Waves are all around us, from sound to light. They have key properties like amplitude, frequency, and wavelength that determine how they behave. Understanding these basics helps us grasp how waves interact and move through different materials.

Wave properties explain everyday phenomena like why some sounds are higher pitched or why light bends in water. They're crucial for technologies we use daily, from radio to X-rays. Knowing how waves work lets us harness their power in countless ways.

Wave Properties

Properties of waves

• Amplitude
  • Maximum displacement of a wave from its equilibrium position measured in units of distance (meters)
  • Determines the energy carried by the wave, with larger amplitudes corresponding to higher energy
• Frequency ($f$)
  • Number of wave cycles that pass a fixed point per unit time measured in Hertz (Hz), where 1 Hz = 1 cycle per second
  • Higher frequency waves have more cycles per second (higher pitch sound, higher energy electromagnetic waves)
• Period ($T$)
  • Time required for one complete wave cycle measured in units of time (seconds)
  • Reciprocal of frequency expressed as $T = \frac{1}{f}$
  • Longer periods correspond to lower frequencies and shorter periods correspond to higher frequencies
• Wavelength ($\lambda$)
  • Distance between two consecutive points on a wave that are in phase (two crests or two troughs) measured in units of distance (meters)
  • Longer wavelengths are associated with lower frequencies and shorter wavelengths with higher frequencies
• Velocity ($v$)
  • Speed at which a wave propagates through a medium measured in units of distance per unit time (meters per second)
  • Determined by the properties of the medium, such as elasticity and density, and is constant for a given medium

Relationships between wave characteristics

• Wave velocity equation: $v = f\lambda$
  • Velocity is directly proportional to both frequency and wavelength
  • If frequency increases and wavelength remains constant, velocity increases
  • If wavelength increases and frequency remains constant, velocity also increases
• Relationship between frequency and period: $f = \frac{1}{T}$
  • Frequency and period are inversely proportional
  • As frequency increases, period decreases, and vice versa
• Relationship between wavelength and period: $\lambda = vT$
  • Wavelength is directly proportional to both velocity and period
  • For a constant velocity, increasing the period will result in a longer wavelength
• Relationship between wavelength and frequency: $\lambda = \frac{v}{f}$
  • Wavelength and frequency are inversely proportional when velocity is constant
  • In a given medium, increasing the frequency will decrease the wavelength, and decreasing the frequency will increase the wavelength
• Wave equation: $y(x,t) = A \sin(kx - \omega t)$
  • Describes the displacement y of a point on a wave as a function of position x and time t
  • A is the amplitude, k is the wave number, and ω is the angular frequency

Wave Interactions

• Superposition: When two or more waves overlap, their amplitudes add algebraically at each point
• Interference: The result of superposition of waves, which can be constructive (amplitudes add) or destructive (amplitudes subtract)
• Standing waves: Formed when two waves of the same frequency traveling in opposite directions interfere
• Dispersion: The phenomenon where waves of different frequencies travel at different speeds in a medium, causing separation of wave components

Applications of wave concepts

1. Determining wave properties in different media

   • Example: Calculate the wavelength of a sound wave with a frequency of 440 Hz traveling through air at a speed of 343 m/s
     • Using $\lambda = \frac{v}{f}$, $\lambda = \frac{343 \text{ m/s}}{440 \text{ Hz}} = 0.78 \text{ m}$
   • Sound waves travel faster in solids than in liquids, and faster in liquids than in gases due to the differences in elasticity and density of the media

2. Analyzing the behavior of waves at boundaries

   • Example: When a wave travels from one medium to another with a different velocity, the frequency remains constant while the wavelength changes
     • If a wave moves from a medium with a higher velocity to one with a lower velocity (water to air), the wavelength decreases
     • If a wave moves from a medium with a lower velocity to one with a higher velocity (air to water), the wavelength increases

3. Applying wave properties to electromagnetic waves

   • Example: Calculate the frequency of a radio wave with a wavelength of 3 meters, given that electromagnetic waves travel at the speed of light ($c = 3 \times 10^8 \text{ m/s}$)
     • Using $f = \frac{c}{\lambda}$, $f = \frac{3 \times 10^8 \text{ m/s}}{3 \text{ m}} = 1 \times 10^8 \text{ Hz} = 100 \text{ MHz}$
   • Different regions of the electromagnetic spectrum have different wavelengths and frequencies (radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays)