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Wave ON String formulas

Master Wave ON String through 22 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Wave ON String, every formula

22 formulas, typeset and free. Print it, or keep it open beside your practice.

Wave equation
$y = A sin (ω t - k x)$
Q1MCQWave equation
A progressive wave on a string is:
- A
  $A\sin(\omega t-kx)$
- B
  $A\sin(\omega t)$
- C
  $Ae^{kx}$
- D
  $A\omega t$
Wave speed
$v = f λ = \frac{ω}{k}$
Q1MCQWave speed
The wave speed is:
- A
  $f\lambda$
- B
  $\dfrac{f}{\lambda}$
- C
  $\dfrac{\lambda}{f}$
- D
  $f+\lambda$
Angular frequency
$ω = 2 π f$
Wave number
$k = \frac{2 π}{λ}$
Q1MCQWave number
The wave number $k$ is:
- A
  $\dfrac{2\pi}{\lambda}$
- B
  $\dfrac{\lambda}{2\pi}$
- C
  $2\pi\lambda$
- D
  $\dfrac{1}{\lambda}$
Speed on a string
$v = \frac{T}{μ}$
Q1MCQSpeed on string
The speed of a wave on a string is:
- A
  $\sqrt{\dfrac{T}{\mu}}$
- B
  $\sqrt{\dfrac{\mu}{T}}$
- C
  $T\mu$
- D
  $\dfrac{T}{\mu}$
Linear mass density
$μ = \frac{m}{L}$
Wave equation (PDE)
$\frac{\partial ^{2} y}{\partial t ^{2}} = v^{2} \frac{\partial ^{2} y}{\partial x ^{2}}$
Q1MCQWave PDE
The one-dimensional wave equation is:
- A
  $\dfrac{\partial^{2}y}{\partial t^{2}}=v^{2}\dfrac{\partial^{2}y}{\partial x^{2}}$
- B
  $\dfrac{\partial y}{\partial t}=v\dfrac{\partial y}{\partial x}$
- C
  $\dfrac{\partial^{2}y}{\partial t^{2}}=v\dfrac{\partial y}{\partial x}$
- D
  $y=vx$
Particle velocity
$v_{p} = \frac{\partial y}{\partial t} = - v \frac{\partial y}{\partial x}$
Q1MCQParticle velocity
The particle velocity of a string element is:
- A
  $-v\dfrac{\partial y}{\partial x}$
- B
  $v\dfrac{\partial y}{\partial x}$
- C
  $v$
- D
  $0$
Power transmitted
$P = 2 π^{2} f^{2} A^{2} μv$
Q1MCQPower
The power transmitted by a wave on a string is:
- A
  $2\pi^{2}f^{2}A^{2}\mu v$
- B
  $f A\mu v$
- C
  $\dfrac{1}{2}\mu v$
- D
  $f^{2}A$
Intensity
$I \propto A^{2} f^{2}$
Q1MCQIntensity
The intensity of a wave is proportional to:
- A
  $A^{2}f^{2}$
- B
  $A f$
- C
  $\dfrac{1}{A^{2}}$
- D
  $f$
Standing wave
$y = 2 A sin (k x) cos (ω t)$
Q1MCQStanding wave
A standing wave on a string is:
- A
  $2A\sin(kx)\cos(\omega t)$
- B
  $A\sin(\omega t-kx)$
- C
  $A\sin(\omega t+kx)$
- D
  $A\cos(kx)$
Node spacing
$Δ x = \frac{λ}{2}$
Q1MCQNode spacing
The distance between two consecutive nodes is:
- A
  $\dfrac{\lambda}{2}$
- B
  $\lambda$
- C
  $\dfrac{\lambda}{4}$
- D
  $2\lambda$
Distance node to antinode
$\frac{λ}{4}$
Q1MCQNode to antinode
The distance between a node and the adjacent antinode is:
- A
  $\dfrac{\lambda}{4}$
- B
  $\dfrac{\lambda}{2}$
- C
  $\lambda$
- D
  $\dfrac{\lambda}{8}$
Fundamental frequency (fixed ends)
$f_{1} = \frac{v}{2 L} = \frac{1}{2 L} \frac{T}{μ}$
Q1MCQFundamental
The fundamental frequency of a string fixed at both ends is:
- A
  $\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$
- B
  $\dfrac{1}{4L}\sqrt{\dfrac{T}{\mu}}$
- C
  $\dfrac{1}{L}\sqrt{\dfrac{T}{\mu}}$
- D
  $\sqrt{\dfrac{T}{\mu}}$
nth harmonic (fixed ends)
$f_{n} = n \frac{v}{2 L}$
Q1MCQHarmonics
A string fixed at both ends produces which harmonics?
- A
  all harmonics
- B
  only odd
- C
  only even
- D
  none
Number of loops
$L = \frac{nλ}{2}$
Reflection from a fixed end
$phase change of π$
Q1MCQFixed-end reflection
A wave reflecting from a fixed end undergoes a phase change of:
- A
  $\pi$
- B
  $0$
- C
  $\dfrac{\pi}{2}$
- D
  $2\pi$
Reflection from a free end
$no phase change$
Q1MCQFree-end reflection
A wave reflecting from a free end undergoes a phase change of:
- A
  $0$
- B
  $\pi$
- C
  $\dfrac{\pi}{2}$
- D
  $2\pi$
Superposition principle
$y = y_{1} + y_{2}$
Q1MCQSuperposition
When two waves overlap, the net displacement is:
- A
  the sum of the individual displacements
- B
  the product
- C
  the larger one
- D
  zero
Speed vs tension
$v \propto T$
Q1MCQSpeed vs tension
If the tension in a string is quadrupled, the wave speed becomes:
- A
  doubled
- B
  halved
- C
  quadrupled
- D
  unchanged
Frequency of a sonometer wire
$f = \frac{p}{2 L} \frac{T}{μ}$
Q1MCQSonometer
The frequency of the $p$th mode of a sonometer wire is:
- A
  $\dfrac{p}{2L}\sqrt{\dfrac{T}{\mu}}$
- B
  $\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$
- C
  $p\sqrt{\dfrac{T}{\mu}}$
- D
  $\dfrac{p}{L}$
Energy density
$u = 2 π^{2} f^{2} A^{2} μ$
Q1MCQEnergy density
The energy density of a wave on a string is proportional to:
- A
  $f^{2}A^{2}$
- B
  $fA$
- C
  $\dfrac{1}{A}$
- D
  $f$

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Wave ON String formulas

Wave ON String, every formula

Wave equation

Wave speed

Angular frequency

Wave number

Speed on a string

Linear mass density

Wave equation (PDE)

Particle velocity

Power transmitted

Intensity

Standing wave

Node spacing

Distance node to antinode

Fundamental frequency (fixed ends)

nth harmonic (fixed ends)

Number of loops

Reflection from a fixed end

Reflection from a free end

Superposition principle

Speed vs tension

Frequency of a sonometer wire

Energy density

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