Waves

Introduction to Waves

1. Definition:

   • A wave is a disturbance that travels through a medium, transferring energy from one point to another without the transport of matter.

2. Types of Waves:

   • Mechanical Waves:
     • Require a medium for propagation.
     • Examples: Sound waves, water waves.
   • Electromagnetic Waves:
     • Do not require a medium; can travel through a vacuum.
     • Examples: Light waves, X-rays.
   • Matter Waves:
     • Associated with particles, based on quantum mechanics.
     • Example: Electron wave.

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Mechanical Waves

1. Classification Based on Motion:

   • Transverse Waves:
     • Particles oscillate perpendicular to the direction of wave propagation.
     • Example: Waves on a string.
   • Longitudinal Waves:
     • Particles oscillate parallel to the direction of wave propagation.
     • Example: Sound waves.

2. Wave Parameters:

   • Wavelength ($\lambda$):
     • Distance between two consecutive crests or troughs (transverse) or compressions or rarefactions (longitudinal).
     • SI Unit: Meter ($m$).
   • Frequency ($f$):
     • Number of oscillations per second.
     • SI Unit: Hertz ($Hz$).
   • Time Period ($T$):
     • Time taken for one complete oscillation.
     • Relation: $T = \frac{1}{f}$.
   • Wave Speed ($v$):
     • Distance traveled by a wave per unit time.
     • Relation: $v = f\lambda$.

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Wave Equation

1. General Form:

   • A traveling wave can be expressed as: $y(x, t) = A \sin(kx - \omega t + \phi)$
     • $A$: Amplitude (maximum displacement).
     • $k$: Wave number ($k = \frac{2\pi}{\lambda}$).
     • $\omega$: Angular frequency ($\omega = 2\pi f$).
     • $\phi$: Phase constant.

2. Differential Equation of a Wave: $\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}$

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Energy in Waves

1. Kinetic Energy:

   • Due to particle motion in the medium.
   • Formula for a small element: $KE = \frac{1}{2} \rho A^2 \omega^2 \sin^2(kx - \omega t)$
     • $\rho$: Density of the medium.

2. Potential Energy:

   • Due to the deformation of the medium.
   • Formula for a small element: $PE = \frac{1}{2} \rho A^2 \omega^2 \cos^2(kx - \omega t)$

3. Total Energy:

   • Sum of kinetic and potential energy.

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Reflection and Transmission of Waves

1. Reflection:

   • When a wave hits a boundary and bounces back into the original medium.
   • At a rigid boundary:
     • Wave inverts (phase change of $180^\circ$).
   • At a free boundary:
     • No phase change.

2. Transmission:

   • When a wave passes through a boundary into a new medium.
   • Speed, wavelength, and amplitude may change.

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Numerical Example

1. Example 1: A wave on a string is represented by $y(x, t) = 0.02 \sin(5x - 50t)$. Find its amplitude, wavelength, frequency, and speed.
   • Amplitude ($A$): $A = 0.02 \, m$
   • Wave Number ($k$): $k = 5 \, rad/m \implies \lambda = \frac{2\pi}{k} = \frac{2\pi}{5} \approx 1.26 \, m$
   • Angular Frequency ($\omega$): $\omega = 50 \, rad/s \implies f = \frac{\omega}{2\pi} = \frac{50}{2\pi} \approx 7.96 \, Hz$
   • Speed ($v$): $v = f\lambda = 7.96 \cdot 1.26 \approx 10 \, m/s$

Properties of Waves

1. Interference:

   • The phenomenon where two or more waves superpose to form a resultant wave.
   • Constructive Interference:
     • Occurs when waves are in phase.
     • Resultant amplitude: $A_{\text{resultant}} = A_1 + A_2$
   • Destructive Interference:
     • Occurs when waves are out of phase.
     • Resultant amplitude: $A_{\text{resultant}} = |A_1 - A_2|$

2. Diffraction:

   • Bending of waves around obstacles or through openings.
   • More pronounced when the size of the opening is comparable to the wavelength.

3. Reflection:

   • When a wave bounces back after hitting a boundary.
   • The angle of incidence equals the angle of reflection: $\theta_i = \theta_r$

4. Refraction:

   • Change in direction of a wave when it passes from one medium to another due to a change in speed.
   • Snell’s Law: $\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2}$
     • $v_1, v_2$: Wave speeds in the two media.

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Standing Waves

1. Formation:

   • Produced by the superposition of two waves of the same frequency and amplitude traveling in opposite directions.
   • Equation: $y(x, t) = 2A \sin(kx) \cos(\omega t)$

2. Nodes and Antinodes:

   • Nodes: Points of zero displacement.
   • Antinodes: Points of maximum displacement.

3. Conditions for Formation:

   • Length of the string ($L$) for standing waves:
     • For fundamental frequency (first harmonic): $L = \frac{\lambda}{2}$
     • For $n$th harmonic: $L = \frac{n\lambda}{2}$

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Sound Waves

1. Definition:

   • Longitudinal mechanical waves that require a medium for propagation.
   • Speed in air: $v = \sqrt{\frac{\gamma R T}{M}}$
     • $\gamma$: Adiabatic constant.
     • $R$: Universal gas constant.
     • $T$: Temperature in Kelvin.
     • $M$: Molar mass of air.

