Prerequisites
Before studying this topic, make sure you understand:
- Simple Harmonic Motion - SHM is the basis of wave motion
- SHM Energy - Energy concepts extend to waves
- Kinematics - For understanding wave propagation
The Hook: From Guitar Strings to Oppenheimer’s Waves
In Oppenheimer (2023), when the atomic bomb explodes, you see the shockwave rippling through the desert — a perfect example of wave motion carrying energy without transporting matter. The light reaches before sound because light waves travel faster!
Closer to home: When you pluck a guitar string, you see waves traveling along it. Drop a stone in water — ripples spread out. Shout in a canyon — sound waves bounce back as echoes.
The Mystery: How can waves carry energy and information across vast distances without anything physically traveling that distance?
What is a Wave?
A wave is a disturbance that propagates through space and time, transferring energy without transferring matter.
In simple terms: A wave is energy’s way of traveling. The medium (water, air, string) doesn’t move forward — it just wiggles up and down or back and forth. But the energy keeps moving forward!
Key Characteristics:
- Transfers energy without transferring matter
- Requires a medium (for mechanical waves)
- Has periodic motion in space and time
- Characterized by wavelength, frequency, amplitude, and speed
Interactive Wave Visualizer
Explore wave motion, superposition, and interference interactively:
- Constructive Interference: Set phase = 0 and watch the resultant amplitude double
- Destructive Interference: Set phase = 180 degrees (pi) and observe cancellation
- Standing Waves: Select “Opposite (Standing)” mode to see nodes and antinodes
- Beats: Select “Beats” mode and set f1 = 1.0 Hz, f2 = 1.1 Hz to observe beat pattern
- Transverse vs Longitudinal: Toggle wave type to see how particle motion differs
Types of Waves
1. Based on Medium
Mechanical Waves:
- Require a material medium to propagate
- Examples: Sound, water waves, seismic waves
- Cannot travel through vacuum
Electromagnetic Waves:
- Do not require a medium
- Can travel through vacuum
- Examples: Light, radio waves, X-rays
- All travel at speed of light in vacuum ($c = 3 \times 10^8$ m/s)
2. Based on Particle Motion
Transverse Waves:
- Particle motion is perpendicular to wave propagation
- Can travel on strings, water surface
- Examples: Light, waves on string, water ripples
Longitudinal Waves:
- Particle motion is parallel to wave propagation
- Creates compressions and rarefactions
- Examples: Sound, pressure waves
Transverse has T-shape → Perpendicular motion
Longitudinal has Line-shape → Parallel motion
Comparison Table:
| Feature | Transverse | Longitudinal |
|---|---|---|
| Particle motion | ⊥ to propagation | ∥ to propagation |
| Can travel in solids | Yes | Yes |
| Can travel in liquids/gases | Only surface waves | Yes |
| Examples | String waves, light | Sound, pressure waves |
| Shows polarization | Yes | No |
Wave Diagram
Wave Parameters
1. Wavelength ($\lambda$)
Distance between two consecutive points in the same phase (e.g., crest to crest).
Unit: meter (m)
2. Frequency ($f$ or $\nu$)
Number of complete waves passing a point per second.
Unit: Hertz (Hz) = s$^{-1}$
3. Time Period ($T$)
Time for one complete wave to pass a point.
$$\boxed{T = \frac{1}{f}}$$4. Amplitude ($A$)
Maximum displacement from equilibrium position.
Unit: meter (m)
Energy $\propto A^2$ (just like in SHM!)
5. Wave Speed ($v$)
Speed at which the wave (disturbance) propagates through the medium.
$$\boxed{v = f\lambda = \frac{\lambda}{T}}$$This is the fundamental wave equation!
