Waves — Complete Formula

Basic Wave Parameters

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Wave Speed	[v = f\lambda]	[v] = wave speed, [f] = frequency, [\lambda] = wavelength	m/s	Most important relation ⭐
Angular Frequency	[\omega = 2\pi f]	[\omega] = angular frequency	rad/s	From SHM ⭐
Wave Number	[k = \dfrac{2\pi}{\lambda}]	[k] = wave number	rad/m	Spatial frequency
Wave Equation	[y = A\sin(kx – \omega t + \phi)]	[A] = amplitude, [\phi] = phase constant	m	General wave form ⭐

💡 Memory Hint:
Wave = SHM travelling in space

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
String Wave Speed	[v = \sqrt{\dfrac{T}{\mu}}]	[T] = tension, [\mu] = mass/length	m/s	Important ⭐
Sound in Gas	[v = \sqrt{\dfrac{\gamma P}{\rho}}]	[\gamma] = adiabatic constant, [P] = pressure, [\rho] = density	m/s	Depends on medium ⭐
Sound (Temperature form)	[v \propto \sqrt{T}]	[T] = absolute temperature	—	Temperature effect ⚠️

💡 Memory Hint:
Heavier medium → slower wave

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Resultant Displacement	[y = y_1 + y_2]	[y_1, y_2] = individual waves	m	Principle of superposition ⭐
Resultant Amplitude	[A = 2A_0 \cos\dfrac{\Delta \phi}{2}]	[\Delta \phi] = phase difference	m	Interference result ⭐

💡 Memory Hint:
Waves add → interference pattern forms

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Node Condition	[x = n\dfrac{\lambda}{2}]	[n] = integer	m	Zero displacement ⭐
Antinode Condition	[x = (2n+1)\dfrac{\lambda}{4}]	—	m	Maximum displacement
Frequency	[f_n = n\dfrac{v}{2L}]	[L] = length	Hz	String fixed at both ends ⭐

💡 Memory Hint:
Nodes → no motion, Antinodes → maximum motion

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Beat Frequency	[f_b = f_1 – f_2]	[f_1, f_2] = frequencies	Hz

💡 Memory Hint:
Beats = interference of close frequencies

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
General Formula	[f’ = f \dfrac{v \pm v_o}{v \mp v_s}]	[f’] = observed freq, [v_o] = observer velocity, [v_s] = source velocity	Hz	Sign convention critical ⚠️
Moving Observer	[f’ = f \left(1 + \dfrac{v_o}{v}\right)]	—	Hz	Towards source ⭐
Moving Source	[f’ = \dfrac{f}{1 – \dfrac{v_s}{v}}]	—	Hz	Towards observer ⭐

💡 Memory Hint:

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Intensity	[I = \dfrac{P}{A}]	[P] = power, [A] = area	W/m²	Energy per area ⭐
Intensity Relation	[I \propto A^2]	—	—	Depends on amplitude
Decibel Scale	[\beta = 10 \log_{10}\left(\dfrac{I}{I_0}\right)]	[I_0] = reference intensity	dB	Logarithmic scale ⭐

💡 Memory Hint:
Loudness ∝ (amplitude)²

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Frequency	[f_n = (2n – 1)\dfrac{v}{4L}]	[n] = 1,2,3…	Hz	Only odd harmonics ⭐

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Frequency	[f_n = n\dfrac{v}{2L}]	—	Hz	All harmonics present ⭐

💡 Memory Hint:
Closed pipe → odd only
Open pipe → all harmonics

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