Oscillations and Waves Formula Sheet
All key Oscillations & Waves Physics formulas - SHM, energy, wave motion, superposition, standing waves, beats & Doppler effect for JEE Main & Advanced quick revision.
Every must-know formula for Oscillations and Waves on one scannable page — built straight from this chapter’s topics. Use it for last-minute revision before JEE Main and Advanced.
Simple Harmonic Motion (SHM)
Defining Equations
$$\boxed{F = -kx} \qquad \boxed{a = -\omega^2 x}$$| Quantity | Formula | Notes |
|---|---|---|
| Displacement | $x = A\sin(\omega t + \phi)$ | Or $x = A\cos(\omega t + \phi')$ |
| Velocity (time) | $v = A\omega\cos(\omega t + \phi)$ | Derivative of $x$ |
| Velocity (position) | $v = \omega\sqrt{A^2 - x^2}$ | Use when time is unknown |
| Acceleration (time) | $a = -A\omega^2\sin(\omega t + \phi)$ | Derivative of $v$ |
| Acceleration (position) | $a = -\omega^2 x$ | Defining relation |
Maximum Values
$$\boxed{v_{max} = A\omega \text{ (at } x=0)} \qquad \boxed{a_{max} = A\omega^2 \text{ (at } x=\pm A)}$$Time Period, Frequency, Angular Frequency
$$\boxed{\omega = 2\pi f = \frac{2\pi}{T}} \qquad f = \frac{1}{T} = \frac{\omega}{2\pi}$$$v_{max}$ occurs at equilibrium ($x=0$); $a_{max}$ occurs at extremes ($x=\pm A$) — opposite locations.
Time period is independent of amplitude for all SHM systems.
Watch the trap: $T = \dfrac{1}{f} = \dfrac{2\pi}{\omega}$, not $\dfrac{2\pi}{f}$.
Phase
| Phase Difference | Relationship |
|---|---|
| $0$ or $2\pi$ | In phase |
| $\pi$ | Out of phase |
| $\dfrac{\pi}{2}$ | Quadrature (90 degrees out) |
SHM Systems: Time Periods
| System | Time Period | Key Feature |
|---|---|---|
| Spring-mass | $T = 2\pi\sqrt{\dfrac{m}{k}}$ | Independent of $g$ |
| Simple pendulum (small angle) | $T = 2\pi\sqrt{\dfrac{l}{g}}$ | Independent of $m$ and amplitude |
| Physical pendulum | $T = 2\pi\sqrt{\dfrac{I}{mgd}}$ | $d$ = pivot to centre of mass |
| Torsional pendulum | $T = 2\pi\sqrt{\dfrac{I}{C}}$ | $C$ = torsional constant |
Spring frequency: $f = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}$ Pendulum frequency: $f = \dfrac{1}{2\pi}\sqrt{\dfrac{g}{l}}$
Combinations of Springs
$$\text{Series: } \frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} \implies k_{eq} = \frac{k_1 k_2}{k_1 + k_2}$$$$\text{Parallel: } k_{eq} = k_1 + k_2$$Pendulum Special Cases
| Situation | Result |
|---|---|
| Pendulum in accelerating car (accel $a$) | $T = 2\pi\sqrt{\dfrac{l}{\sqrt{g^2 + a^2}}}$, $g_{eff} = \sqrt{g^2 + a^2}$ |
| Pendulum on the Moon ($g_{moon} = g/6$) | $T_{moon} = \sqrt{6}\,T_{earth}$ |
Energy in SHM
$$\boxed{KE = \frac{1}{2}m\omega^2(A^2 - x^2) = \frac{1}{2}k(A^2 - x^2)}$$$$\boxed{PE = \frac{1}{2}m\omega^2 x^2 = \frac{1}{2}kx^2}$$$$\boxed{E_{total} = \frac{1}{2}m\omega^2 A^2 = \frac{1}{2}kA^2}$$| Position | Displacement | KE | PE |
|---|---|---|---|
| Equilibrium | $0$ | $E$ (100%) | $0$ |
| Midpoint | $A/2$ | $3E/4$ | $E/4$ |
| Equal-energy point | $A/\sqrt{2}$ | $E/2$ | $E/2$ |
| Extreme | $\pm A$ | $0$ | $E$ (100%) |
Equal-energy point ($KE = PE$): $\boxed{x = \dfrac{A}{\sqrt{2}} \approx 0.707A}$
Averages over one cycle: $\langle KE \rangle = \langle PE \rangle = \dfrac{E}{2}$
Energy vs time (with $x = A\sin\omega t$): both KE and PE oscillate at frequency $2f$, 180 degrees out of phase (antiphase).
