JEE Physics Waves Notes Set 02

Progressive Waves

A progressive wave carries energy forward through a medium without carrying matter with it. Each particle only oscillates about its mean position. In a transverse wave the particles move at right angles to the direction of travel (crests and troughs). In a longitudinal wave they move along it (compressions and rarefactions). Sound in a fluid is longitudinal, and waves on a string are transverse.

A wave travelling along +x is written \( y = A\sin(\omega t - kx + \phi) \) and one travelling along -x as \( y = A\sin(\omega t + kx + \phi) \). Opposite signs of \( \omega t \) and \( kx \) mean travel along +x.

Two points whose phases differ by 60 degrees are \( \lambda/6 \) apart. Points \( \lambda/2 \) apart are in opposite phase.

Speed of Waves in Different Media

1. In a gas the speed is proportional to \( \sqrt{T} \) (absolute temperature) and inversely proportional to \( \sqrt{M} \). It does not depend on pressure at constant temperature.
2. Humidity increases the speed of sound in air, because water vapour is lighter than dry air.
3. When a wave goes from one medium to another, frequency stays the same while speed and wavelength change in proportion.

Intensity and Energy

1. Average energy density of a wave \( u = \tfrac12\rho\omega^2A^2 \).
2. Intensity \( I = \tfrac12\rho v\omega^2A^2 = 2\pi^2\rho v f^2A^2 \). So \( I\propto A^2f^2 \) in a given medium.
3. From a point source \( I\propto 1/r^2 \) and amplitude \( \propto 1/r \).
4. Sound level \( \beta = 10\log_{10}(I/I_0) \) decibel, with \( I_0 = 10^{-12} \) W/m\( ^2 \). A power change from \( P_1 \) to \( P_2 \) changes the level by \( 10\log_{10}(P_2/P_1) \) dB.

Superposition and Interference

When two waves of the same frequency overlap, the displacements add. If their amplitudes are \( a_1 \) and \( a_2 \) and the phase difference is \( \phi \):

\[ A_R = \sqrt{a_1^2 + a_2^2 + 2a_1a_2\cos\phi},\qquad I_R = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\phi \]

For equal sources of intensity I the maximum intensity is 4I and the minimum is zero.

Beats

1. Two sources of nearly equal frequencies \( f_1 \) and \( f_2 \) give periodic waxing and waning of loudness. Beat frequency \( = |f_1 - f_2| \).
2. The resultant is \( 2a\cos(\pi\Delta f\,t)\sin(2\pi f_{avg}t) \) for equal amplitudes.
3. Loading a fork with wax lowers its frequency and filing it raises its frequency. Observing how the beat count changes tells which fork is higher.
4. Quick check: if two forks differ by x percent, the beat frequency is x percent of the frequency.

Stationary Waves

Two identical waves travelling in opposite directions form a stationary wave, \( y = 2a\sin(kx)\cos(\omega t) \) for a wave with a node at x = 0. Amplitude depends on position, and all particles between two nodes move in phase.

1. Nodes (zero amplitude) and antinodes (maximum amplitude \( 2a \)) alternate, and neighbouring nodes or antinodes are \( \lambda/2 \) apart.
2. A fixed end or closed end is a node, and a free end or open end is an antinode.
3. No energy is transferred along the wave. Energy is stored alternately as kinetic energy (at nodes the strain is largest) and potential energy.
4. At the instant all particles pass the mean position, the energy is entirely kinetic.

Vibrating Strings

A string of length L fixed at both ends supports the frequencies

\[ f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}},\qquad n = 1, 2, 3,\dots \]

1. All harmonics are present: fundamental \( f_1 \), first overtone \( 2f_1 \), second overtone \( 3f_1 \) and so on.
2. Laws of a vibrating string: \( f\propto 1/L \), \( f\propto\sqrt{T} \), \( f\propto 1/\sqrt{\mu} \), and \( f\propto 1/r \) for the same material and tension.
3. Sonometer works on resonance: the wire vibrates with large amplitude when its frequency equals the fork frequency. Melde's experiment verifies that \( p^2T \) is constant, where p is the number of loops.

Organ Pipes and Resonance Column

1. For a closed pipe, the first overtone is the third harmonic and the second overtone is the fifth harmonic.
2. An open pipe closed suddenly has its fundamental halved.
3. End correction: \( e = 0.6r \) for a pipe of radius r. The effective length is \( L + e \) for a closed pipe.
4. In the resonance column experiment, the first resonance is at \( L_1 + e = \lambda/4 \) and the second at \( L_2 + e = 3\lambda/4 \). So \( \lambda = 2(L_2 - L_1) \) and \( e = (L_2 - 3L_1)/2 \).
5. Kundt's tube shows heaps of powder at nodes, and the distance between two heaps is \( \lambda/2 \). Quincke's tube is an interference experiment, not a beats experiment.

Doppler Effect

\[ f' = f\,\frac{v \pm v_L}{v \mp v_S} \]

Use the upper signs in the numerator for a listener moving towards the source, and the lower signs for moving away. In the denominator use the minus sign for a source moving towards the listener, and plus for moving away. Velocities are measured along the line joining source and listener, and v is the speed of sound in the medium.

1. Moving source changes the wavelength in the medium, and a moving listener changes only the rate at which crests are met.
2. For light, with source speed v much smaller than c, \( \Delta f/f = \Delta\lambda/\lambda = v/c \). A receding source gives a red shift.
3. A source moving along a circle gives extreme frequencies at the points where the line to the observer is tangent to the circle.
4. If the source and the observer are on either side of a wind blowing at speed w, replace v by \( v\pm w \) for the corresponding motion.

Solved Examples

1. A wave on a string of linear density 0.04 kg/m has \( y = 0.02\sin 2\pi(t/0.04 - x/0.50) \). Then \( v = \lambda f = 0.50\times25 = 12.5 \) m/s and \( T = \mu v^2 = 0.04\times156.25 = 6.25 \) N.
2. Two forks of 256 Hz and 260 Hz are in phase at t = 0. They are next in phase after \( 1/4 \) s. At \( t = 1/16 \) s the phase difference is \( 2\pi\times4\times\tfrac1{16} = \pi/2 \).
3. A closed pipe has fundamental 100 Hz. The next two resonances are 300 Hz and 500 Hz.
4. An earthquake has P waves at 9 km/s and S waves at 6 km/s, with a gap of 5 minutes. \( d(1/6 - 1/9) = 300 \), so d = 5400 km.