Waves formulas

Class 11 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 14, the special cases questions are built on, and diagrams where they help.

41 formulas8 sectionsClass 11 · Chapter 143 of 8 sections free

By Sreeraj P, M.Sc Physics · 10+ years teaching NEET and JEE

Most used formulasOther formulas and cases

1Progressive waves

$$\begin{array}{l}\displaystyle y=A\sin(\omega t-kx+\phi)\\[5pt]\displaystyle k=\frac{2\pi}\lambda,\quad \omega=2\pi n,\quad v=n\lambda=\frac\omega k\end{array}$$

$$\begin{array}{l}\displaystyle y=A\sin(\omega t-kx+\phi)\\[6pt]\displaystyle k=\frac{2\pi}\lambda\\[6pt]\displaystyle \omega=2\pi n\\[6pt]\displaystyle v=n\lambda=\frac\omega k\end{array}$$

$-kx$: wave moves along $+x$; $+kx$: along $-x$. Any $f(x\mp vt)$ is a travelling wave (e.g. $\dfrac{a}{b+(x-vt)^2}$); $x^2-v^2t^2$ or $\sin(4x^2-9t^2)$ is not. Speed from $f(ax+bt)$: $v=b/a$. Wave equation: $\dfrac{\partial^2y}{\partial t^2}=v^2\dfrac{\partial^2y}{\partial x^2}$.

$$\Delta\phi=\frac{2\pi}\lambda\Delta x,\qquad \Delta\phi=\frac{2\pi}T\Delta t$$

$$\begin{array}{l}\displaystyle \Delta\phi=\frac{2\pi}\lambda\Delta x\\[6pt]\displaystyle \Delta\phi=\frac{2\pi}T\Delta t\end{array}$$

Phase difference ↔ path difference ↔ time difference. Transverse: needs rigidity (solids, liquid surface), can be polarised. Longitudinal: compressions and rarefactions, travels in solids, liquids and gases, cannot be polarised.

$$v_p=\frac{\partial y}{\partial t}=-v\frac{\partial y}{\partial x},\qquad v_{p,\max}=A\omega,\quad a_{p,\max}=A\omega^2$$

$$\begin{array}{l}\displaystyle v_p=\frac{\partial y}{\partial t}=-v\frac{\partial y}{\partial x}\\[6pt]\displaystyle v_{p,\max}=A\omega,\quad a_{p,\max}=A\omega^2\end{array}$$

Particle velocity = −(wave speed) × (slope of the wave shape). $\dfrac{v_{p,\max}}{v}=Ak$. On the snapshot, particles ahead of a crest move up (wave to the right).

$$\begin{array}{l}\displaystyle u=\tfrac12\rho\omega^2A^2,\quad P=\tfrac12\mu\omega^2A^2v\\[5pt]\displaystyle I=\tfrac12\rho\omega^2A^2v=2\pi^2n^2A^2\rho v\end{array}$$

$$\begin{array}{l}\displaystyle u=\tfrac12\rho\omega^2A^2\\[6pt]\displaystyle P=\tfrac12\mu\omega^2A^2v\\[6pt]\displaystyle I=\tfrac12\rho\omega^2A^2v=2\pi^2n^2A^2\rho v\end{array}$$

Energy density, power on a string, intensity. $I\propto A^2n^2$. Point source: $I\propto1/r^2$, $A\propto1/r$; line source: $I\propto1/r$, $A\propto1/\sqrt r$.

2Speed of waves

$$v=\sqrt{\frac T\mu}=\sqrt{\frac{T}{\rho A}}=\sqrt{\frac{\text{stress}}\rho}$$

Transverse wave on a string ($\mu$ = mass per unit length). Independent of amplitude and frequency; the frequency is set by the source and does not change from one medium to another, only $\lambda$ and $v$ change.