$$KE = \frac{E}{2}[1 + \cos 2\omega t], \qquad PE = \frac{E}{2}[1 - \cos 2\omega t]$$Power in SHM: $P = -\dfrac{1}{2}kA^2\omega\sin(2\omega t)$, average power $\langle P \rangle = 0$.
Pendulum total energy: $E = mgl(1 - \cos\theta_{max}) \approx \dfrac{1}{2}mgl\theta_{max}^2$; speed at lowest point $v = \sqrt{2gl(1 - \cos\theta_0)}$.
$E \propto A^2$ — double the amplitude gives quadruple the energy.
$KE = PE$ at $x = A/\sqrt{2}$, not at $A/2$ (don’t drop the $\sqrt{2}$).
Vertical spring: measure energy from the new equilibrium (shifted by $x_0 = mg/k$); then only $\tfrac{1}{2}kA^2$ matters.
Wave Motion
Wave Parameters
$$\boxed{v = f\lambda = \frac{\lambda}{T} = \frac{\omega}{k}}$$$$k = \frac{2\pi}{\lambda}, \qquad \omega = 2\pi f = \frac{2\pi}{T}, \qquad T = \frac{1}{f}$$Travelling Wave Equation
$$\boxed{y(x,t) = A\sin(kx - \omega t + \phi)}$$| Direction / Type | Equation |
|---|---|
| Wave along $+x$ | $y = A\sin(kx - \omega t)$ |
| Wave along $-x$ | $y = A\sin(kx + \omega t)$ |
| Sound (displacement) | $s = s_0\sin(kx - \omega t)$ |
| Sound (pressure) | $\Delta P = P_0\cos(kx - \omega t)$ (90 degrees out of phase with displacement) |
General differential wave equation: $\dfrac{\partial^2 y}{\partial t^2} = v^2\dfrac{\partial^2 y}{\partial x^2}$
Wave Speed in Different Media
| Medium | Speed | Symbols |
|---|---|---|
| String (transverse) | $v = \sqrt{\dfrac{T}{\mu}}$ | $T$ = tension, $\mu = m/l$ |
| Sound in gas | $v = \sqrt{\dfrac{\gamma RT}{M}} = \sqrt{\dfrac{\gamma P}{\rho}}$ | $\gamma\approx1.4$ (air), $T$ in K |
| Sound in solid (rod) | $v = \sqrt{\dfrac{E}{\rho}}$ | $E$ = Young’s modulus |
| Sound in liquid | $v = \sqrt{\dfrac{B}{\rho}}$ | $B$ = bulk modulus |
Sound in air (empirical): $v \approx 331 + 0.6t$ m/s ($t$ in degrees C); $\approx 343$ m/s at 20 degrees C.
Particle vs Wave Velocity
$$v_{particle} = \frac{\partial y}{\partial t} = -A\omega\cos(kx - \omega t), \qquad v_{particle,\,max} = A\omega$$$$v_{wave} = \frac{\omega}{k} = f\lambda \quad (\text{constant for a given medium})$$Energy, Power, Intensity
$$\boxed{u = \frac{1}{2}\rho\omega^2 A^2} \qquad \boxed{P_{avg} = \frac{1}{2}\rho v\omega^2 A^2}$$$$I = \frac{P}{A_{area}}, \qquad I_{spherical} = \frac{P}{4\pi r^2} \propto \frac{1}{r^2}$$Power $\propto A^2$, $\propto \omega^2$, $\propto v$.
