Chapter 7 NGSS: Wave Motion & Sound Waves

Introduction to Waves

  • Waves involve the movement of energy through matter and space.

  • Water waves appear to move water towards the shore, but it's the energy that's moving, not the water itself.

  • Waves carry energy.

Wave Phenomenon

  • A wave is a disturbance that moves from one point to another through a medium.

  • Demonstration using a Slinky:

    • A single traveling pulse can be produced by moving one end of the Slinky back and forth.

    • The pulse moves through the Slinky, with portions of the Slinky moving as the pulse passes.

    • After the pulse, the Slinky returns to its original position.

Definition of a Wave

  • A wave is a movement of energy through matter and space.

  • It's a disturbance that moves from one point to another through a medium.

Types of Waves

  • Mechanical Waves

    • Require a medium to travel.

    • Examples: water waves, sound waves, waves on a rope.

  • Electromagnetic Waves

    • Do not require a medium to travel.

    • Examples: light waves, radio waves, X-rays, infrared waves.

Electromagnetic Spectrum

  • The electromagnetic spectrum encompasses a wide range of wavelengths and frequencies.

  • From longer to shorter wavelengths: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays.

  • Frequency and energy increase as wavelength decreases.

  • Examples of sources and uses for different types of electromagnetic waves are provided.

Wave Properties

  • Periodic Wave: A series of longitudinal pulses with equal time intervals.

  • Period (T): The time between pulses.

  • Frequency (f): The number of pulses or cycles per unit of time; f = 1/T.

  • Wavelength (\lambda ): The distance between the same points on successive pulses.

  • Wave Speed (v): The distance a pulse travels in one period; v = \frac{\lambda}{T} = f\lambda.

Classwork Problem

  • A wave traveling in the positive x-direction has a frequency of f = 20.0 \, Hz.

  • Find:

    • Wavelength

    • Amplitude

    • Period

    • Velocity

Wave Speed on a Rope

  • The speed of a pulse on a rope depends on how quickly succeeding segments can be accelerated.

  • By Newton's second law, acceleration is proportional to force and inversely proportional to mass: a = \frac{F}{m}.

  • A larger tension produces a larger acceleration.

  • Wave speed increases with tension (F) and decreases with mass per unit length (\mu = \frac{m}{L}):

    • v = \sqrt{\frac{F}{\mu}}

Types of Mechanical Waves

  • Longitudinal Waves

  • Transverse Waves

Longitudinal Waves

  • The direction of the wave is parallel to the direction of the vibration.

  • Example: Sound wave.

  • Consist of compressions (regions of high density) and rarefactions (regions of low density).

Transverse Waves

  • The direction of the wave is perpendicular to the direction of the vibration.

  • Examples: string waves, water waves.

Visual Representation of Waves

  • (a) Longitudinal wave

  • (b) Transverse wave

  • Wavelength is shown for both types of waves.

Wavelength of Longitudinal Waves

  • The distance from:

    • Compression-to-compression

    • Rarefaction-to-rarefaction

Wavelength of Transverse Waves

  • The distance from:

    • Crest-to-crest

    • Trough-to-trough

  • Wavelength is written as the Greek letter lambda (\lambda).

  • Measured as the distance between two equal points on a wave.

Wave Components

  • Crest: The highest point of a transverse wave.

  • Trough: The lowest point of a transverse wave.

  • Amplitude: The maximum displacement from the equilibrium position.

  • v = \frac{\lambda}{T}

Interference and Standing Waves

  • When a periodic wave is reflected, the reflected wave interferes with the incoming wave.

  • Interference: The process in which two or more waves combine.

Types of Interference

  • Constructive Interference: Occurs when crests of two waves overlap, resulting in a larger amplitude.

  • Destructive Interference: Occurs when a crest and trough combine, resulting in cancellation or a smaller amplitude.

Principle of Superposition

  • When two or more waves combine, the resulting disturbance is equal to the sum of the individual disturbances.

Water Wave Interference

  • Theoretical drawing of an interference pattern compared to an actual interference pattern.

Standing Waves

  • Occur when a wave reflects upon itself and interference causes a stable pattern.

  • Nodes: Points that remain stationary due to complete destructive interference.

  • Antinodes: Points halfway between nodes where the largest amplitude occurs.

Standing Wave Properties

  • Formed when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere.

  • Node: Point in a standing wave that undergoes complete destructive interference and is stationary.

  • Antinode: Point in a standing wave, halfway between two nodes, at which the largest amplitude occurs.