“Very Fast Lightning”
$$v = f \lambda$$Velocity = Frequency × Lambda (wavelength)
6. Angular Frequency ($\omega$) and Wave Number ($k$)
$$\omega = 2\pi f = \frac{2\pi}{T}$$ $$k = \frac{2\pi}{\lambda}$$(spatial frequency)
Alternative wave equation:
$$\boxed{v = \frac{\omega}{k}}$$Mathematical Description of Waves
Transverse Wave on String
A sinusoidal wave traveling in the positive $x$-direction:
$$\boxed{y(x, t) = A\sin(kx - \omega t + \phi)}$$Or: $y(x, t) = A\sin\left(\frac{2\pi}{\lambda}x - \frac{2\pi}{T}t + \phi\right)$
Where:
- $y$ = displacement at position $x$ and time $t$
- $A$ = amplitude
- $k = \frac{2\pi}{\lambda}$ = wave number
- $\omega = 2\pi f$ = angular frequency
- $\phi$ = initial phase
For wave traveling in negative $x$-direction:
$$y(x, t) = A\sin(kx + \omega t + \phi)$$- Minus sign $(kx - \omega t)$: Wave traveling in +x direction
- Plus sign $(kx + \omega t)$: Wave traveling in -x direction
Memory: Think of the wave “chasing” the minus sign!
Longitudinal Wave (Sound)
Displacement wave:
$$s(x, t) = s_0\sin(kx - \omega t)$$Pressure wave:
$$\Delta P(x, t) = P_0\cos(kx - \omega t)$$Note: Displacement and pressure are 90° out of phase!
Wave Speed in Different Media
1. Speed of Transverse Wave on String
$$\boxed{v = \sqrt{\frac{T}{\mu}}}$$Where:
- $T$ = tension in string (N)
- $\mu = \frac{m}{l}$ = mass per unit length (kg/m)
Key Insights:
- Higher tension → faster wave
- Heavier string → slower wave
When you tighten a guitar string (increase $T$), the pitch goes higher because wave speed increases, which increases frequency (since $v = f\lambda$ and $\lambda$ is fixed by string length).
Thicker strings (larger $\mu$) produce lower notes because waves travel slower!
2. Speed of Sound in Gases
$$\boxed{v = \sqrt{\frac{\gamma RT}{M}}}$$Where:
- $\gamma$ = adiabatic constant (1.4 for air)
- $R$ = universal gas constant (8.314 J/mol·K)
- $T$ = absolute temperature (K)
- $M$ = molar mass (kg/mol)
Simplified for air:
$$v \approx 331 + 0.6t$$m/s
where $t$ is temperature in °C.
At 20°C: $v \approx 343$ m/s
Key Insights:
- Speed increases with temperature
- Speed is independent of pressure (at constant $T$)
- Lighter gases → faster sound (helium voice effect!)
3. Speed of Sound in Solids
$$\boxed{v = \sqrt{\frac{E}{\rho}}}$$Where:
- $E$ = Young’s modulus (Pa)
- $\rho$ = density (kg/m³)
For longitudinal waves in a rod.
4. Speed of Sound in Liquids
$$\boxed{v = \sqrt{\frac{B}{\rho}}}$$Where:
- $B$ = bulk modulus (Pa)
- $\rho$ = density (kg/m³)
Comparison:
| Medium | Approximate Speed |
|---|---|
| Air (20°C) | 343 m/s |
| Water | 1480 m/s |
| Steel | 5000 m/s |
| Aluminum | 6400 m/s |
Waves travel faster in solids than liquids, and faster in liquids than gases!
Wave Equation
The general differential equation for wave motion:
$$\boxed{\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}}$$Any function of the form $y(x, t) = f(x - vt)$ or $y(x, t) = g(x + vt)$ satisfies this equation.
Solutions:
- $f(x - vt)$: wave traveling in +x direction with speed $v$
- $g(x + vt)$: wave traveling in -x direction with speed $v$
Phase and Phase Difference
Phase of a wave at position $x$ and time $t$:
$$\phi = kx - \omega t + \phi_0$$Phase difference between two points $x_1$ and $x_2$ at same time:
$$\Delta\phi = k(x_2 - x_1) = \frac{2\pi}{\lambda}(x_2 - x_1)$$Path difference:
$$\Delta x = x_2 - x_1 = \frac{\lambda}{2\pi}\Delta\phi$$Important Relations:
| Path Difference | Phase Difference | Interference |
|---|---|---|
| $0, \lambda, 2\lambda, ...$ | $0, 2\pi, 4\pi, ...$ | Constructive |
| $\frac{\lambda}{2}, \frac{3\lambda}{2}, ...$ | $\pi, 3\pi, ...$ | Destructive |
| $\frac{\lambda}{4}, \frac{3\lambda}{4}, ...$ | $\frac{\pi}{2}, \frac{3\pi}{2}, ...$ | Intermediate |
Energy in Waves
Energy Density
Energy per unit length (or volume) in a wave:
$$\boxed{u = \frac{1}{2}\rho\omega^2 A^2}$$Where $\rho$ is the mass per unit length (or volume).