2. Intensity and Loudness:

   • Intensity ($I$): $I = \frac{P}{A}$
     • $P$: Power, $A$: Area.
   • Loudness:
     • Perceived intensity, depends on the sensitivity of the ear.

3. Doppler Effect:

   • The change in frequency of a sound wave due to the relative motion of the source and observer.
   • Formula: $f' = f \frac{v \pm v_o}{v \mp v_s}$
     • $v$: Speed of sound.
     • $v_o$: Velocity of the observer.
     • $v_s$: Velocity of the source.
     • Use $+$ when moving towards, $-$ when moving away.

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Beats

1. Definition:

   • Beats are the periodic variation in sound intensity due to the interference of two waves of slightly different frequencies.
   • Beat frequency: $f_{\text{beat}} = |f_1 - f_2|$

2. Applications:

   • Used to tune musical instruments by matching frequencies.

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Numerical Examples

1. Example 1: Two waves of frequencies $250 \, Hz$ and $255 \, Hz$ interfere. Find the beat frequency.

   • Formula: $f_{\text{beat}} = |f_1 - f_2|$
   • Substituting values: $f_{\text{beat}} = |255 - 250| = 5 \, Hz$

2. Example 2: A sound wave with frequency $500 \, Hz$ travels through air at $20^\circ C$. Find its wavelength. (Speed of sound in air at $20^\circ C$: $343 \, m/s$).

   • Formula: $\lambda = \frac{v}{f}$
   • Substituting values: $\lambda = \frac{343}{500} = 0.686 \, m$

Resonance in Waves

1. Definition:

   • Resonance occurs when a system is driven by a periodic force at its natural frequency, resulting in maximum amplitude.

2. Examples:

   • Vibrating tuning forks.
   • Resonance in bridges (e.g., Tacoma Narrows bridge collapse).

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Energy and Power in Waves

1. Energy in a Wave:

   • The energy carried by a wave is proportional to the square of its amplitude: $E \propto A^2$

2. Power Transmitted by a Wave:

   • Formula: $P = \frac{1}{2} \rho v A^2 \omega^2$
     • $\rho$: Density of the medium.
     • $v$: Speed of the wave.
     • $A$: Amplitude.
     • $\omega$: Angular frequency.

3. Intensity ($I$):

   • Power per unit area: $I = \frac{P}{A} \propto A^2$

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Wave Speed in Different Media

1. Speed in a String:

   • The speed of a wave on a stretched string is given by: $v = \sqrt{\frac{T}{\mu}}$
     • $T$: Tension in the string.
     • $\mu$: Linear mass density ($\mu = \frac{m}{L}$).

2. Speed in a Solid:

   • Formula: $v = \sqrt{\frac{E}{\rho}}$
     • $E$: Young’s modulus.
     • $\rho$: Density.

3. Speed in a Fluid:

   • Formula: $v = \sqrt{\frac{B}{\rho}}$
     • $B$: Bulk modulus.
     • $\rho$: Density.

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Applications of Waves

1. Sonar:

   • Uses ultrasonic waves to measure distances underwater.
   • Time taken for an echo to return gives the distance: $d = \frac{v \cdot t}{2}$

2. Musical Instruments:

   • Produce sound through resonance and standing waves.
   • Example: String instruments (guitar, violin).

3. Communication:

   • Radio waves and microwaves are used for transmitting signals.

4. Medical Imaging:

   • Ultrasonography uses sound waves to image internal organs.

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Important Phenomena in Waves

1. Doppler Effect:

   • Change in observed frequency due to relative motion between source and observer.
   • Applications:
     • Radar speed guns.
     • Astronomy (redshift/blueshift).

2. Polarization:

   • Restriction of wave vibrations to a single plane.
   • Only transverse waves can be polarized.
   • Applications:
     • Sunglasses to reduce glare.
     • Optical instruments.

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Numerical Examples

1. Example 1: A string of length $1.5 \, m$ is stretched under a tension of $100 \, N$ with a mass of $0.05 \, kg$. Find the speed of the wave in the string.

   • Formula: $v = \sqrt{\frac{T}{\mu}}$
     • $\mu = \frac{m}{L} = \frac{0.05}{1.5} = 0.0333 \, kg/m$ $v = \sqrt{\frac{100}{0.0333}} = \sqrt{3000} \approx 54.77 \, m/s$

2. Example 2: A sonar sends a sound wave and receives its echo after $4 \, s$. If the speed of sound in water is $1500 \, m/s$, calculate the depth of the object.

   • Formula: $d = \frac{v \cdot t}{2}$
     • Substituting values: $d = \frac{1500 \cdot 4}{2} = 3000 \, m$

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Recap: Key Points to Remember

• Waves transfer energy without the transport of matter.
• Mechanical waves require a medium; electromagnetic waves do not.
• Standing waves form due to the superposition of two waves traveling in opposite directions.
• Resonance, interference, diffraction, and polarization are critical phenomena of waves with numerous applications.