Waves

Fundamentals of Waves and Energy Transfer

• Definition of a Wave:

  • A wave is a disturbance that transfers energy from one place to another without transferring matter.

  • Examples of waves include Wi-Fi, radio waves, and sound.

• Energy Source and Movement:

  • All waves require an energy source.

  • Waves transfer energy away from the source (for example, sound energy traveling away from a human mouth).

  • The source of wave energy is known as a vibration.

  • A vibration is defined as a back-and-forth movement of an object.

Mechanical Waves and Mediums

• Mechanical Waves Definition:

  • Mechanical waves are waves that travel through matter.

  • Matter includes solids, liquids, and gases (e.g., sound traveling through air).

  • Mechanical waves cannot travel through a vacuum, which is defined as a space where no matter is found.

• The Medium:

  • A medium is the material/matter in which a wave travels.

  • Examples of mediums include air, water, solids, and strings.

  • Mechanical waves can propagate through solids, liquids, and gases.

• Motion Patterns of Mechanical Waves:

  • Mechanical waves move in two primary patterns:

  • Transverse waves (characterized by up-and-down motion).

  • Longitudinal waves (characterized by back-and-forth motion).

Transverse Waves: Structure and Characteristics

• Definition:

  • A transverse wave is a wave in which the disturbance is perpendicular (at a right angle) to the direction the wave travels (sideways movement of energy).

• Anatomy of a Transverse Wave:

  • Crest: The highest point of the wave.

  • Trough: The lowest point of the wave.

  • Equilibrium: The middle or resting position of the wave.

  • Amplitude: The distance from the equilibrium line to either the crest or the trough.

  • Wavelength: The distance between two identical points on consecutive waves:

  • Distance from crest to the next crest = $1\text{ wavelength}$.

  • Distance from trough to the next trough = $1\text{ wavelength}$.

Longitudinal Waves: Compressions and Rarefactions

• Definition:

  • A longitudinal wave causes the particles in a medium to move parallel to the direction that the wave travels.

  • Particles transfer energy by vibrating back and forth within the medium, moving parallel to the wave's path of motion while the energy moves forward (e.g., left to right).

• Particle States and Wave Anatomy:

  • Rest Position: Prior to a longitudinal wave traveling through a medium, all particles of matter are spaced at equal distances apart. This uniform spacing is the rest position of the medium, which alters once a wave passes through.

  • Compressions: The regions of a longitudinal wave where the particles in the medium are closest together, corresponding to areas of high pressure.

  • Rarefactions: The regions of a longitudinal wave where the particles in the medium are spread furthest apart, corresponding to areas of low pressure.

  • Wavelength in Longitudinal Waves: The distance between the center of one compression to the next compression, or from one rarefaction to the next rarefaction.

• Graphical Representation:

  • Longitudinal waves (such as sound waves) are frequently represented graphically as transverse waves because transverse wave diagrams are significantly easier to measure and visualize.

  • In these graphical conversions, compressions correspond to crests and rarefactions correspond to troughs.

Wave Characteristics and Energy Relationships

• Amplitude and Energy:

  • Amplitude is the distance measured from the equilibrium (resting position) to the crest or the trough.

  • For sound, when the volume changes, the amplitude changes.

  • The higher the amplitude of a wave, the greater the energy it carries.

  • In a uniform wave train, the amplitude of the crests and troughs remains constant throughout the wave.

• Wavelength:

  • Wavelength is the physical distance between two identical points on adjacent waves (e.g., crest to crest or trough to trough).

  • The larger the wavelength, the lower the frequency of the wave.

• Frequency and Energy:

  • Frequency is the number of waves that pass a given point in one second ($1\text{ s}$).

  • It is measured in Hertz ($\text{Hz}$).

  • More waves passing a point corresponds to a higher frequency; fewer waves correspond to a lower frequency.

  • The smaller the frequency value, the bigger (longer) the wave.

  • The amount of energy transferred is directly tied to the frequency of the wave: if the frequency of a wave doubles, the energy transported by the wave also doubles.

Mechanical Waves in Nature: Sound and Water

• Sound Waves:

  • Wave Classification: Longitudinal wave.

  • Source: Vibrating objects.

  • Particle Movement:

  • With each single vibration, the energy source moves back and forth once.

  • The forward motion pushes particles in the medium closer together, forming an area of high pressure called a compression.

