Oscillations and Waves — Revision Notes

⚡ 30-Second Revision

SHM: — Restoring force $F = -kx$. Differential equation: $\frac{d^2x}{dt^2} + \omega^2x = 0$.
Displacement: — $x(t) = A \sin(\omega t + \phi)$.
Velocity: — $v(t) = A\omega \cos(\omega t + \phi) = \pm \omega\sqrt{A^2 - x^2}$. Max $v_{max} = A\omega$.
Acceleration: — $a(t) = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$. Max $a_{max} = A\omega^2$.
Angular Frequency: — $\omega = \sqrt{k/m}$ (spring), $\omega = \sqrt{g/L}$ (pendulum).
Period: — $T = 2\pi/\omega$.
Total Energy (SHM): — $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$.
Wave Equation: — $v = f\lambda$.
Transverse Wave: — Particle oscillation $\perp$ wave direction.
Longitudinal Wave: — Particle oscillation $||$ wave direction.
Standing Waves (String fixed ends): — $\lambda_n = 2L/n$, $f_n = nv/(2L)$.
Standing Waves (Open pipe): — $\lambda_n = 2L/n$, $f_n = nv/(2L)$.
Standing Waves (Closed pipe): — $\lambda_n = 4L/(2n-1)$, $f_n = (2n-1)v/(4L)$ (only odd harmonics).
Doppler Effect: — $f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$. (Numerator + for observer towards, - for away; Denominator - for source towards, + for away).

2-Minute Revision

Oscillations are repetitive motions, with Simple Harmonic Motion (SHM) being the simplest, characterized by a restoring force proportional to displacement ($F=-kx$). Key SHM parameters are amplitude ($A$), period ($T$), and frequency ($f$).

Remember the period formulas: $T = 2pi\sqrt{m/k}$ for a spring-mass system and $T = 2pi\sqrt{L/g}$ for a simple pendulum (small angles). Velocity in SHM is maximum at equilibrium and zero at extremes, while acceleration is maximum at extremes and zero at equilibrium.

Total mechanical energy is conserved in ideal SHM, converting between kinetic and potential, given by $E = \frac{1}{2}kA^2$. Waves are disturbances transferring energy without matter. The fundamental wave equation is $v = f\lambda$.

Differentiate between transverse (particle motion perpendicular to wave) and longitudinal (particle motion parallel to wave) waves. The principle of superposition explains interference and standing waves.

For standing waves in strings and pipes, remember the conditions for nodes and antinodes and the formulas for harmonic frequencies. The Doppler effect describes the apparent change in frequency due to relative motion between source and observer; master its formula and sign conventions.

5-Minute Revision

Begin by solidifying Simple Harmonic Motion (SHM). Recall that SHM is defined by a linear restoring force, $F = -kx$. This leads to sinusoidal displacement $x(t) = A \sin(\omega t + \phi)$. From this, derive velocity $v(t) = A\omega \cos(\omega t + \phi)$ and acceleration $a(t) = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x(t)$.

Understand that maximum velocity is $A\omega$ at equilibrium ($x=0$), and maximum acceleration is $A\omega^2$ at extremes ($x=\pm A$). The angular frequency $\omega$ is $\sqrt{k/m}$ for a spring-mass system and $\sqrt{g/L}$ for a simple pendulum.

The total mechanical energy in SHM is conserved, $E = \frac{1}{2}kA^2$, and continuously exchanges between kinetic and potential forms. For example, if a spring-mass system has $k=100,\text{N/m}$ and $A=0.

1, ext{m}$,$E = \frac{1}{2}(100)(0.1)^2 = 0.5, ext{J}$.

Next, move to waves. A wave is a disturbance propagating energy without matter. The core relationship is $v = f\lambda$. Distinguish between transverse waves (e.g., light, waves on a string where particles move perpendicular to wave direction) and longitudinal waves (e.

g., sound, where particles move parallel). The principle of superposition is vital for understanding interference (constructive/destructive) and standing waves. For standing waves in a string fixed at both ends or an open organ pipe, the allowed frequencies are $f_n = nv/(2L)$, where $n=1,2,3...

