JEE Physics Waves Notes Set 01

Wave Motion and Progressive Waves

A wave transfers energy and momentum without transferring matter. Waves may be transverse (particles move perpendicular to the direction of travel; possible only in solids and on liquid surfaces) or longitudinal (particles move parallel to the direction; sound in gases). A progressive wave is \( y = A\sin(\omega t - kx + \phi) \) with \( k = 2\pi/\lambda \), \( v = f\lambda = \omega/k \), phase difference \( \Delta\phi = 2\pi\,\Delta x/\lambda \) and maximum particle speed \( A\omega \). On a string \( v = \sqrt{T/\mu} \).

Sound Waves

The speed of sound in a gas is \( v = \sqrt{\gamma P/\rho} \) (Laplace), larger than Newton's isothermal value \( \sqrt{P/\rho} \). It is independent of pressure at constant temperature, rises as \( \sqrt T \), is greater in lighter and moist air, and does not depend on frequency. Intensity is proportional to \( A^2f^2 \), and intensity level is \( \beta = 10\log_{10}(I/I_0) \) dB. At a boundary the frequency stays the same while speed and wavelength change.

Superposition, Interference and Beats

Waves add algebraically. For two waves of amplitudes \( a_1 \) and \( a_2 \) with phase difference \( \phi \), \( A = \sqrt{a_1^2 + a_2^2 + 2a_1a_2\cos\phi} \). Constructive interference occurs for path difference \( n\lambda \) and destructive for \( (2n+1)\lambda/2 \); \( I_{max}/I_{min} = \left(\frac{a_1+a_2}{a_1-a_2}\right)^2 \). Two waves of nearly equal frequency produce beats at \( |f_1 - f_2| \) per second. Loading a fork with wax lowers its frequency, while filing raises it.

Stationary Waves

Two identical waves moving in opposite directions form a standing wave with fixed nodes and antinodes, with no net energy flow. A string fixed at both ends has \( f_n = \dfrac{n}{2L}\sqrt{T/\mu} \), with all harmonics. An open pipe has \( f_n = nv/2L \) (all harmonics); a closed pipe has \( f = (2n-1)v/4L \) (odd harmonics only). In a closed pipe the lowest resonance length is \( \lambda/4 \) and the next is \( 3\lambda/4 \); end correction is added to the length. For a sonometer wire \( f \propto \frac{1}{L}\sqrt{T} \) and \( f \propto \frac{1}{d}\sqrt{T/\rho} \).

Doppler Effect

The apparent frequency heard is \( f' = f\,\dfrac{v \pm v_o}{v \mp v_s} \), taking the upper sign for the observer moving towards the source and the lower sign in the denominator for the source moving towards the observer. It depends on the relative motion along the line joining source and observer, not on their separation. For light, a receding source shows a red shift with \( \Delta\lambda/\lambda = v/c \).