Power Transmitted by Wave
Average power transmitted:
$$\boxed{P_{avg} = \frac{1}{2}\rho v\omega^2 A^2}$$For a string: $\rho$ is linear mass density $\mu$ For sound: $\rho$ is volume density
Key Insights:
- Power $\propto A^2$ (amplitude squared)
- Power $\propto \omega^2$ (frequency squared)
- Power $\propto v$ (wave speed)
Intensity of Wave
Intensity = Power per unit area:
$$\boxed{I = \frac{P}{A_{area}}}$$For spherical waves spreading from a point source:
$$I = \frac{P}{4\pi r^2}$$So intensity decreases as $\frac{1}{r^2}$ (inverse square law).
Particle Velocity vs Wave Velocity
Particle velocity: Velocity of a particle of the medium (perpendicular to direction for transverse waves)
$$v_{particle} = \frac{\partial y}{\partial t} = -A\omega\cos(kx - \omega t)$$Wave velocity: Velocity at which the disturbance propagates
$$v_{wave} = \frac{\omega}{k} = f\lambda$$Wrong: Wave speed = particle speed
Right:
- Particle velocity: How fast particles oscillate (max = $A\omega$)
- Wave velocity: How fast the wave pattern moves (= $f\lambda$)
These are completely different! Particle velocity varies with time, wave velocity is constant (for given medium).
Reflection and Transmission of Waves
At Fixed End
- Wave reflects with phase change of $\pi$ (inverted)
- Amplitude remains same (no energy loss)
At Free End
- Wave reflects without phase change
- Amplitude remains same
At Boundary Between Media
When wave travels from medium 1 (with speed $v_1$) to medium 2 (with speed $v_2$):
Reflected amplitude:
$$A_r = \frac{v_2 - v_1}{v_2 + v_1}A_i$$Transmitted amplitude:
$$A_t = \frac{2v_2}{v_2 + v_1}A_i$$Note: Frequency remains same, but wavelength changes (since $v = f\lambda$).
Doppler Effect (Introduction)
When source or observer is moving, the observed frequency changes.
General formula:
$$\boxed{f' = f\frac{v + v_o}{v - v_s}}$$Where:
- $f'$ = observed frequency
- $f$ = source frequency
- $v$ = wave speed (in medium)
- $v_o$ = observer velocity (+ if toward source)
- $v_s$ = source velocity (+ if toward observer)
(More details in the Doppler Effect chapter!)
Common Mistakes to Avoid
Wrong: Wavelength stays same when wave enters new medium
Right: Frequency stays same, wavelength changes!
$$v_1 = f\lambda_1, \quad v_2 = f\lambda_2$$ $$\frac{\lambda_2}{\lambda_1} = \frac{v_2}{v_1}$$Wrong: Higher frequency → higher wave speed on string
Right: Wave speed depends only on medium properties!
$$v = \sqrt{\frac{T}{\mu}}$$(independent of $f$ or $\lambda$)
Frequency and wavelength adjust to maintain $v = f\lambda$
Wrong: Energy $\propto A$
Right: Energy $\propto A^2$
Double the amplitude → quadruple the energy!
Wrong: $\Delta\phi = \Delta x$
Right: $\Delta\phi = \frac{2\pi}{\lambda}\Delta x$
Always include the $\frac{2\pi}{\lambda}$ factor!
Memory Tricks & Patterns
The VFL Triangle
“Very Fast Lightning”
$$v = f\lambda$$All wave problems start here!
String Tension Memory
“Tighter string, Terrific speed”
$$v = \sqrt{\frac{T}{\mu}}$$More tension → faster wave
Temperature and Sound
“Hotter air, Higher speed”
$$v \propto \sqrt{T}$$Higher temperature → faster sound
Phase Path Relation
“Two Pi Over Lambda”
$$\Delta\phi = \frac{2\pi}{\lambda}\Delta x$$Practice Problems
Level 1: Foundation (NCERT Style)
A wave has frequency 50 Hz and wavelength 2 m. Find its speed.
