Waves and Wavelengths

Learning Objectives

• By the end of this section, you will be able to:

  • Describe important physical features of waveforms.

  • Show how physical properties of sound waves are associated with perceptual experience.

  • Show how physical properties of light waves are associated with perceptual experience.

Overview

• Visual and auditory stimuli occur in the form of waves. These waves are distinct in their medium of propagation and composition: sound waves are mechanical waves requiring a medium (like air or water) and are characterized by vibrations of particles, while light waves are electromagnetic waves that can travel through a vacuum and consist of oscillating electric and magnetic fields.

• Despite differing compositions, waveforms share similar characteristics that are essential for visual and auditory perception, allowing us to quantify and understand these stimuli.

• This section will detail the physical properties of waves, such as amplitude, wavelength, and frequency, and the corresponding perceptual experiences, including loudness, pitch, brightness, and color.

Key Physical Characteristics of Waves

1. Amplitude

   • Definition: The amplitude of a wave is defined as the distance from the center line (or equilibrium position) to the maximum displacement of the wave, either to the top point of the crest or the bottom point of the trough. It represents the intensity or energy of the wave. For sound waves, higher amplitude means greater pressure variations, and for light waves, higher amplitude means a stronger electric and magnetic field.

2. Wavelength

   • Definition: Wavelength ($\lambda$) refers to the length of one complete wave cycle, measured from a specific point on one wave to the identical point on the next consecutive wave.

   • Measurement: Wavelength is typically measured from peak to peak (crest to crest) or from trough to trough. It is commonly expressed in units of meters (m), nanometers (nm) for light, or centimeters (cm).

Relationship between Wavelength and Frequency

• Wavelength is inversely related to frequency for waves traveling at a constant speed. This means that if the wavelength is long, the frequency is low, and vice-versa.

• Frequency

  • Definition: Frequency ($f$) refers to the number of wave cycles that pass a given point in a given unit of time. It quantifies how often a wave repeats itself.

  • Unit: Expressed in hertz (Hz) or cycles per second. One hertz ($1 \text{ Hz}$) means one cycle per second.

  • Inverse Relationship: Longer wavelengths correspond to lower frequencies, while shorter wavelengths correspond to higher frequencies.

  • Mathematical Relationship: The speed of a wave ($v$) is related to its wavelength ($\lambda$) and frequency ($f$) by the formula: $v = \lambda f$. For light waves in a vacuum, $v$ is the speed of light ($c \approx 3 \times 10^8 \text{ m/s}$).

Sound Waves

• Characteristics of sound waves, which are longitudinal mechanical waves, directly relate to our perceptions of sound. Sound is produced by vibrations that create pressure oscillations in a medium.

• Frequency and Pitch

  • The frequency of a sound wave is the primary determinant of its perceived pitch.

  • High-frequency sound waves (more rapid pressure oscillations) are perceived as high-pitched sounds.

  • Low-frequency sound waves (slower pressure oscillations) are perceived as low-pitched sounds.

  • Audible Range in Humans: The human audible range typically spans from approximately 20 Hz (a very low rumble) to 20,000 Hz ($20 \text{ kHz}$), which is a very high-pitched whine.

  • Greatest sensitivity is found in the middle of this range, generally between 1,000 Hz and 5,000 Hz, where human speech typically falls.

Variability in Audible Ranges Across Species

• The range of frequencies an animal can hear varies significantly based on its biology and ecological niche.

• Chickens: 125 - 2000 Hz (limited range)

• Mice: 1000 - 91,000 Hz (extends into ultrasonic range)

• Beluga Whales: 1000 - 123,000 Hz (excellent ultrasonic hearing for echolocation)

• Dogs: 70 - 45,000 Hz (can hear higher pitches than humans, useful for detecting distant sounds)

• Cats: 45 - 64,000 Hz (also extends into ultrasonic range, aiding in hunting small prey)

Loudness and Amplitude

• The loudness of sound is closely associated with the amplitude of the sound wave, which reflects the intensity of the pressure variations.

• Higher amplitudes (greater pressure changes) correlate with louder sounds. Conversely, lower amplitudes result in softer sounds.

• Loudness Measurement: Loudness is measured in decibels (dB), which is a logarithmic unit representing the ratio of a given sound intensity to a reference intensity. This logarithmic scale mirrors how humans perceive sound intensity – a tenfold increase in sound intensity typically corresponds to an increase of 10 dB.

  • Example of Decibel Levels:

    • Rustling leaves: 10 dB

    • Whispering: 30 dB

    • Typical conversation: 60 dB (often at 10 million times the intensity of the quietest audible sound)

    • Vacuum cleaner: 70 dB

    • Rock concert: 120 dB (approaching the threshold of pain)

Potential for Hearing Damage

• Sustained or intense exposure to sounds ranging from 80 dB to 130 dB or higher poses significant risks for temporary or permanent hearing damage, including noise-induced hearing loss (NIHL) and tinnitus.