Phase and Path Difference
$$\boxed{\Delta\phi = \frac{2\pi}{\lambda}\Delta x} \qquad \Delta x = \frac{\lambda}{2\pi}\Delta\phi$$Reflection / Transmission at a Boundary
| End / Boundary | Result |
|---|---|
| Fixed end | Reflects with phase change of $\pi$ (inverted) |
| Free end | Reflects with no phase change |
| Medium 1 to 2 | $A_r = \dfrac{v_2 - v_1}{v_2 + v_1}A_i$, $A_t = \dfrac{2v_2}{v_2 + v_1}A_i$ |
Frequency stays constant across a boundary; wavelength changes: $\dfrac{\lambda_2}{\lambda_1} = \dfrac{v_2}{v_1}$.
Wave speed on a string depends only on $T$ and $\mu$ — not on frequency.
Energy $\propto A^2$, never $\propto A$.
Across media: frequency fixed, wavelength changes.
Superposition, Interference and Standing Waves
Principle of Superposition
$$\boxed{y_{resultant} = y_1 + y_2 + y_3 + \cdots}$$Interference (same frequency)
$$\boxed{A = \sqrt{A_1^2 + A_2^2 + 2A_1 A_2\cos\phi}} \qquad \tan\alpha = \frac{A_2\sin\phi}{A_1 + A_2\cos\phi}$$| Type | Phase Difference | Path Difference | Amplitude |
|---|---|---|---|
| Constructive | $0, 2\pi, 4\pi, \dots$ | $n\lambda$ | $A_1 + A_2$ |
| Destructive | $\pi, 3\pi, 5\pi, \dots$ | $(2n+1)\dfrac{\lambda}{2}$ | $\lvert A_1 - A_2\rvert$ |
For $A_1 = A_2 = A_0$: amplitude is $2A_0$ at $\phi=0$, $A_0\sqrt{2}$ at $\phi=\pi/2$, and $0$ at $\phi=\pi$.
Standing (Stationary) Waves
Formed by two identical waves travelling in opposite directions:
$$\boxed{y = 2A\sin(kx)\cos(\omega t)}$$| Feature | Formula |
|---|---|
| Position-dependent amplitude | $A(x) = 2A\sin(kx)$ |
| Node positions ($\sin kx = 0$) | $x = n\dfrac{\lambda}{2}$ |
| Antinode positions ($\sin kx = \pm1$) | $x = (2n+1)\dfrac{\lambda}{4}$ |
| Node-to-node spacing | $\dfrac{\lambda}{2}$ |
| Antinode-to-antinode spacing | $\dfrac{\lambda}{2}$ |
| Node-to-nearest-antinode | $\dfrac{\lambda}{4}$ |
Standing waves transfer no net energy.
Normal Modes: Strings and Pipes
| System | Boundary | Harmonics | Fundamental | $f_n$ |
|---|---|---|---|---|
| String (both fixed) | Node-Node | All | $\dfrac{v}{2L}$ | $f_n = n\dfrac{v}{2L}$ |
| String (one free) | Node-Antinode | Odd only | $\dfrac{v}{4L}$ | $f_n = (2n-1)\dfrac{v}{4L}$ |
| Closed pipe (one end) | Node-Antinode | Odd only | $\dfrac{v}{4L}$ | $f_n = (2n-1)\dfrac{v}{4L}$ |
| Open pipe (both ends) | Antinode-Antinode | All | $\dfrac{v}{2L}$ | $f_n = n\dfrac{v}{2L}$ |
For a string fixed at both ends, $v = \sqrt{\dfrac{T}{\mu}}$ and $f_n = \dfrac{n}{2L}\sqrt{\dfrac{T}{\mu}} = nf_1$.
