WavesNEET Physics · Class 11 · NCERT Chapter 14

Introduction

A wave is a disturbance that moves through a medium, transferring energy without permanently displacing the medium itself. A pebble dropped in a pond, sound from a guitar string, light from the Sun: all are waves.

For NEET 2027 you can expect 1 to 2 questions from this chapter. Repeated favourites: wave speed on a string, frequency of standing waves on strings and in pipes, beats and the Doppler effect. Five formulas in the cheat sheet cover almost everything.

Mechanical waves: transverse and longitudinal

A mechanical wave needs a medium (a string, air, water). There are two main kinds:

• Transverse: particles oscillate perpendicular to the direction the wave moves. Waves on a string and electromagnetic waves are transverse.
• Longitudinal: particles oscillate along the direction the wave moves. Sound waves in air are longitudinal (compressions and rarefactions).

The wave equation

A sinusoidal wave traveling in the positive x direction:

Each symbol:

• $A$: amplitude. The maximum displacement of any particle from its rest position.
• $\omega =2\pi f$: angular frequency, in rad/s.
• $k=2\pi /\lambda$: wave number, in rad/m. Tells how rapidly the phase changes with x.
• $\lambda$: wavelength, the distance between two adjacent crests.
• $T=1/f$: time period.

Wave speed:

Watch a traveling wave: y(x, t) = A sin(omega t minus k x). Slide the wave speed and wavelength to see how the pattern moves.

Speed of a wave on a string

For a stretched string of tension T and linear mass density mu (mass per unit length, in kg per metre):

Memory hook: bigger tension speeds the wave up (string snaps tighter), heavier string slows it down. Speed is a property of the medium, not of the wave that runs through it. Two different waves on the same string move at the same speed.

Speed of a transverse wave on a stretched string depends on tension and mass per unit length, nothing else.

Speed of sound in a medium

For a fluid, Newton derived $v=\sqrt{B/\rho }$, treating the compression as isothermal. This underestimates the actual speed.

Laplace pointed out that the rapid compressions in a sound wave are closer to adiabatic. Replacing the isothermal bulk modulus with the adiabatic one gives the Newton-Laplace formula:

For air at 0°C (γ = 1.4, P = 101325 Pa, ρ = 1.29 kg/m³): v ≈ 332 m/s. Speed in solids is much higher because Y (Young's modulus) replaces γP and is much larger.

Principle of superposition

When two or more waves overlap in a medium, the resulting displacement at every point is the algebraic sum of the displacements due to each wave alone. This single rule explains:

• Interference: constructive (in phase) or destructive (180° out of phase).
• Standing waves: two waves of equal amplitude going in opposite directions.
• Beats: two waves of slightly different frequencies, same direction.

Two waves of the same frequency, different amplitudes and a phase difference. The third trace is their sum.

Standing waves on a string

Fix a string at both ends with tension T and linear density mu. Allowed wavelengths and frequencies:

n = 1 is the fundamental (lowest pitch). n = 2 is the second harmonic, and so on. The two fixed ends are always nodes.

A string fixed at both ends supports standing waves with allowed wavelengths lambda_n = 2L over n. Pick n to see the n-th harmonic.

Standing waves in pipes

For air columns, the pattern depends on whether each end is open or closed.

Open at both ends

Both ends are displacement antinodes (pressure nodes). All harmonics are allowed:

Closed at one end

Closed end is a displacement node, open end is an antinode. Only odd harmonics are allowed:

So a closed pipe of length L has fundamental v/(4L), the same as an open pipe of length 2L. NEET PYQs love this comparison.

Standing waves in air columns. Open pipe (both ends open) supports all harmonics. Closed pipe (one end closed) supports only odd harmonics.

Beats

When two waves of slightly different frequencies $f_1$ and $f_2$ superpose, the resulting amplitude rises and falls slowly. The number of beats per second equals the difference between the frequencies:

Beats are easy to hear when the two frequencies are within a few hertz of each other. Past about 10 Hz, the ear stops perceiving them as separate beats.

Two waves at slightly different frequencies superpose to give beats. The beat frequency equals the difference between the two frequencies.

Doppler effect

When source and observer move relative to each other, the observed frequency differs from the emitted frequency. For sound, the general formula is:

Sign convention used in NEET (one of several common conventions, but the safest one to memorise):

• $v_o$ is positive when the observer moves toward the source.
• $v_s$ is positive when the source moves toward the observer.
• Approach raises the pitch (numerator goes up, denominator goes down).
• Recede lowers the pitch.

Real-world examples: pitch of a passing ambulance siren, redshift of light from distant galaxies (using the relativistic version), radar guns measuring car speeds.

Frequency observed when source and observer move relative to each other. Sign convention used here: positive v_s means source approaches observer; positive v_o means observer moves toward source.

Worked NEET problems

Summary cheat sheet

• Wave equation: $y=A\sin (\omega t-kx)$, $\omega =2\pi f$, $k=2\pi /\lambda$.
• Wave speed: $v=f\lambda =\omega /k$.
• String: $v=\sqrt{T/\mu }$.
• Sound: $v=\sqrt{\gamma P/\rho }$ (Newton-Laplace).
• Standing on string (both ends fixed): $f_n=nv/(2L)$, all harmonics.
• Open pipe: $f_n=nv/(2L)$, all harmonics.
• Closed pipe: $f_n=(2n-1)v/(4L)$, only odd harmonics.
• Beats: $f_{\text{beat}}=|f_1-f_2|$.
• Doppler: $f^{\prime }=f(v+v_o)/(v-v_s)$.
• Approach: pitch rises. Recede: pitch falls.

Next: try the interactive widgets for wave equation, standing waves, beats and Doppler effect, or work through the 32 NEET PYQs with full solutions. To time yourself, take the free 10-question mock test.