• Examples of High Decibel Sounds and Risk:

  • Food processor: $\sim80 \text{ dB}$ (risk with prolonged exposure over several hours)

  • Heavy truck (25 feet away): $\sim90 \text{ dB}$ (begin to risk hearing damage after 8 hours)

  • Subway train (20 feet away): $\sim100 \text{ dB}$ (risk after about 2 hours)

  • Live rock music and jackhammers can exceed 120 dB (risk after minutes of exposure).

Statistics on Hearing Loss

• One-third of all acquired hearing loss in adults is attributable to noise exposure, making it a prevalent and preventable condition.

• The louder the sound, the shorter the exposure time required to cause hearing damage. This relationship is non-linear and cumulative.

  • Example: Listening through earbuds at maximum volume (100–105 dB) can induce noise-induced hearing loss after just 15 minutes of continuous exposure daily.

• Higher volumes can accelerate and increase the risk of age-related hearing loss (presbycusis) by damaging the delicate hair cells in the cochlea over time.

Threshold for Pain

• The pain threshold for sound typically occurs around 130 dB. Sounds at or above this level can cause immediate physical discomfort and rapid, irreversible hearing damage.

• Examples: A jet plane taking off from short distance or a revolver firing at close range produce sounds at or above this critical threshold.

Perception Interaction of Frequency and Amplitude

• Although amplitude is generally the primary factor linked with perceived loudness, the interplay of frequency and amplitude can significantly affect perceived loudness due to the non-uniform sensitivity of the human ear across different frequencies.

• Example: A 10 Hz sound wave is inaudible regardless of its amplitude because it falls below the lower limit of the human audible range (20 Hz).

• A 1000 Hz sound wave, which is near the peak of human hearing sensitivity, will appear significantly louder with increasing amplitude compared to a sound wave of the same amplitude at the extremes of the audible range.

Timbre of Sound

• Timbre refers to the unique quality or "color" of a sound, which allows us to distinguish between different musical instruments or voices, even when they are playing the same note (same fundamental frequency) at the same loudness level (same amplitude).

• Timbre is influenced by the complex interplay of a sound wave's fundamental frequency, the presence and relative intensity of its overtones or harmonics (multiples of the fundamental frequency), and the attack and decay characteristics of the sound. This complexity creates the richness and distinctiveness of various sounds.

Light Waves

• The visible spectrum is the portion of the electromagnetic spectrum detectable to humans, meaning these are the wavelengths of light that our eyes can perceive as color.

• Electromagnetic Spectrum: This vast spectrum comprises all forms of electromagnetic radiation, which are waves of oscillating electric and magnetic fields that propagate through space at the speed of light. It ranges from very short wavelength, high-energy gamma rays to very long wavelength, low-energy radio waves, including: gamma rays, x-rays, ultraviolet light, visible light, infrared light, microwaves, and radio waves.

• Human visible spectrum: For humans, visible light corresponds to wavelengths ranging from approximately 380 to 740 nm ($1 \text{ nm} = 1 \text{ billionth of a meter or } 10^{-9} \text{ m}$).

Animal Perception of Light

• Other species have evolved to detect different parts of the electromagnetic spectrum, providing them with unique sensory abilities crucial for their survival and environmental interaction.

• Honeybees: Can see ultraviolet light (wavelengths shorter than human vision), which helps them locate nectar guides on flowers that are invisible to humans.

• Snakes: Possess specialized pits that can detect infrared radiation (wavelengths longer than human vision), allowing them to "see" the body heat of prey in total darkness.

Wavelength and Color Perception

• The wavelength of light is the most crucial physical property that dictates our perception of color. Different wavelengths are perceived as different colors.

• Associations (from longest to shortest visible wavelengths):

  • Longer wavelengths (approx. 620-740 nm) correlate with the perception of color red.

  • Intermediate-long wavelengths (approx. 570-620 nm) correlate with orange and yellow.

  • Intermediate wavelengths (approx. 495-570 nm) correlate with the perception of color green.

  • Shorter wavelengths (approx. 450-495 nm) correlate with the perception of blue.

  • Very short wavelengths (approx. 380-450 nm) correlate with the perception of indigo and violet.

  • Mnemonic for order of colors from longest to shortest wavelength: ROYGBIV (Red, Orange, Yellow, Green, Blue, Indigo, Violet).

Amplitude and Brightness

• The amplitude of light waves correlates with our perception of brightness or intensity.

• Higher amplitude light waves mean a greater intensity of the electric and magnetic fields, which translates to perceiving a light source as brighter. This corresponds to a higher number of photons detected by the eye per unit of time.

• Conversely, lower amplitude light waves result in a dimmer appearance.