Standing Wave Example

Supposing that a wave is produced when a rope is moved up and down in a second.
What is the frequency of the wave? ( How many cycles can you see in the picture above)?
If the rope is 14m, calculate the wavelength.
Calculate the velocity of the wave

Musical Instruments and Standing Waves

  • Stringed instruments (guitars, pianos) produce music using standing waves on strings.

  • The frequency of the sound wave equals the string's frequency of oscillation and is related to the musical pitch.

  • A higher frequency corresponds to a higher-pitched sound.

Harmonics on a String Fixed at Both Ends

  • The simplest standing wave, the fundamental or first harmonic has nodes at both ends and an antinode in the middle.

  • The wavelength is determined by the length of the string.

  • The distance between nodes is half the wavelength, so the wavelength must be twice the length of the string (\lambda = 2L).

  • The wave speed is determined by the tension in the string and the mass per unit length of the string.

  • The frequency can be found using the relationship v = f\lambda \implies f = \frac{v}{\lambda} = \frac{v}{2L}.

Harmonics and Frequency

  • A longer string length (L) results in a lower frequency.

  • The effective length can be shortened by placing a finger on the string, producing a higher-pitched tone.

  • The third harmonic has four nodes and three antinodes, with a wavelength equal to two-thirds L.

  • The resulting frequency is three times the fundamental.

Example: Guitar String Harmonics

  • A guitar string has a mass of 4g, a length of 74cm and a tension of 400N, producing a wave speed of 274m/s.

    • (A) What is its fundamental frequency?

    • (B) What is the frequency of the second harmonic?

    • (C) Calculate the wavelength of the 1st, 2nd, and 3rd harmonics.

  • Solution:

    • (A) f_1 = \frac{v}{2L} = \frac{274}{2(0.74)} = 185 \, Hz

    • (B) f2 = \frac{v}{L} = \frac{274}{0.74} = 370 \, Hz \quad \text{or} \quad 2f1 = 2(185) = 370 \, Hz

    • (C) \lambda1 = 2L = 2(0.74) = 1.48 \, m \lambda2 = L = 0.74 \, m
      \lambda_3 = \frac{2}{3}L = \frac{2}{3}(0.74) = 0.49 \, m

Harmonics Formulas Summary

  • For a string fixed at both ends:

    • L = \frac{\lambda}{2}, \lambda = 2L, f_1 = \frac{v}{\lambda} = \frac{v}{2L}

    • L = \lambda, f_2 = \frac{v}{\lambda} = \frac{v}{L}

    • L = \frac{3}{4}\lambda, \lambda = \frac{4}{3}L, f_3 = \frac{v}{\lambda} = \frac{3v}{4L}

Standing Waves in Tubes

  • Standing-wave patterns for the first three harmonics for a tube open at one end and closed at the other:

    • The first harmonic (fundamental) has a wavelength four times longer than the length of the tube (\lambda = 4L).

    • The wavelength of the second harmonic is equal to four-thirds of the length of the tube.

    • The wavelength of the third harmonic is equal to four-fifths of the length of the tube.

Musical Intervals and Harmonics

  • Musical scales and intervals are based upon the ratios between the higher harmonics in the notes.

  • Octave: The frequency of the second harmonic is twice that of the first harmonic (e.g., from one note C to the next higher C).

  • Fifth: The frequency of the third harmonic is 3/2 that of the second harmonic (e.g., between C and G).

  • Fourth: The frequency of the fourth harmonic is 4/3 that of the third harmonic (e.g., between C and F).

Musical Tuning

  • The frequency of the fifth harmonic is 5/4 that of the fourth harmonic. This interval is a major third, between the first and third notes (C and E).

  • Just tuning involves tuning an instrument so that in one key, all the intervals have simple frequency ratios.

  • Equally-tempered tuning is a compromise so that the ratios are all approximately correct, but not perfect. The ratios between adjacent half steps on the scale are all identical, so the scales sound correct regardless of what key you are in.

Frequency Calculation Example (Sol)

  • A C-major scale begins with do on middle C having a frequency of approximately 264 Hz. Assuming that they have been tuned to the perfect ratios for the intervals in question, what should the frequency be for sol (G)?

    • Sol is a fifth above do with a ratio of 3/2: f = \frac{3}{2}(264 \, Hz) = 396 \, Hz

Frequency Calculation Example (Fa)

  • A C-major scale begins with do on middle C having a frequency of approximately 264 Hz. Assuming that they have been tuned to the perfect ratios for the intervals in question, what should the frequency be for fa (F)?

    • Fa is a fourth above do with a ratio of 4/3: f = \frac{4}{3}(264 \, Hz) = 352 \, Hz

Classwork Problems

  1. A guitar string has an overall length of 1.25m and a total mass of 40g before it was strung in a guitar. There is a distance of 64cm between the two fixed ends. The string is tightened to a tension of 720N.