Solution:
$$v = f\lambda = 50 \times 2 = $$100 m/s
A string has mass per unit length 0.01 kg/m and tension 40 N. Find the speed of transverse wave on it.
Solution:
$$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{40}{0.01}} = \sqrt{4000} = 63.2$$m/s
Or: $v = \sqrt{4000} = 20\sqrt{10} \approx$ 63.2 m/s
A wave equation is $y = 0.05\sin(10\pi x - 200\pi t)$ (SI units). Find amplitude, wavelength, frequency, and speed.
Solution: Comparing with $y = A\sin(kx - \omega t)$:
- $A = 0.05$ m = 5 cm
- $k = 10\pi$, so $\lambda = \frac{2\pi}{k} = \frac{2\pi}{10\pi} = 0.2$ m = 20 cm
- $\omega = 200\pi$, so $f = \frac{\omega}{2\pi} = \frac{200\pi}{2\pi} = 100$ Hz
- $v = f\lambda = 100 \times 0.2 = $ 20 m/s
Or: $v = \frac{\omega}{k} = \frac{200\pi}{10\pi} = $ 20 m/s ✓
Level 2: JEE Main
Two waves are given by: $y_1 = 5\sin(4x - 60t)$ $y_2 = 5\sin(4x - 60t + \frac{\pi}{3})$
Find the phase difference and path difference. (SI units)
Solution: Phase difference: $\Delta\phi = \frac{\pi}{3}$ rad = 60°
From the equations: $k = 4$ rad/m So: $\lambda = \frac{2\pi}{k} = \frac{2\pi}{4} = \frac{\pi}{2}$ m
Path difference:
$$\Delta x = \frac{\lambda}{2\pi}\Delta\phi = \frac{\pi/2}{2\pi} \times \frac{\pi}{3} = \frac{\pi}{12}$$m
Or: $\Delta x = \frac{\lambda\Delta\phi}{2\pi} = \frac{(\pi/2) \times (\pi/3)}{2\pi} = \frac{\pi^2/6}{2\pi} = \frac{\pi}{12}$ m ≈ 0.262 m
A string of length 1 m and mass 5 g is under tension 80 N. What is the speed of transverse wave on it?
Solution:
$$\mu = \frac{m}{l} = \frac{5 \times 10^{-3}}{1} = 5 \times 10^{-3}$$kg/m
$$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{80}{5 \times 10^{-3}}} = \sqrt{16000} = 126.5$$m/s
Or: $v = \sqrt{16000} = 40\sqrt{10} \approx$ 126.5 m/s
Sound travels at 340 m/s in air. What is the wavelength of a 256 Hz tuning fork?
Solution:
$$\lambda = \frac{v}{f} = \frac{340}{256} = 1.328$$m ≈ 1.33 m
Level 3: JEE Advanced
A wave $y = 0.02\sin(30x - 400t)$ travels on a string with mass per unit length $\mu = 0.05$ kg/m. Find: (a) Wave speed (b) Tension in string (c) Average power transmitted
Solution: From equation: $k = 30$ rad/m, $\omega = 400$ rad/s, $A = 0.02$ m
(a) $v = \frac{\omega}{k} = \frac{400}{30} = 13.33$ m/s
(b) $v = \sqrt{\frac{T}{\mu}}$
$$T = \mu v^2 = 0.05 \times (13.33)^2 = 0.05 \times 177.78 = 8.89$$N
(c) $P_{avg} = \frac{1}{2}\mu v\omega^2 A^2$
$$P_{avg} = \frac{1}{2}(0.05)(13.33)(400)^2(0.02)^2$$ $$= \frac{1}{2}(0.05)(13.33)(160000)(0.0004)$$ $$= 0.025 \times 13.33 \times 64 = 21.3$$W
A wave travels from a string with $\mu_1 = 0.01$ kg/m to another string with $\mu_2 = 0.04$ kg/m. Both are under same tension $T$. If incident amplitude is $A$, find reflected and transmitted amplitudes.