Allowed wavelengths: both-same ends $\lambda_n = \dfrac{2L}{n}$; one-different end $\lambda_n = \dfrac{4L}{2n-1}$.
Beats
$$\boxed{f_{beats} = |f_1 - f_2|}$$Resultant: $y = [2A\cos(2\pi f_{beat}t)]\sin(2\pi f_{avg}t)$, with $f_{avg} = \dfrac{f_1+f_2}{2}$ and $f_{beat} = \dfrac{|f_1-f_2|}{2}$. Audible when $|f_1 - f_2| < \sim 10$ Hz. Frequency $\propto \sqrt{T}$ on a string, so a fractional tension change $\delta$ shifts frequency by $\approx \tfrac{\delta}{2}$.
“Closed is Odd, Open is All” — closed pipe gives odd harmonics ($f_1, 3f_1, 5f_1, \dots$); open pipe gives all.
Same-to-same (N-N or A-A) = $\lambda/2$; change-type (N-A) = $\lambda/4$.
Beats = difference of frequencies, never the sum.
Open-pipe fundamental is twice the closed-pipe fundamental of equal length.
Doppler Effect
Sound — General Formula
$$\boxed{f' = f\frac{v + v_o}{v - v_s}}$$Sign convention (TAPS): Toward Adds for oPserver (numerator $+v_o$), Toward Subtracts for Source (denominator $-v_s$).
| Scenario | Formula |
|---|---|
| Stationary observer, moving source | $f' = f\dfrac{v}{v - v_s}$ |
| Moving observer, stationary source | $f' = f\dfrac{v + v_o}{v}$ |
| Both stationary / zero relative velocity | $f' = f$ |
| Reflection off moving surface | $f' = f\dfrac{v + v_r}{v - v_r}$ |
| Not along line of sight | $f' = f\dfrac{v + v_o\cos\alpha}{v - v_s\cos\beta}$ |
For a reflector with $v_r \ll v$: $\dfrac{\Delta f}{f} = \dfrac{2v_r}{v}$.
Light / EM Waves
| Regime | Formula |
|---|---|
| Non-relativistic ($v \ll c$) | $f' = f\left(1 + \dfrac{v}{c}\right)$, $\dfrac{\Delta f}{f} = \dfrac{v}{c}$ |
| Relativistic | $f' = f\sqrt{\dfrac{c + v}{c - v}} = f\sqrt{\dfrac{1+\beta}{1-\beta}}$, $\beta = v/c$ |
| Radar reflection ($v \ll c$) | $\Delta f = 2f\dfrac{v}{c}$ |
Redshift parameter: $\boxed{z = \dfrac{\lambda' - \lambda}{\lambda} = \dfrac{\Delta\lambda}{\lambda}}$; for $v \ll c$, $z \approx \dfrac{v}{c}$.
- Approaching: $f' > f$, $\lambda' < \lambda$ (blueshift)
- Receding: $f' < f$, $\lambda' > \lambda$ (redshift)
Shock Waves / Sonic Boom (source faster than wave, $v_s > v$)
$$\boxed{M = \frac{v_s}{v}} \qquad \boxed{\sin\theta = \frac{v}{v_s} = \frac{1}{M}}$$$M<1$ subsonic, $M=1$ sonic, $M>1$ supersonic.
Moving source and moving observer (same speed) give different shifts — source motion has the larger effect.
Never use the sound formula for light — for light only relative velocity matters.
For any reflection problem, remember the factor of 2.
Chapter Map
graph TD
A[Oscillations and Waves] --> B[SHM]
A --> C[Wave Motion]
B --> B1[Time periods: spring, pendulum]
B --> B2[Energy: KE, PE, total]
C --> C1[Wave speed and equation]
C --> C2[Superposition and interference]
C2 --> C3[Standing waves: strings, pipes]
C2 --> C4[Beats]
C --> C5[Doppler effect]