MechanicsHigh Yield (2 Qs in JEE)

Wave Motion & Sound Waves Formulas

Wave equation, speed of sound in gases, standing waves in strings and organ pipes, beats, and Doppler effect.

7 Formulas•642+ Questions in Bank

Core Formulas & Mathematical Identities

y(x, t) = A\sin(kx \pm \omega t + \phi), \quad v = \frac{\omega}{k} = f\lambda = \sqrt{\frac{T}{\mu}} \; (\text{String})

Wave traveling with wavenumber k = 2π/λ and angular frequency ω = 2πf.

Examiner Tip: Minus sign (-ωt) signifies propagation in positive x-direction.

v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T}{M}}

Speed of acoustic waves in ideal gas under adiabatic compressions.

Examiner Tip: Speed of sound increases with temperature: v ∝ √T (in Kelvin).

f_n = n \frac{v}{2L} \; (n=1,2,3\dots, \text{Open Pipe}), \quad f_n = (2n - 1) \frac{v}{4L} \; (n=1,2,3\dots, \text{Closed Pipe})

Harmonics in resonant air columns.

Examiner Tip: Fundamental frequency of open pipe is twice that of closed pipe of identical length.

f_{\text{beat}} = |f_1 - f_2|, \quad f' = f_0 \left( \frac{v \pm v_o}{v \mp v_s} \right)

Periodic amplitude modulation from interfering sound waves, and frequency shift due to relative motion.

Examiner Tip: Use upper signs when observer and source approach each other.

I = \frac{1}{2}\rho v \omega^2 A^2 = 2\pi^2 \rho v f^2 A^2, \quad \beta = 10\log_{10}\left(\frac{I}{I_0}\right) \text{ dB} \; (I_0 = 10^{-12} \text{ W/m}^2)

Energy transmission per unit area per second and logarithmic decibel scale.

Examiner Tip: A 10-fold increase in intensity corresponds to a +10 dB loudness increase.

e = 0.6 r, \quad \lambda = 2(L_2 - L_1), \quad v = 2f(L_2 - L_1)

Acoustic end correction for organ pipe of inner radius r. Eliminates end correction error by subtracting first and second resonance lengths.

Examiner Tip: End correction applies at every open end: open pipe has 2e, closed pipe has 1e.

f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} = n f_1 \quad (n = 1, 2, 3\dots)

Natural vibration frequencies of string clamped at both ends under tension T with linear mass density μ.

Examiner Tip: Nodes occur at fixed ends; anti-nodes occur at centers of vibration loops.

Frequently tested misconceptions in Wave Motion & Sound Waves

•Open organ pipe produces ALL harmonics (1, 2, 3...); closed organ pipe produces ONLY ODD harmonics (1, 3, 5...).
•In Doppler effect, shift is in perceived frequency, NOT the actual wave speed in the medium.