Waves

WavesNEET Physics · Class 11 · NCERT Chapter 14

Overview Notes NEET Questions Interactive Learning

7 interactive concept widgets for Waves. Drag any slider, change any number, and watch the formula and the answer update live. Built so you understand how each NEET problem actually works, not just the final number.

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Traveling wave visualiser

Wave speed on a stretched string

Superposition of two waves

Standing wave on a stretched string

Standing waves in pipes (open and closed)

Beats from two close frequencies

Doppler effect for sound

Wave equation and speed

A traveling sinusoidal wave plus the textbook speed formula for transverse waves on a string.

Wave equation

Traveling wave visualiser

A live sinusoidal wave with adjustable amplitude, wavelength and speed. The numbers on the right update automatically.

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.

Amplitude A: 1.00 m

Wavelength λ: 2.00 m

Wave speed v: 2.00 m/s

Frequency f

1.000 Hz

Period T

1.000 s

Wave number k

3.14 rad/m

Angular ω

6.28 rad/s

y (x, t) = A sin (ω t - k x), v = ω / k = f λ

Try this

Pause the animation. Now drag wavelength: the spatial pattern stretches or compresses, freezing in place.
Increase v at fixed lambda: the wave moves faster, frequency rises (f = v/lambda).
Halve lambda at fixed v: f doubles, k doubles, omega doubles.

Wave on a string

Wave speed on a stretched string

Move the sliders for tension and mass per unit length. Watch how each one shifts the wave speed.

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

Tension T: 50 N

Linear mass density μ: 1.00 g/m

Wave speed

223.6 m/s

v = \frac{T}{μ}

Try this

Quadruple the tension: speed doubles. v scales as sqrt(T).
Heavier strings are slower at the same tension. v scales as 1/sqrt(mu).
Tightening a guitar string raises pitch because v rises and so does the fundamental frequency.

Superposition

Superposition of two waves

The single rule that explains interference, beats and standing waves. Adjust amplitudes and phase difference to see the resultant.

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

Amplitude A₁: 1.00

Amplitude A₂: 1.00

Phase difference φ: 0°

Resultant amplitude

2.000

A_{R} = A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} cos ϕ

Try this

Phase = 0°: constructive interference. A_R = A₁ + A₂.
Phase = 180°: destructive interference. A_R = |A₁ − A₂|.
Phase = 90°: A_R = sqrt(A₁² + A₂²) (Pythagorean combination).

Standing waves

Standing waves on a fixed string and standing waves in open or closed air columns.

Standing wave on string

Standing wave on a stretched string

A live standing wave on a string fixed at both ends. Pick the harmonic and watch the nodes stay still.

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.

Length L: 1.00 m

Wave speed v: 100 m/s

Wavelength λ

1.000 m

Frequency f_n

100.0 Hz

λ_{n} = \frac{2 L}{n}, f_{n} = \frac{n v}{2 L}

Red dots are nodes (always at rest). The bumps in between are antinodes (largest oscillation).

Try this

n = 1 is the fundamental: one antinode in the middle, two nodes (the fixed ends).
Higher n: more nodes and antinodes. The frequency rises proportionally.
Doubling tension speeds up v by sqrt(2), so each f_n rises by sqrt(2). The harmonic ratios stay the same.

Standing wave in pipe

Standing waves in air columns

Open vs closed pipes have very different harmonic spectra. Compare them side by side.

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

Length L: 0.50 m

Sound speed v: 340 m/s

Frequency f

340.0 Hz

λ = 1.000 m

f_{n} = \frac{n v}{2 L}

Curve shows displacement amplitude. Both ends are antinodes.

Try this

Open pipe: harmonics are 1f, 2f, 3f, ... All present.
Closed pipe: harmonics are 1f, 3f, 5f, ... Only odd ones.
Closed pipe of length L has the same fundamental as an open pipe of length 2L.

Beats and Doppler effect

Two close frequencies beat together; relative motion shifts the observed frequency.

Beats

Beats from two close frequencies

When two waves of similar frequency overlap, the loudness rises and falls at the beat frequency, equal to the difference between them.

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

f₁: 20 Hz

f₂: 22 Hz

Beat frequency

2.00 Hz

f_{beat} = ∣ f_{1} - f_{2} ∣

Solid: y₁ + y₂. Dashed: amplitude envelope (modulating cos).

Try this

Equal frequencies: no beats (envelope is flat).
Larger frequency difference: faster beat rate, harder for the ear to hear as "beats".
Tuning instruments: musicians match strings by listening for the beat to slow to zero.

Doppler effect

Doppler effect for sound

The classic NEET formula with source and observer both able to move. Slide them and watch the observed frequency change.

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.

Source frequency f: 440 Hz

Sound speed v: 340 m/s

Source velocity v_s: 20 m/s (+ toward observer)

Observer velocity v_o: 0 m/s (+ toward source)

Observed frequency f'

467.5 Hz

Shift: 27.5 Hz (6.25%)

f^{'} = f \cdot \frac{v + v _{o}}{v - v _{s}}

Approach: f' > f (higher pitch)

Try this

Stationary observer, source approaching at 30 m/s: pitch goes up about 9% (e.g., a passing siren).
Source stationary, observer approaching at 30 m/s: pitch goes up about 9% too.
Both moving away: f' < f (lower pitch). Both moving together at the same speed: f' = f.
For light, you would replace v with c and use the relativistic formula instead.

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