Ch 16 pt 1 & Ch 17

Waves

• A wave is an organized disturbance traveling with a well-defined wave speed.
• Waves transfer energy but not material.
• Mechanical Waves:
  • Require a material medium (e.g., air, water, a stretched string).
  • Examples: sound waves, ripples on a lake.
• Electromagnetic Waves:
  • Do not require a material medium and can travel through a vacuum.
  • Examples: visible light, microwaves, x-rays, gamma rays.

Transverse Waves

• The disturbance occurs perpendicular to the direction of wave propagation.
• Light waves are transverse waves.

Longitudinal Waves

• The disturbance occurs parallel to the direction of wave propagation.
• Sound waves are longitudinal waves.

Periodic Nature of Waves

• Waves are caused by the simple harmonic motion of the medium or electromagnetic field.
• The period $T$ of one cycle is related to the frequency $f$ by: $f = \frac{1}{T}$
• Wave speed $v$ is: $v = \frac{\lambda}{T} = f\lambda$

Wave Speed of a String's Transverse Wave

• Wave speed depends on the acceleration of string particles due to tension force.
• Newton's 2nd Law relates force and acceleration.
• Wave speed equation: $v = \sqrt{\frac{T}{m/L}}$
  • $T$: Tension along the string.
  • $m$: Mass of the string.
  • $L$: Length of the string.
  • $m/L$: String's linear mass density.

Wave Speed

• Wave speed of a transverse wave on a string: $v = \sqrt{\frac{T}{m/L}}$
• Speed of a single particle of the string: $v_{\text{particle}} = A\omega \sin(\omega t)$
• Each particle undergoes simple harmonic motion at angular frequency $\omega$.

Principle of Linear Superposition

• When two or more waves are present simultaneously, the resultant disturbance is the sum of the disturbances of the individual waves.

Sound Waves

• Condensation: Region of higher pressure.
• Rarefaction: Region of lower pressure.
• Sound waves are longitudinal waves consisting of consecutive condensations and rarefactions.

Interference of Sound Waves

• Constructive Interference:
  • Condensations overlap with condensations, rarefactions with rarefactions.
  • Waves are in phase, resulting in louder sound.
• Destructive Interference:
  • Condensations overlap with rarefactions.
  • Waves are out of phase, potentially resulting in no sound.

Difference in Path-Lengths Determines Type of Interference

• Path-length $r$: Distance from source to listener.
• Path-length difference $\Delta r$ determines interference type.
• For in-phase sources:
  • Constructive: $\Delta r = m\lambda$, where $m = 0, 1, 2, 3, …$
  • Destructive: $\Delta r = (m + \frac{1}{2})\lambda$, where $m = 0, 1, 2, 3, …$
• For out-of-phase sources:
  • Constructive: $\Delta r = (m + \frac{1}{2})\lambda$, where $m = 0, 1, 2, 3, …$
  • Destructive: $\Delta r = m\lambda$, where $m = 0, 1, 2, 3, …$

Standing Waves

• Occur when two transverse waves of same $f$, $\lambda$, and $A$ travel in opposite directions and overlap.
• Standing waves do not travel.

Standing Waves on a String

• Condition for standing wave: $T = \frac{1}{f<em>1} = \frac{2L}{v}$, therefore, $f</em>1 = \frac{v}{2L}$
• Fundamental frequency (first harmonic): $f_1 = \frac{v}{2L}$
• Higher harmonics: $f<em>n = n f</em>1 = n \frac{v}{2L}$, where $n = 1, 2, 3, …$
• Nodes: Points of destructive interference, spaced $\lambda/2$ apart.
• Antinodes: Points of constructive interference, spaced $\lambda/2$ apart.
• Each mode is numbered by the number of antinodes.