Wave Motion & Sound Phenomena: Master Notes

Chapter Overview

Wave motion governs acoustic and mechanical energy propagation. In IIT-JEE (Advanced), top problems focus on standing wave boundary conditions (Organ Pipes & Strings), Doppler Effect with moving reflectors / wind, acoustic intensity levels in decibels ($\beta =10{\log }_{10}(I/I_0)$), and beat frequencies ($f_{\text{beat}}=|f_1-f_2|$).

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1. The 1D Wave Equation & Wave Kinematics

$$\boxed{{\displaystyle \frac{{\partial }^{2}y}{\partial x^2}=\frac{1}{v^2}\frac{{\partial }^{2}y}{\partial t^2}}}$$

Any function of the form $y(x,t)=f(x\mp vt)$ represents a travelling wave:

• $f(x-vt)$: Travelling in $+x$ direction.
• $f(x+vt)$: Travelling in $-x$ direction.

1.1 Speed of Mechanical & Sound Waves

• Transverse Wave on a Stretched String: $v=\sqrt{\frac{T}{\mu }}$ ($\mu =\text{mass per unit length}$)
• Longitudinal Sound Wave in Fluid/Gas (Newton-Laplace):$$\boxed{{\displaystyle v=\sqrt{\frac{B_{\text{adiabatic}}}{\rho }}=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{\gamma RT}{M}}}}$$(Speed of sound depends only on temperature $T$, not pressure $P$ at constant temperature!)

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2. Standing Waves & Resonant Modes

$$y(x,t)=2A\sin (kx)\cos (\omega t)$$

Multi-Mode DiagramWave Motion & Standing Wave Harmonics

Option 1: Publication-Grade Scientific Vector SVG

Standing wave nodes and antinodes in fixed strings and acoustic organ pipes.

Standing Wave: $y(x,t) = 2 A \sin(kx) \cos(\omega t)$

Option 2: Interactive 60 FPS Canvas Physics Engine

Simulate standing wave resonance modes (n = 1, 2, 3, 4).

Harmonic Mode n: 3

Nodes Count ($N$):4

Option 3: Interactive 3D Spatial Viewport

3D circular membrane standing modes.

+X

+Y

+Z

👆 Click and drag anywhere to rotate in 3D space

Azimuth Angle (ψ):35°

Elevation Angle (θ):-22°

Harmonic Frequency:$f_n = n \frac{v}{2L}$

Option 5: Reactive Parametric Physics with Live Calculus

Resonance frequency calculator for strings.

Harmonic n: 2

: 45

Frequency f_n ($f_n$):$220 Hz$

Active Mode:Option 1 (Print & Derivation Vector SVG)Switch modes using the tabs above to explore different visualizations.

Resonator System	Allowed Wavelengths ${\lambda }_n$	Resonant Frequencies $f_n$	Harmonic Series
String Fixed at Both Ends	${\lambda }_n=\frac{2L}{n}$	$f_n=n\left(\frac{v}{2L}\right),n=1,2,3,\dots$	All integer harmonics present ($1f_1,2f_1,3f_1,\dots$)
Open-Open Organ Pipe	${\lambda }_n=\frac{2L}{n}$	$f_n=n\left(\frac{v}{2L}\right),n=1,2,3,\dots$	All integer harmonics present ($1f_1,2f_1,3f_1,\dots$)
Closed-Open Organ Pipe	${\lambda }_n=\frac{4L}{2n-1}$	$f_n=(2n-1)\left(\frac{v}{4L}\right),n=1,2,3,\dots$	Odd harmonics only ($1f_1,3f_1,5f_1,\dots$)

• End Correction for Pipe of Radius $r$:
  • Closed-Open: $L_{\text{eff}}=L+0.6r$
  • Open-Open: $L_{\text{eff}}=L+1.2r$

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3. The Doppler Effect Master Formula

$$\boxed{{\displaystyle f^{\prime }=f_0\left(\frac{v\pm v_{\text{wind}}\mp v_{\text{observer}}}{v\pm v_{\text{wind}}\mp v_{\text{source}}}\right)}}$$

Universal Mnemonic:

• Observer Moving Toward Source: Increases frequency $\to +v_o$ in numerator.
• Source Moving Toward Observer: Increases frequency $\to -v_s$ in denominator.
• Wind in Direction of Sound: Increases effective wave speed $\to (v+v_w)$.

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4. Authentic Previous Years Questions (PYQs)

PYQ 1: JEE Advanced 2022 (Paper 1) — Doppler Effect with Moving Wall Reflector

Question:
A stationary sound source emits sound of frequency $f_0=1000\text{ Hz}$. A car moving directly toward a vertical rigid wall with speed $u=20\text{ m/s}$ is located between the source and the wall. The speed of sound in air is $v=340\text{ m/s}$. Find the beat frequency $f_{\text{beat}}$ heard by the driver of the car.