Solution: Wave speeds:
$$v_1 = \sqrt{\frac{T}{\mu_1}} = \sqrt{\frac{T}{0.01}}$$ $$v_2 = \sqrt{\frac{T}{\mu_2}} = \sqrt{\frac{T}{0.04}}$$ $$\frac{v_2}{v_1} = \sqrt{\frac{\mu_1}{\mu_2}} = \sqrt{\frac{0.01}{0.04}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$$So $v_2 = \frac{v_1}{2}$
Reflected amplitude:
$$A_r = \frac{v_2 - v_1}{v_2 + v_1}A = \frac{\frac{v_1}{2} - v_1}{\frac{v_1}{2} + v_1}A = \frac{-\frac{v_1}{2}}{\frac{3v_1}{2}}A = -\frac{1}{3}A$$Magnitude: $|A_r| = \frac{A}{3}$ (negative means inverted)
Transmitted amplitude:
$$A_t = \frac{2v_2}{v_2 + v_1}A = \frac{2 \times \frac{v_1}{2}}{\frac{v_1}{2} + v_1}A = \frac{v_1}{\frac{3v_1}{2}}A = \frac{2}{3}A$$The equation of a standing wave is $y = 2A\sin(kx)\cos(\omega t)$. Find the distance between two consecutive nodes.
Solution: Nodes occur where $\sin(kx) = 0$
$$kx = 0, \pi, 2\pi, 3\pi, ...$$ $$x = 0, \frac{\pi}{k}, \frac{2\pi}{k}, \frac{3\pi}{k}, ...$$Distance between consecutive nodes:
$$\Delta x = \frac{\pi}{k} = \frac{\pi}{2\pi/\lambda} = \frac{\lambda}{2}$$Answer: Distance between consecutive nodes = $\frac{\lambda}{2}$
(We’ll explore standing waves more in Superposition of Waves!)
Quick Revision Box
| Wave Type | Equation | Speed Formula |
|---|---|---|
| String wave | $y = A\sin(kx - \omega t)$ | $v = \sqrt{\frac{T}{\mu}}$ |
| Sound (gas) | $s = s_0\sin(kx - \omega t)$ | $v = \sqrt{\frac{\gamma RT}{M}}$ |
| Sound (solid) | — | $v = \sqrt{\frac{E}{\rho}}$ |
| Sound (liquid) | — | $v = \sqrt{\frac{B}{\rho}}$ |
Universal Relations:
- $v = f\lambda = \frac{\omega}{k}$
- $k = \frac{2\pi}{\lambda}$, $\omega = 2\pi f$
- Energy $\propto A^2$
- Power $\propto A^2\omega^2$
- Phase difference: $\Delta\phi = \frac{2\pi}{\lambda}\Delta x$
Related Topics
Within Oscillations & Waves
- Simple Harmonic Motion — SHM is basis of wave motion
- SHM Energy — Energy concepts extend to waves
- Superposition of Waves — Interference and standing waves
- Doppler Effect — Frequency change due to motion
Connected Chapters
- Sound Waves — Application of wave motion
- Optics — Light as electromagnetic waves
- Electromagnetic Waves — Waves without medium
Math Connections
- Trigonometry — Sine and cosine functions
- Partial Derivatives — For wave equation
- Differential Equations — Wave equation solutions
Teacher’s Summary
Waves transfer energy, not matter. The medium oscillates, the energy propagates.
Fundamental equation: $v = f\lambda$ connects all wave parameters
Two types by motion:
- Transverse: perpendicular motion (light, string waves)
- Longitudinal: parallel motion (sound)
Wave speed depends on medium properties:
- String: $v = \sqrt{\frac{T}{\mu}}$ (tension and mass density)
- Sound in gas: $v = \sqrt{\frac{\gamma RT}{M}}$ (temperature)
- General: stiffer/lighter medium → faster wave
Energy in waves:
- Energy $\propto A^2\omega^2$
- Intensity $\propto \frac{1}{r^2}$ for spherical waves
JEE Strategy: Always start with $v = f\lambda$. Identify wave type to use correct speed formula. Watch for phase vs path difference conversions!
“Waves are nature’s way of sharing energy without giving away matter — the universe’s ultimate courier service.”