Chapter 8: Waves

Classification of Waves
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- Mechanical Waves: Require a medium for propagation (e.g., sound waves, string waves, water waves).
- Electromagnetic Waves: Do not require a medium for propagation (e.g., radio waves, heat waves, light waves).
- Matter Waves: Associated with moving particles.
Progressive/Traveling Waves
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- Propagates or distributes pulses in space along a specific direction.
- Examples: Waves on a string, waves on a water surface.
Periodic Waves
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- Repeated in regular intervals of time.
- May be transverse or longitudinal.
Transverse Waves:
- Displacement of medium is perpendicular to wave propagation.
- Examples: Ripple in a pond, wave on a string.
- Cannot propagate in gases or liquids.
Longitudinal Waves:
- Displacement of medium is parallel to wave propagation.
- Examples: Wave in a spring, sound waves in air.
For all waves: $v = f\lambda$.
Particles separated by $n\lambda$ are in phase.
Particles separated by $(n+\frac{1}{2})\lambda$ are out of phase.

Produced by vibrating bodies ($\lambda \approx 1m$).
Essentials for sound: Vibrating body, medium for propagation, listener.
Longitudinal waves with 3D propagation in air.
Consist of compressions (max crowding) and rarefactions (min crowding).
Exhibit Reflection, Refraction, Diffraction, Interference, but NOT Polarization (because they are longitudinal).

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Speed of any mechanical wave: $v=\sqrt{\frac{E}{\rho}}$. ($E$: modulus of elasticity, $\rho$: density).
For gases (compressional wave): $v=\sqrt{\frac{E_{bulk}}{\rho}}$.
Newton's formula (isothermal assumption): $v=\sqrt{\frac{P}{\rho}}$.
- At S.T.P: $v \approx 281~m/s$.
- Experimental value: $332~m/s$.
- Percentage error in Newton's calculation: 16%.
Laplace's formula (adiabatic assumption): $v=\sqrt{\frac{\gamma P}{\rho}}$.
- $\gamma = \frac{C_p}{C_v} = 1.42$ (for air).
Speed increases by $0.61~m/s$ (or $61~cm/s$) for each $1^\circ C$ rise in temperature: $v_t = v_o + 0.61t$.
Effect of Moisture: Presence of moisture reduces air density, increasing sound speed.
Dependence:
- Independent of pressure.
- $v \propto \frac{1}{\sqrt{\rho}}$.
- $v \propto \sqrt{T}$.
- $\frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}}$.
Speed of sound in hydrogen is 4 times more than in oxygen at same temperature.

When two or more waves reach a point simultaneously, the resultant displacement is the sum of individual displacements.
$Y_{total} = Y_1 + Y_2 + Y_3 + ... + Y_n$.

Applications of Superposition
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- Interference: Superposition of two waves of same frequency, traveling in same direction.
- Beats: Produced by two waves of slightly different frequencies traveling in the same direction.
- Stationary Waves: Produced by two waves of equal frequency and amplitude traveling along same line in opposite direction.

Mixing of two identical sound waves propagating along same direction.
Conditions: coherent waves, same medium, same direction, identical waves, sources close.
Constructive Interference: Waves reinforce, louder sound.
- Condition: Path difference $= n\lambda$ (where $n=0, \pm1, \pm2, ...$).
- Echoing zone.
Destructive Interference: Waves cancel, fainter sound.
- Condition: Path difference $= (n+\frac{1}{2})\lambda$ (where $n=0, \pm1, \pm2, ...$).
- Silence zone.
Path difference: Difference between lengths of paths traveled by two waves.

All kinds of waves show reflection.
From denser boundary: Phase reverses ($180^\circ$ change). Reflection coefficient maximum, transmission coefficient almost zero.
From rarer boundary: No phase reversal. Reflection and transmission coefficients have considerable values.
Reflected wave has same wavelength and frequency, but phase may change.

Superposition of two identical waves traveling opposite to each other in the same medium.
Involves both constructive and destructive interference.
Antinodes: Points of constructive interference (maximum amplitude). Energy is wholly potential at extreme position, wholly kinetic at equilibrium.
Nodes: Points of destructive interference (minimum/zero amplitude), stationary points.
Distance between two consecutive nodes or antinodes: $\lambda/2$.
Distance between a node and consecutive antinode: $\lambda/4$.

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	Pipes open or closed at both ends / strings fixed at both ends	Pipes closed at one end
Length L	$L = n\frac{\lambda}{2}$	$L = (2n-1)\frac{\lambda}{4}$
Fundamental	$f = v/2L$	$f = v/4L$
Harmonics	$2f, 3f, ... , nf$	$3f, 5f, ... , (2n-1)f$

Fundamental frequency of open pipe = $2 \times$ fundamental frequency of closed pipe.
Number of harmonics in open pipe = $2 \times$ Number of harmonics in closed pipe.

Apparent change in pitch (frequency) of sound due to relative motion of source and observer.
Discovered by Doppler in 1845.
$v$: true speed of sound. $u_o$: speed of listener. $u_s$: speed of source.

Introduction to Waves