String case	Result
Load $M$ hanging	$v=\sqrt{Mg/\mu}$; lift accelerating up: $\sqrt{M(g+a)/\mu}$
Load (density $\rho$) immersed in liquid $\sigma$	$v=\sqrt{\dfrac{Mg(1-\sigma/\rho)}\mu}$; relative density of load $\dfrac{v_1^2}{v_1^2-v_2^2}$ (air $v_1$, water $v_2$)
Rope hanging (own weight), height $x$ from bottom	$v=\sqrt{gx}$; time to travel length $l$: $2\sqrt{l/g}$
Rope of mass $m$ with block $M$ at the bottom	$\dfrac{\lambda_{\text{top}}}{\lambda_{\text{bottom}}}=\sqrt{\dfrac{M+m}M}$
Wire clamped, cooled by $\Delta\theta$	$T=YA\alpha\Delta\theta$, $v=\sqrt{\dfrac{Y\alpha\Delta\theta}\rho}$
Wire stretched by $\Delta l$	$T=YA\dfrac{\Delta l}l$, $v=\sqrt{\dfrac{Y\Delta l}{\rho l}}$
Ring rotating ($\omega$, radius $R$), pulse on it	$T=\mu\omega^2R^2$, $v=\omega R$
Non-uniform string $\mu(x)$	$t=\displaystyle\int\frac{dx}{\sqrt{T/\mu(x)}}$

$$v=\sqrt{\frac Y\rho}\ (\text{rod}),\qquad v=\sqrt{\frac B\rho}\ (\text{fluid}),\qquad v=\sqrt{\frac{\gamma P}\rho}=\sqrt{\frac{\gamma RT}M}$$

$$\begin{array}{l}\displaystyle v=\sqrt{\frac Y\rho}\ (\text{rod})\\[6pt]\displaystyle v=\sqrt{\frac B\rho}\ (\text{fluid})\\[6pt]\displaystyle v=\sqrt{\frac{\gamma P}\rho}=\sqrt{\frac{\gamma RT}M}\end{array}$$

Longitudinal waves. Newton (isothermal, $\sqrt{P/\rho}$) gave 280 m/s; Laplace's adiabatic correction gives 332 m/s. $v_{\text{solid}}>v_{\text{liquid}}>v_{\text{gas}}$.

$v\propto\sqrt T$ (kelvin); $v_t\approx v_0+0.61\,t$ m/s ($t$ in °C). Independent of pressure at constant $T$. Humid air: faster (lower density). $v\propto1/\sqrt M$: $v_{\text{H}_2}=4v_{\text{O}_2}$.
$v_{\text{sound}}=\sqrt{\dfrac\gamma3}\,v_{rms}$. Gas mixture: $\gamma_{\text{mix}}$ from $\dfrac{n_1+n_2}{\gamma-1}=\dfrac{n_1}{\gamma_1-1}+\dfrac{n_2}{\gamma_2-1}$, $M_{\text{mix}}=\dfrac{n_1M_1+n_2M_2}{n_1+n_2}$.

3Sound: pressure, intensity, loudness

$$\begin{array}{l}\displaystyle \Delta P=-B\frac{\partial y}{\partial x},\quad P_0=BAk=\rho v\omega A\\[5pt]\displaystyle \Delta\rho_{\max}=\rho Ak=\frac{\rho}{B}P_0\end{array}$$

$$\begin{array}{l}\displaystyle \Delta P=-B\frac{\partial y}{\partial x}\\[6pt]\displaystyle P_0=BAk=\rho v\omega A\\[6pt]\displaystyle \Delta\rho_{\max}=\rho Ak=\frac{\rho}{B}P_0\end{array}$$

Pressure (and density) wave is $\pi/2$ out of phase with the displacement wave: pressure extreme where displacement is zero.

$$I=\frac{P_0^2}{2\rho v},\qquad \beta=10\log_{10}\frac I{I_0}\ \text{dB},\quad I_0=10^{-12}\ \text{W m}^{-2}$$

$$\begin{array}{l}\displaystyle I=\frac{P_0^2}{2\rho v}\\[6pt]\displaystyle \beta=10\log_{10}\frac I{I_0}\ \text{dB}\\[6pt]\displaystyle I_0=10^{-12}\ \text{W m}^{-2}\end{array}$$

$\beta_2-\beta_1=10\log\dfrac{I_2}{I_1}$: intensity ×10 → +10 dB; ×2 → +3 dB; distance ×2 (point source) → −6 dB. Threshold of pain ≈ 1 W m⁻² (120 dB).

Infrasonic < 20 Hz < audible < 20 kHz < ultrasonic. Loudness ↔ intensity, pitch ↔ frequency, quality (timbre) ↔ overtones present.

5 more sections and 19 formulas in the full chapter

4Reflection, transmission and superposition4 formulas · 1 case table
5Standing waves and strings4 formulas · 1 case table · 1 diagram
6Organ pipes and resonance tube8 formulas · 1 case table
7Beats2 formulas
8Doppler effect1 formula · 1 case table

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