$(all harmonics are present). For a closed organ pipe, only odd harmonics are present,$f_n = (2n-1)v/(4L)$, where$n=1,2,3...$. For example, the fundamental frequency of an open pipe of length$L$is$v/(2L)$, while for a closed pipe of the same length, it's$v/(4L)$.

Finally, master the Doppler effect for sound: $f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$. Remember the sign convention: for observer ($v_o$), '+' if moving towards source, '-' if away. For source ($v_s$), '-' if moving towards observer, '+' if away.

Practice applying this to various scenarios, including those with wind or reflections, by adjusting the effective speed of sound or treating the reflector as a virtual source/observer.

Prelims Revision Notes

1
Periodic Motion: — Repeats after a fixed time interval. Oscillatory motion is a type of periodic motion (back and forth about equilibrium).
2
Simple Harmonic Motion (SHM): — Restoring force $F = -kx$. Equation: $m\frac{d^2x}{dt^2} + kx = 0 \implies \frac{d^2x}{dt^2} + \omega^2x = 0$.

* Displacement: $x(t) = A \sin(\omega t + \phi)$ or $A \cos(\omega t + \phi)$. * Velocity: $v = \frac{dx}{dt} = A\omega \cos(\omega t + \phi) = \pm \omega\sqrt{A^2 - x^2}$. Max velocity $v_{max} = A\omega$ at $x=0$.

* Acceleration: $a = \frac{dv}{dt} = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$. Max acceleration $a_{max} = A\omega^2$ at $x=\pm A$. * Angular Frequency: $\omega = \sqrt{k/m}$ (spring-mass), $\omega = \sqrt{g/L}$ (simple pendulum).

* Period: $T = 2\pi/\omega = 1/f$. * Total Energy: $E = KE + PE = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$. Energy is conserved.

1
Waves: — Disturbance transferring energy without matter.

* Wave Speed: $v = f\lambda$. * Types: * Transverse: Particle oscillation perpendicular to wave direction (e.g., light, string waves). Can be polarized. * Longitudinal: Particle oscillation parallel to wave direction (e.

g., sound waves). Cannot be polarized. * Speed of Sound: In gas $v = \sqrt{\gamma P/\rho}$ or $v = \sqrt{\gamma RT/M}$. In solid $v = \sqrt{Y/\rho}$. In liquid $v = \sqrt{B/\rho}$. * Principle of Superposition: Resultant displacement is vector sum of individual displacements.

* Interference: Constructive (phase diff $2n\pi$) and Destructive (phase diff $(2n+1)\pi$). * Standing Waves: Formed by superposition of two identical waves travelling in opposite directions.

Nodes (zero displacement) and Antinodes (max displacement). * String fixed at both ends: $L = n\frac{\lambda}{2}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$, all harmonics). * Open Organ Pipe: $L = n\frac{\lambda}{2}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...

$, all harmonics). * **Closed Organ Pipe:**$L = (2n-1)\frac{\lambda}{4}$,$f_n = (2n-1)\frac{v}{4L}$($n=1,2,3...$, only odd harmonics). * Beats: Formed by superposition of two waves of slightly different frequencies.

Beat frequency $f_{beat} = |f_1 - f_2|$. * Doppler Effect: Apparent frequency $f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$. * $v_o$: observer speed. '+' if towards source, '-' if away. * $v_s$: source speed.

'-' if towards observer, '+' if away. * $v$: speed of sound in medium.

Vyyuha Quick Recall

For Doppler Effect signs: 'O'bserver 'T'owards 'A'dds (Numerator +). 'S'ource 'T'owards 'S'ubtracts (Denominator -). If away, reverse the sign. (OTA, STS)