\frac{T}{\mu}}$$

$T$ = Tension in the string (in Newtons)
$\mu$ = Mass per unit length (Linear mass density = $m / L$)

B. Speed of Longitudinal Wave (Sound) in Mediums:

In Solids (Rod): $v = \sqrt{\frac{Y}{\rho}}$ (where $Y$ = Young's Modulus, $\rho$ = Density)
In Fluids (Liquids/Gases): $v = \sqrt{\frac{B}{\rho}}$ (where $B$ = Bulk Modulus)

C. Newton's Formula & Laplace's Correction (Speed of Sound in Gas):

Newton's Assumption (Isothermal Process): $$v = \sqrt{\frac{P}{\rho}} \quad \text{(Gives } \approx 280 \text{ m/s in air - Incorrect)}$$
Laplace Correction (Adiabatic Process): $$v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T}{M}}$$
- $\gamma = C_p / C_v$ (Adiabatic index, for air $\gamma \approx 1.41$)
- $P$ = Pressure, $\rho$ = Density, $T$ = Temperature in Kelvin, $M$ = Molar mass.

Factors Affecting Speed of Sound in Gas:

Temperature: $v \propto \sqrt{T}$ $\implies \frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}}$
- Temperature coefficient: $v_t = v_0 + 0.61 t$ (where $t$ is in °C).
Humidity: Speed increases with increase in humidity ($\rho_{\text{moist air}} < \rho_{\text{dry air}}$).
Pressure: No effect on speed of sound at constant temperature.

5. Principle of Superposition & Standing Waves (अप्रगामी तरंगें)

Principle of Superposition: When two or more waves overlap, the resultant displacement is the vector sum of individual displacements: $$\vec{y} = \vec{y}_1 + \vec{y}_2 + \dots + \vec{y}_n$$
Standing Waves: Formed when two identical waves travel in opposite directions along the same line.
- Equation: $y = (2A \sin kx) \cos \omega t$
- Nodes (निस्पंद): Points of zero amplitude ($x = 0, \frac{\lambda}{2}, \lambda, \dots$)
- Antinodes (प्रस्पंद): Points of maximum amplitude ($x = \frac{\lambda}{4}, \frac{3\lambda}{4}, \dots$)
- Distance between two consecutive nodes or antinodes = $\frac{\lambda}{2}$
- Distance between adjacent node and antinode = $\frac{\lambda}{4}$

6. Modes of Vibration in Organ Pipes & Strings

A. Stretched String (Fixed at both ends):

Fundamental Frequency (1st Harmonic): $$f_1 = \frac{v}{2L} = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$$
$n^{\text{th}}$ Harmonic Frequency: $$f_n = n \cdot f_1 = \frac{n}{2L} \sqrt{\frac{T}{\mu}} \quad (n = 1, 2, 3, \dots)$$
Harmonic Ratio: $f_1 : f_2 : f_3 : \dots = 1 : 2 : 3 : \dots$ (All harmonics present)

B. Organ Pipe Closed at One End:

Fundamental Frequency (1st Harmonic): $$f_1 = \frac{v}{4L}$$
$n^{\text{th}}$ Harmonic Frequency: $$f_n = (2n - 1) f_1 = (2n - 1) \frac{v}{4L} \quad (n = 1, 2, 3, \dots)$$
Harmonic Ratio: $f_1 : f_3 : f_5 : \dots = 1 : 3 : 5 : \dots$ (Only odd harmonics present)

C. Organ Pipe Open at Both Ends:

Fundamental Frequency (1st Harmonic): $$f_1 = \frac{v}{2L}$$
$n^{\text{th}}$ Harmonic Frequency: $$f_n = n \cdot f_1 = n \frac{v}{2L} \quad (n = 1, 2, 3, \dots)$$
Harmonic Ratio: $f_1 : f_2 : f_3 : \dots = 1 : 2 : 3 : \dots$ (All harmonics present)

End Correction ($e$):

For Closed Pipe: $L_{\text{effective}} = L + 0.3d$ (where $d$ = inner diameter)
For Open Pipe: $L_{\text{effective}} = L + 0.6d$

7. Beats (विस्पंद)

Definition: Periodic variation in intensity of sound produced by superposition of two sound waves of slightly different frequencies ($\Delta f < 10\text{ Hz}$).
Beat Frequency ($f_b$): Number of beats heard per second. $$f_b = |f_1 - f_2|$$
Time Period of Beats ($T_b$): $$T_b = \frac{1}{|f_1 - f_2|}$$

8. Doppler Effect (डॉप्लर प्रभाव)

Definition: The apparent change in frequency/pitch of sound heard by an observer due to relative motion between the source of sound and the observer.

General Formula for Apparent Frequency ($f'$):

$$f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$$

Where:

$f$ = Actual frequency of source
$v$ = Speed of sound wave in medium
$v_o$ = Speed of observer
$v_s$ = Speed of source

Sign Convention Rule:

Numerator ($v \pm v_o$): Use + if observer moves towards source; - if moving away.
Denominator ($v \mp v_s$): Use - if source moves towards observer; + if moving away.

Summary Cases:

Motion Condition	Formula for Apparent Frequency ($f'$)
1. Source moving towards stationary Observer ($v_o = 0$)	$f' = f \left( \frac{v}{v - v_s} \right)$
2. Source moving away from stationary Observer ($v_o = 0$)	$f' = f \left( \frac{v}{v + v_s} \right)$
3. Observer moving towards stationary Source ($v_s = 0$)	$f' = f \left( \frac{v + v_o}{v} \right)$
4. Observer moving away from stationary Source ($v_s = 0$)	$f' = f \left( \frac{v - v_o}{v} \right)$
5. Both moving towards each other	$f' = f \left( \frac{v + v_o}{v - v_s} \right)$
6. Both moving away from each other	$f' = f \left( \frac{v - v_o}{v + v_s} \right)$

Quick Exam Tips for MP Board Students:

Definitions: Memorize definitions of Transverse vs Longitudinal waves, Laplace Correction, and Beats.
Derivations: Practice derivation of fundamental frequency for Open and Closed organ pipes.
Numericals: Expect numerical problems based on $v = \sqrt{T/\mu}$, Beat frequency ($f_b = |f_1 - f_2|$), and Doppler effect cases.

Waves

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