  • The subsequent backward movement leaves behind a region of low pressure with fewer particles, called a rarefaction.

• Water Waves:

  • Wave Classification: A combination of transverse waves and longitudinal waves.

  • Source: Friction generated between wind and water at sea.

  • Particle Movement:

  • Water particles move forward and backward (longitudinal component) while simultaneously moving up and down (transverse component).

  • This combined motion results in water particles following a circular path.

  • This circular path becomes progressively smaller as the wave approaches deeper regions or moves away from the surface disturbance.

Sound Perception: Loudness, Intensity, and Pitch

• Loudness and Amplitude:

  • Sound waves are longitudinal mechanical waves.

  • The more energy a sound wave carries, the larger its amplitude becomes, and the louder the sound is perceived to be.

  • Loudness is defined as the human perception of the energy carried by a sound wave.

• Sound Intensity:

  • Intensity is defined as the amount of sound energy that passes through one square meter ($1\,\text{m}^2$) of space in one second.

  • As a sound wave travels farther away from its source, the area of air particles sharing the wave's total energy expands.

  • Because the surface area increases with distance from the source, the amount of energy passing through any single square meter decreases.

  • Consequently, the sound wave becomes less intense over distance.

  • Directly proportional relationship: As intensity increases or decreases, amplitude increases or decreases, and perceived loudness increases or decreases.

• Decibels ($\text{dB}$):

  • The decibel ($\text{dB}$) is the standard unit of measurement used to describe the intensity or loudness of sound.

• Pitch and Ultrasound:

  • Pitch is the perception of how high or how low a sound seems.

  • A higher frequency wave produces a higher perceived pitch; a lower frequency wave produces a lower pitch.

  • A healthy human ear can detect sound frequencies ranging between approximately $20\,\text{Hz}$ and $20\,000\,\text{Hz}$.

  • Frequencies that lie above the human hearing limit ($> 20\,000\,\text{Hz}$) are classified as ultrasound.

Wave Interactions with Matter: Reflection, Absorption, and Transmission

• General Behavior:

  • Waves change direction when they travel from one medium to another.

  • As waves travel through matter, some of the energy they carry can be transferred to the matter.

• Reflection:

  • Reflection occurs when a wave bounces off matter, a surface, or an object.

  • All waves reflect when they strike an impermeable surface, causing the wave to change direction.

  • An echo is an everyday example of a sound wave reflecting off a surface.

• Absorption:

  • Absorption is the process where wave energy is taken in by matter.

  • Specifically, it is the transfer of energy by a wave to the particles of the medium through which it travels.

  • As a wave travels through air, some of its energy is continuously transferred to air particles; therefore, the energy carried by the wave decreases over distance.

  • The total amount of energy absorbed depends on both the type of wave and the physical medium through which it propagates.

• Transmission:

  • Transmission is the passage of a wave through a medium or barrier.

  • When matter does not absorb all of a wave's energy, it transmits the remaining energy through to the opposite side of the barrier, allowing the wave to continue traveling.

  • Mechanical waves can be transmitted through solids, liquids, and gases.

Practical Applications and Questions & Discussion

• Question: If we sent a radio up into space, would we be able to hear the song playing on the radio? Why or why not?

  • Response: No. There is no matter in the vacuum of space. Radios produce sound waves, and sound waves are mechanical waves, which strictly require a physical medium of matter to travel and cannot propagate through a vacuum.

• Question: An echo sounds softer compared to the original sound. What interaction of waves could cause this?

  • Response: Absorption. As sound waves propagate through the air toward a surface and return, a portion of their energy is absorbed and transferred to air particles and the reflecting surface, diminishing the wave's total energy and resulting in lower amplitude and softer loudness.

• Application: Ocean Depth Mapping (Bathymetry):

  • Ships use echoes to locate and calculate the depth of the ocean floor.

  • A specialized shipboard device called a transducer emits a sound wave (frequently high-frequency ultrasound) directed straight down through the water column.

  • The sound pulse hits the hard ocean floor and reflects upward as an echo.

  • A receiver positioned on the ship detects the returning echo and measures the total elapsed travel time.

  • The measured time is then used to calculate the precise depth of the ocean floor.

• Application: Biological Echolocation:

  • Certain animals utilize sound reflection to navigate and hunt.

  • Both dolphins and bats rely on echolocation to detect objects, gauge distances, and perceive their environments.