Oscillations and Waves — NEET Physics

An oscillation repeats in time; a wave carries that oscillation through space. Simple harmonic motion (SHM) — where the restoring force is proportional to displacement and directed back toward equilibrium — is the model behind pendulums, springs, sound and light. This chapter builds SHM from its defining equation through its energy, then extends to travelling waves, the all-important $v=f\lambda$, standing waves on strings and pipes, beats and the Doppler effect — with the derivations and worked examples you need to solve, not just recognise.

1. What makes motion simple harmonic

A motion is SHM when the restoring force (or acceleration) is proportional to the displacement and points back toward the mean position:

$F=-kx⟺a=-{\omega }^{2}x,\omega =\frac{2\pi }{T}=2\pi f$

The negative sign is the whole story: the further the body strays, the harder it is pulled back — producing a smooth, repeating oscillation.

2. Displacement, velocity and acceleration

$x=A\sin (\omega t+\varphi ),v=A\omega \cos (\omega t+\varphi )=\omega \sqrt{A^2-x^2},a=-{\omega }^{2}x$

• At the mean position ($x=0$): velocity is maximum ($v_{\max }=A\omega$), acceleration is zero.
• At the extremes ($x=\pm A$): velocity is zero, acceleration is maximum ($a_{\max }={\omega }^{2}A$).

Velocity and acceleration are exactly out of step: where one peaks, the other vanishes. This is the single most-tested idea in the chapter.

3. The two standard oscillators

$T_{\text{spring}}=2\pi \sqrt{\frac{m}{k}},T_{\text{pendulum}}=2\pi \sqrt{\frac{L}{g}}$

• A pendulum's period depends only on length and $g$ — not on mass or (for small swings) amplitude. Quadruple its length and the period doubles.
• On the Moon ($g_{\text{Moon}}=g/6$) a pendulum's period grows by $\sqrt{6}\approx 2.45$, so pendulum clocks run slow there.

Worked example 3.1. A 2 kg mass on a spring of $k=8$ N/m: $T=2\pi \sqrt{2/8}=2\pi \times 0.5=\pi \approx 3.14$ s.

4. Energy in SHM

Energy shuttles between kinetic and potential while the total stays fixed:

$KE=\frac{1}{2}m{\omega }^2(A^2-x^2),PE=\frac{1}{2}m{\omega }^2{x}^2,E=\frac{1}{2}m{\omega }^2{A}^2=\frac{1}{2}k{A}^2$

• Total energy is constant and proportional to amplitude squared — double the amplitude, quadruple the energy.
• KE is maximum at the centre, PE at the extremes.

Worked example 4.1. At $x=A/2$, the ratio $KE:PE=(A^2-x^2):x^2=\frac{3A^2}{4}:\frac{A^2}{4}=3:1$.

5. Wave motion and the wave relation

A wave transfers energy and phase without transporting matter. Two families:

• Transverse — particles oscillate perpendicular to travel (waves on a string, light, all EM waves).
• Longitudinal — particles oscillate along the direction of travel (sound, compression waves).

The universal relation ties speed, frequency and wavelength:

$v=f\lambda$

• Sound of 500 Hz with wavelength 0.68 m travels at $v=500\times 0.68=340$ m/s.
• Wave speed is fixed by the medium; change the frequency and the wavelength adjusts to keep $v$ constant.

6. Speed of waves in a medium

• On a stretched string: $v=\sqrt{\frac{T}{\mu }}$ (tension $T$, linear mass density $\mu$) — tighten the string and waves travel faster.
• Sound travels fastest in solids, slower in liquids, slowest in gases; in air it is ~340 m/s and rises with temperature ($\propto \sqrt{T}$).

7. Superposition: standing waves and harmonics

When two identical waves travel in opposite directions they form a standing wave with fixed nodes (no motion) and antinodes (maximum motion).

• String fixed at both ends (or a pipe open at both ends): fundamental $f_1=\frac{v}{2L}$, with harmonics $f_n=\frac{nv}{2L}$ (all integer multiples).
• Pipe closed at one end: only odd harmonics, fundamental $f_1=\frac{v}{4L}$.

Worked example 7.1. A 1 m string carries waves at 200 m/s. Its fundamental frequency is $f_1=\frac{v}{2L}=\frac{200}{2}=100$ Hz; the next harmonic is 200 Hz.

8. Beats

Two waves of slightly different frequency alternately reinforce and cancel, producing a throbbing loudness. The beat frequency is the difference:

$f_{\text{beat}}=|f_1-f_2|$

• Tuning forks at 256 Hz and 260 Hz give 4 beats per second — the basis of tuning instruments by ear.

9. The Doppler effect

Relative motion between source and observer shifts the observed frequency:

$f^{\prime }=f\left(\frac{v\pm v_o}{v\mp v_s}\right)$

• Approaching raises the pitch (upper signs); receding lowers it — the falling pitch of a passing siren.
• Used in medical ultrasound (blood-flow Doppler), radar speed guns and astronomy (redshift).

10. Common traps NEET sets here

• Confusing where $v$ and $a$ peak — velocity max at the mean, acceleration max at the extremes.
• Thinking pendulum period depends on mass or amplitude — it depends only on $L$ and $g$.
• Total SHM energy scaling — it goes as $A^2$, not $A$.
• Sound as a transverse wave — sound is longitudinal; only its speed changes with medium, not its frequency.
• Applying both-ends-open harmonics to a closed pipe — a closed pipe gives only odd harmonics with $f_1=v/4L$.

11. Memory aids

• "Fast at the middle, hard at the ends" — velocity max at mean, acceleration max at extremes.
• "Pendulum: only L and g" — mass and amplitude don't matter.
• "Energy loves amplitude squared" — $E=\frac{1}{2}k{A}^2$.
• "v = fλ, always" — the anchor of every wave question.
• "Closed pipe skips the evens" — only odd harmonics, $f_1=v/4L$.

12. Exam protocol

1. Identify SHM by $a=-{\omega }^{2}x$; velocity max at the centre, acceleration max at the extremes.
2. Use $T=2\pi \sqrt{m/k}$ (spring) and $2\pi \sqrt{L/g}$ (pendulum); pendulum ignores mass and amplitude.
3. Recall SHM energy $=\frac{1}{2}k{A}^2\propto A^2$, constant, with the $KE:PE$ split $(A^2-x^2):x^2$.
4. Anchor every wave question on $v=f\lambda$; wave speed is set by the medium.
5. String/open pipe: $f_n=nv/2L$; closed pipe: odd harmonics, $f_1=v/4L$.
6. Beat frequency is the difference of the two frequencies.
7. Doppler: approaching raises pitch, receding lowers it.