Waves

Learning Objectives

• Understand the distinction between transverse and longitudinal waves, focusing on the direction of oscillations relative to wave propagation.
• Calculate the displacement relation in a progressive wave using the equation $y(x, t) = a \sin(kx - \omega t + \Phi)$, identifying each parameter.
• Determine the speed of a traveling wave on a stretched string using $v = \sqrt{T/\mu}$, where $T$ is tension and $\mu$ is linear mass density.
• Apply the principle of superposition to analyze the resultant displacement when multiple waves traverse the same medium.
• Analyze standing waves and normal modes, identifying nodes and antinodes, and calculating frequencies for strings and air columns.
• Examine the reflection of waves at rigid and open boundaries, noting phase changes.
• Investigate the phenomenon of beats, calculating beat frequency from two waves of slightly different frequencies.
• Evaluate the speed of sound in different media using $v = \sqrt{B/\rho}$ for fluids and $v = \sqrt{\gamma P/\rho}$ for gases.
• Explore wave parameters such as wavelength, frequency, period, amplitude, angular frequency, wave number, and phase relations.
• Differentiate between constructive and destructive interference based on phase differences in superposed waves.
• Understand the conditions for resonance in strings and air columns, and how external frequencies can match natural frequencies.

Chapter Notes

Transverse and Longitudinal Waves

• Transverse Waves: Particles oscillate perpendicular to the direction of wave propagation.
• Longitudinal Waves: Particles oscillate parallel to the direction of wave propagation.

Displacement Relation in a Progressive Wave

• The displacement in a sinusoidal wave propagating in the positive X direction: $y(x, t) = a \sin(kx - \omega t + \Phi)$
  • $a$: Amplitude
  • $k$: Angular wave number
  • $\omega$: Angular frequency
  • $\Phi$: Phase constant

Speed of a Travelling Wave

• Transverse Wave on a Stretched String: $v = \sqrt{\frac{T}{\mu}}$
  • $T$: Tension
  • $\mu$: Linear mass density
• Longitudinal Wave (Sound in Fluids): $v = \sqrt{\frac{B}{\rho}}$
  • $B$: Bulk modulus
  • $\rho$: Density

Principle of Superposition of Waves

• The displacement of any element is the sum of displacements due to each wave: $y(x, t) = y_1(x, t) + y_2(x, t)$

Standing Waves and Normal Modes

• Formed by interference of two waves traveling in opposite directions.
• Nodes: Points of zero displacement.
• Antinodes: Points of maximum displacement.
• Fixed-end String:
  • Fundamental frequency: $\nu = \frac{v}{2L}$
  • Allowed wavelengths: $\lambda = \frac{2L}{n}$

Reflection of Waves

• Rigid Boundary: Phase change of $\pi$.
• Open Boundary: No phase change.

Beats

• Occur when two waves of slightly different frequencies interfere.
• Beat frequency: $\nu_{\text{beat}} = |\nu_1 - \nu_2|$

Speed of Sound in Different Media

• In Fluids: $v = \sqrt{\frac{B}{\rho}}$
• In Gases: $v = \sqrt{\frac{\gamma P}{\rho}}$
  • $\gamma$: Adiabatic index

Wave Parameters and Phase Relations

• Wavelength ($\lambda$), Frequency ($\nu$), Period ($T$), Amplitude ($a$), Angular Frequency ($\omega$), Wave Number ($k$).
• Relation: $\lambda \nu = v$

Constructive and Destructive Interference

• Constructive: Waves in phase, amplitudes add.
• Destructive: Waves out of phase, amplitudes subtract.

Standing Waves in Stretched Strings

• Normal Modes: $\nu_n = \frac{n v}{2L}$
• Harmonics: Fundamental, second, third, etc.

Standing Waves in Air Columns

• Open Pipe: All harmonics present.
• Closed Pipe: Only odd harmonics present.

Resonance in Strings and Air Columns

• Occurs when external frequency matches natural frequency of the system.