Waves

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Waves — Key Facts Wave equation: $v = f\lambda$; wave speed $v = \sqrt{\frac{T}{\mu}}$ for stretched string ($T$ = tension, $\mu$ = linear mass density) Transverse: displacement perpendicular to propagation direction (string waves, light); Longitudinal: displacement parallel (sound, spring waves) Doppler Effect: $f’ = f \times \frac{(v \pm v_0)}{(v \mp v_s)}$; source moving toward observer → higher frequency Standing waves: nodes (zero amplitude) and antinodes (maximum amplitude); fixed end → node ⚡ Exam tip: Doppler Effect applies to ALL waves (sound, light, water); formula differs for light at high speeds

---

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of wave motion and behaviour.

Waves — CUET Physics Study Guide

A wave is a disturbance that transfers energy from one point to another without transferring matter. Waves are classified by the direction of particle displacement relative to propagation direction: transverse (perpendicular) and longitudinal (parallel).

Wave on a String: For a stretched string, wave speed $v = \sqrt{\frac{T}{\mu}}$, where $T$ is tension and $\mu = \frac{m}{L}$ is linear mass density. This formula shows that tighter strings and lighter strings support faster waves — used in musical instruments (guitar, sitar) where tension and length determine pitch.

Sound Waves: In a gas, sound speed $v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M}}$. At STP air, $v \approx 332$ m/s. At temperature $T$ (in Kelvin): $v_T = 332\sqrt{\frac{T}{273}}$ m/s. Sound is longitudinal — particles oscillate parallel to wave propagation, creating compressions and rarefactions.

Doppler Effect: When source or observer moves, observed frequency changes. The general formula: $$f’ = f \times \frac{v \pm v_{\text{observer}}}{v \mp v_{\text{source}}}$$ Use upper signs for motion toward each other, lower for motion away. For light at relativistic speeds, $f’ = f\sqrt{\frac{1+\beta}{1-\beta}}$ where $\beta = v/c$ (source approaching).

Standing Waves on Strings:

• Both ends fixed: $L = \frac{n\lambda}{2}$, $f_n = \frac{n v}{2L}$ ($n = 1, 2, 3…$)
• Fundamental ($n=1$): $f_1 = \frac{v}{2L}$, first harmonic
• Second harmonic ($n=2$): $f_2 = \frac{2v}{2L} = 2f_1$

Standing Waves in Pipes:

• Open pipe (both ends open): $L = \frac{n\lambda}{2}$, $f_n = \frac{nv}{2L}$ (all harmonics)
• Closed pipe (one end closed): $L = \frac{n\lambda}{4}$, $f_n = \frac{nv}{4L}$ (only odd harmonics: $n = 1, 3, 5…$)

Example: A string $1$ m long, mass $0.01$ kg, under tension $100$ N. Find fundamental frequency. $\mu = \frac{0.01}{1} = 0.01$ kg/m; $v = \sqrt{\frac{100}{0.01}} = \sqrt{10000} = 100$ m/s. $f_1 = \frac{v}{2L} = \frac{100}{2 \times 1} = 50$ Hz.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Waves — Complete CUET Physics Notes

Wave Intensity: For a progressive wave, intensity $I = \frac{P}{A} = \frac{1}{2}\rho A v \omega^2 a^2$ (for sound), where $\rho$ is medium density, $A$ is cross-sectional area, $\omega = 2\pi f$, and $a$ is amplitude. Intensity is proportional to amplitude squared ($I \propto a^2$) and frequency squared ($I \propto f^2$).

Energy in a Wave: At any point in a travelling wave, kinetic energy and potential energy are in phase for transverse waves (both maximum at zero displacement). Energy flows with the wave at speed $v$ — the medium particles do not travel with the wave; they oscillate in place.

Beats: When two waves of slightly different frequencies $f_1$ and $f_2$ interfere, the resultant amplitude varies at beat frequency $f_{\text{beat}} = |f_1 - f_2|$. Used for tuning musical instruments — adjust until beats disappear. Beat period $T_{\text{beat}} = \frac{1}{|f_1 - f_2|}$.

Doppler for Light: At non-relativistic speeds, the formula is approximately the same as for sound. But at relativistic speeds ($v \approx c$), special relativity changes it: for source approaching at speed $v = \beta c$, observed frequency is $f’ = f\sqrt{\frac{1+\beta}{1-\beta}}$. This has been experimentally confirmed by redshift measurements of receding galaxies (Hubble’s law: $v = H_0 d$).

Shock Waves: When a source moves faster than the wave speed (source speed > $v$), a conical shock wave (Mach cone) is produced. The Mach number $M = \frac{v_{\text{source}}}{v_{\text{wave}}}$. The half-angle of the Mach cone $\sin\alpha = \frac{v}{v_s} = \frac{1}{M}$. Sonic boom occurs when an aircraft breaks the sound barrier — a sudden pressure change heard as a loud boom.

Wave Propagation Formulas:

• Sound in gas: $v = \sqrt{\frac{\gamma P}{\rho}}$ (adiabatic bulk modulus)
• Sound in liquid: $v = \sqrt{\frac{B}{\rho}}$ (B = bulk modulus)
• Sound in solid rod: $v = \sqrt{\frac{Y}{\rho}}$ (Y = Young’s modulus)
• Surface water waves: $v = \sqrt{\frac{g\lambda}{2\pi}\tanh\frac{2\pi d}{\lambda}}$ (complicated, depends on depth $d$ and wavelength $\lambda$)

Qualities of Musical Sound:

• Pitch: determined by frequency (higher $f$ = higher pitch)
• Loudness: determined by intensity (amplitude squared)
• Timbre: determined by waveform (overtones/harmonics — why same note sounds different on violin vs piano)

Resonance: When driving frequency equals natural frequency, amplitude is maximum. Every musical instrument is a resonance system: air columns (open/closed pipes), strings (fixed ends). The frequency of nth mode is determined by the physical dimensions.

CUET Exam Patterns (2022–2024):

• Doppler effect (source/observer moving) is very frequently tested (1–2 marks)
• Standing wave frequencies for strings and pipes appear every year
• Beat frequency is a common MCQ
• Wave speed formula for strings ($v = \sqrt{T/\mu}$) appears in numerical problems
• Common mistakes: using wrong sign in Doppler formula, confusing open and closed pipe formulas

⚡ Key insight: For standing waves, always identify boundary conditions first. Fixed end → node (zero displacement). Open end → antinode (maximum displacement). For pipes, a closed pipe has a node at the closed end and antinode at the open end, giving only odd harmonics ($f_1, 3f_1, 5f_1…$). An open pipe has antinodes at both ends, giving all harmonics ($f_1, 2f_1, 3f_1…$). The length $L$ relates to wavelength by $L = n(\lambda/2)$ for both ends antinode/node, or $L = n(\lambda/4)$ for one node and one antinode.

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.