    • A) What is the mass per unit length of the string?

    • B) What is the wave speed for waves on the tightened string?

    • C) What is the wavelength of the travelling waves that interfere to form the fundamental standing wave?

  2. A pipe that is open at both ends will form standing waves. Suppose we have an open pipe 50cm of length.

    • A) What is the possible longest wavelength for the traveling waves that could interfere to form a standing wave on this pipe?

    • B) If the speed of sound is 340m/s, What is the frequency of the first and second harmonic?

  3. An organ pipe closed at one end and open at the other has a length of 0.8m

    • A) Calculate the second longest possible wavelength

    • B) Calculate the frequency of the first three harmonics if the velocity is 340m/s

Sound Waves Introduction

  • Sound waves can be generated in various ways, including oscillating strings.

  • Sound waves travel through the air to reach our ears.

Nature and Properties of Sound

  • How is sound created?

    • Sound is created by vibrations that produce waves with compressions and rarefactions.

  • At room temperature, sound travels at 340 m/s.

  • Audible range: 20 Hz - 20 kHz

  • Ultrasonic: higher than 20 kHz

  • Infrasonic: lower than 20 Hz

Pitch

  • Pitch refers to how high or low we perceive a sound to be, depending on the frequency of the sound wave.

  • High frequency = high pitch

  • Low frequency = low pitch

Doppler Effect

  • Refers to the change in frequency when there is relative motion between an observer of waves and the source of the waves.

Doppler Effect Explained

  • A moving source of sound, like a car horn, changes pitch depending on its motion relative to the listener.

  • As a car passes a stationary observer, the horn's pitch changes from a higher pitch to a lower pitch.

Doppler Effect: Wavefronts

  • Comparing wavefronts for a stationary vs. moving car horn illustrates why the pitch changes.

    • When the car is approaching, the wavefronts reaching the observer are closer together.

    • When the car is moving away, the wavefronts reaching the observer are farther apart.

Doppler Effect Summary

  • The Doppler effect is the change in frequency due to the source or receiver's motion.

  • Greater speed of the source = greater Doppler effect.

  • Blue Shift: Increased frequency (source approaching).

  • Red Shift: Decreased frequency (source receding).

Wave Properties and Interactions

  • Reflection

  • Refraction

  • Dispersion

  • Diffraction

Reflection

  • Reflection is the bouncing of an incident wave when it reaches a boundary.

  • Law of Reflection: The angle of incidence equals the angle of reflection.

Reflection Illustration

Normal
Reflected ray
Incident ray
Angle of incidence
Angle of reflection
Mirror

Clear vs. Diffuse Reflection

  • Smooth, shiny surfaces have a clear reflection.

  • Rough, dull surfaces have a diffuse reflection (light is scattered in different directions).

Refraction

  • Bending of light due to a change in speed.

  • Index of Refraction: The amount by which a material refracts light.

  • Prisms: Glass that bends light. Different frequencies are bent different amounts, breaking light into different colors.

Refraction Details

  • When light passes from one medium to another, it is refracted due to the change in speed.

    • From high to low speed: the angle of refraction is less than the angle of incidence (bent toward the normal).

    • From low to high speed: bent away from the normal.

  • Light moves slower in denser media (water, glass).

Refraction Diagrams

Diagram depicting light bending toward the normal and away from the normal.

Daily Phenomena of Refraction

  • Swimming pools and ponds appear shallower than they are.

  • Objects are at a deeper depth than where they appear to be.

  • Bent objects in liquids.

Refraction Illustration (Fish in Water)

  • Diagram illustrating how light bends as it passes from water to air, causing the perceived image of a fish to be different from its actual position.

Diffraction

  • Waves (longer wavelengths or lower frequencies) diffract more effectively than high-frequency waves.

  • Diffraction patterns are determined by the size of the opening and the wavelength.

  • Diffraction spreading out of plane waves as they pass through a hole.

Dispersion

  • The index of refraction of a material varies with the wavelength of light. Different wavelengths are bent by different amounts.

  • Wavelength is associated with color. Blue light is bent the most, and red light is bent the least.

  • Light is bent toward the normal as it enters a prism and away from the normal as it exits the prism.

Total Internal Reflection

  • Occurs when light passes from a denser medium to a less dense medium.

  • The light ray is unable to exit the medium.

  • The angle of incidence in the denser medium is greater than the critical angle.

Total Internal Reflection Illustration

Diagram showing light being reflected internally within a medium (water).