<!-- Sound waves toward car --> <path d="M 115 50 Q 125 60 115 70" fill="none" stroke="#3b82f6" stroke-width="2"/> <path d="M 125 45 Q 140 60 125 75" fill="none" stroke="#3b82f6" stroke-width="2"/> <!-- Moving Car --> <rect x="200" y="45" width="80" height="35" rx="5" fill="#10b981" stroke="var(--vp-c-text-1)" stroke-width="1.5"/> <text x="215" y="67" fill="#fff" font-size="12" font-weight="700">Car (u)</text> <line x1="285" y1="62" x2="330" y2="62" stroke="#10b981" stroke-width="2.5" marker-end="url(#arrow-green)"/> <text x="295" y="55" fill="#10b981" font-size="12" font-weight="700">u = 20 m/s</text> <!-- Rigid Wall --> <line x1="440" y1="20" x2="440" y2="100" stroke="var(--vp-c-text-1)" stroke-width="4"/> <g stroke="var(--vp-c-text-3)" stroke-width="1.5"> <line x1="440" y1="30" x2="455" y2="20"/> <line x1="440" y1="50" x2="455" y2="40"/> <line x1="440" y1="70" x2="455" y2="60"/> <line x1="440" y1="90" x2="455" y2="80"/> </g> <text x="410" y="115" fill="var(--vp-c-text-2)" font-size="11" font-weight="700">Wall</text> 

Figure Doppler effect geometry: Moving observer receives direct frequency from stationary source and reflected frequency from rigid wall ahead.

Step-by-Step Solution:

The driver receives two frequencies:

1. Direct sound from stationary source behind the car (Car moving away from source):$$f_1=f_0\left(\frac{v-u}{v}\right)=1000\left(\frac{340-20}{340}\right)=1000\left(\frac{320}{340}\right)\approx 941.18\text{ Hz}$$
2. Reflected sound from the stationary wall ahead:
   • Frequency incident on the wall: $f_{\text{wall}}=f_0=1000\text{ Hz}$ (both source and wall are stationary!).
   • Frequency heard by driver moving toward the wall:$$f_2=f_{\text{wall}}\left(\frac{v+u}{v}\right)=1000\left(\frac{340+20}{340}\right)=1000\left(\frac{360}{340}\right)\approx 1058.82\text{ Hz}$$
3. Beat Frequency:$$f_{\text{beat}}=|f_2-f_1|=f_0\left[\frac{(v+u)-(v-u)}{v}\right]=f_0\left(\frac{2u}{v}\right)$$$$f_{\text{beat}}=1000\left(\frac{2\times 20}{340}\right)=1000\left(\frac{40}{340}\right)=\frac{400}{34}\approx \boxed{{\displaystyle 117.65\text{ Hz}\approx 118\text{ Hz}}}$$

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PYQ 2: JEE Advanced 2020 — Resonance Column with End Correction

Question:
In a resonance tube experiment, the first two successive resonance lengths are ${\ell }_1=16.0\text{ cm}$ and ${\ell }_2=50.0\text{ cm}$ when excited by a tuning fork of frequency $f=500\text{ Hz}$. Find:

1. The speed of sound in air $v$.
2. The end correction $e$ of the tube.

Step-by-Step Solution:

1. For a closed organ pipe:$${\ell }_1+e=\frac{\lambda }{4},{\ell }_2+e=\frac{3\lambda }{4}$$
2. Subtracting the two equations eliminates the end correction $e$:$${\ell }_2-{\ell }_1=\frac{3\lambda }{4}-\frac{\lambda }{4}=\frac{\lambda }{2}⟹\lambda =2({\ell }_2-{\ell }_1)=2(50.0-16.0)=2(34.0)=68.0\text{ cm}=0.68\text{ m}$$
3. Speed of Sound:$$v=f\lambda =500\times 0.68=\boxed{{\displaystyle 340\text{ m/s}}}$$
4. End Correction:$$e=\frac{\lambda }{4}-{\ell }_1=\frac{68.0}{4}-16.0=17.0-16.0=\boxed{{\displaystyle 1.0\text{ cm}}}$$

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5. High-Yield Formula Sheet

Entity	Formula	Notes
Speed in Gas	$v=\sqrt{\frac{\gamma RT}{M}}$	$T$ in Kelvin
Doppler Effect	$f^{\prime }=f_0\frac{v\pm v_o}{v\mp v_s}$	Moving source/observer
Beats Frequency	$f_{\text{beat}} =	f_1 - f_2
Sound Level (dB)	$\beta =10{\log }_{10}\left(\frac{I}{I_0}\right)$	$I_0={10}^{-12}{\text{ W/m}}^2$
Resonance Tube	$\lambda =2({\ell }_2-{\ell }_1)$	Independent of end correction