~Waves and Sound~
1. Production of Mechanical Waves
A wave is produced by a disturbance that causes a vibration
Work must be done (force over a distance) to create a vibration
This work transfers energy through a material as a mechanical wave
Mechanical wave
→ Transfer of energy through a medium due to vibration
Medium
→ The material the wave travels through (solid, liquid, gas)
2. Types of Waves
Mechanical Waves
Require a medium
Energy moves, particles only vibrate
Examples: sound, water waves, waves on a string
Electromagnetic Waves
Do NOT need a medium
Made of changing electric and magnetic fields
Created by accelerating charges
Examples: light, radio waves
3. Wave Characteristics
Waveform – shape of the wave
Equilibrium position – rest position before production of wave
Amplitude (A) – max displacement from equilibrium
Bigger amplitude → more energy
Crest – highest point
Trough – lowest point
Wavelength (λ)
→ Distance between identical points (crest to crest)
Frequency & Period
Frequency (f) – cycles per second
Period (T) – time for one cycle
FORMULAS:
f = N / Δt
T = Δt / N
f = 1 / T
T = 1 / f
Units:
Frequency → Hz (1/s)
Period → seconds
4. Propagation of Mechanical Waves
Why waves can travel through a medium
Particles in a medium are connected by intermolecular forces (forces between neighbouring molecules).
These forces allow a vibration to pass from one particle to the next.
Intermolecular Forces
The strength of the intermolecular forces determines how efficiently the vibration travels.
Stronger forces → wave travels more efficiently.
Vibration
Caused by the net motion of particles in the medium.
Net Motion
The displacement of a particle over a certain time interval.
Defined as the difference between the particle’s initial and final positions.
5. Simple Harmonic Motion (SHM)
Simple Harmonic Motion
Motion that repeats itself at regular intervals.
The vibration does not lose energy as heat.
The amplitude stays constant over time.
Damped Harmonic Motion
A type of periodic (repeating) motion.
Amplitude decreases over time.
Energy decreases over time due to heat, friction, or air resistance.
This is what happens in real-world vibrations and mechanical waves.
Examples: pendulum, mass–spring system
6. Particle Behavior in Different Media
Ideal Conditions
No frictional forces
No energy loss
What happens to the particles in ideal conditions
After the wave passes, particles return to their original positions.
There is no net motion once the particles stop vibrating.
Energy
No energy is lost to the medium.
Energy is transferred by the wave, but particles do not move with the wave.
Wave Behavior
The wave can continue indefinitely in ideal conditions.
Effectiveness of a Medium
A medium’s ability to transmit vibrations depends on:
Molecular structure
Density
Temperature
Elasticity
Elasticity: the ability of a medium to return to its original shape after being disturbed.
If a medium returns to its original shape, it is elastic.
More elastic mediums are better at transporting vibrations as waves.
Solids
Strong intermolecular forces
Most efficient wave transfer
Waves travel faster, farther, longer
More rigid = more elastic
Liquids
Molecules close together
Sound travels ~5× faster in water than air
Gases
Molecules far apart
Least efficient for wave transfer
7. Types of Mechanical Waves
Transverse Waves
Particle motion ⟂ wave direction
Example: waves on a string
Longitudinal Waves
Particle motion ∥ wave direction
Example: sound waves
Have compressions (high pressure) and rarefactions (low pressure)
Complex Wave Motion
You can produce both transverse and longitudinal waves from a vibration in a medium
8. Sound Waves
Categories
Audible: 20 Hz – 20 kHz
Infrasonic: below 20 Hz
Ultrasonic: above 20 kHz
Sound Intensity
Depends on amplitude
Bigger amplitude → louder sound
Sound Intensity (I)
→ Power per area (W/m²)
Human hearing: 10⁻¹² to 1 W/m²
Sound level measured in decibels (dB)
Every 10× increase in intensity → +10 dB
>100 dB for long periods damages hearing
9. Wave Speed
Wave Speed
The speed of a wave depends on the properties of the medium it travels through.
Applies to all waves (mechanical and electromagnetic).
Universal Wave Equation
vm/s = fHz λm
Also:
f = 1 / T
v = λ / T
10. Waves on a String
Wave Speed on a String
Effect of Density
The density of a material affects the speed of a wave.
Higher density → particles are harder to accelerate → slower wave speed.
Linear Density (μ)
Definition
Linear density (μ): mass per unit length of a string.
Linear Density Formula
μ=m/L
m = mass of the string kg
L = length of the string m
Effect on Wave Speed
Greater μ → greater inertia
Greater inertia → slower wave speed
Effect of Tension
Tension in the string also affects wave speed.
Greater tension → string becomes more rigid → faster wave speed.
Wave Speed Formula for a String
v=FTμv=μFT
vv = wave speed
FTFT = tension in the string
μμ = linear density
Increasing tension → speed increases
Increasing linear density → speed decreases
Changing Wave Speed
The only way to change the speed of a wave is by changing the medium through which it is travelling.
When the speed of a wave changes, it is the wavelength that changes, not the frequency.
The frequency of a wave is determined by the source of the waves (the rate of the initial vibration producing the waves).
11. Speed of Sound in Air
The speed of sound in air depends on air density and temperature.
Waves travel faster in hotter gas than in cooler gas because the molecules move faster and transfer their kinetic energy better.
Formula:
v = 331.4 + (0.606)T
T in °C
Hotter air → faster sound
12. Mach Number
Used to compare the speed of an object to the speed of sound.
Usually used to describe the speed of aircraft.
Mach number: the ratio of the airspeed of an object to the local speed of soun
Mach = v_object / v_sound
Mach 1 = speed of sound
Used for aircraft
Sonic boom occurs when object ≥ speed of sound
13. Waves at Media Boundaries
Media boundary: where two media meet
Some wave reflects, some transmits
Speed changes → wavelength changes
Reflections
Fixed end: reflected wave inverted
Free end: reflected wave upright same orientation
Refraction
Change in wave speed when entering new medium
Faster → slower: reflected wave inverted
Slower → faster: reflected wave upright
14. Interference of Waves
Principle of Superposition
Resulting amplitude = sum of individual amplitudes
Constructive Interference
Waves in phase
Larger amplitude
Destructive Interference
Waves out of phase
Smaller or zero amplitude
Phase shift: a shift of an entire wave along the x-axis with respect to an identical wave.
Waves have the same frequency and amplitude but start at different points in their cycle.
In Phase:
Two vibrating objects always move in the same direction at the same time.
Two identical waves that have the same phase shift.
Out of Phase:
At any point in their cycles, the two vibrating objects are moving in opposite directions.
Two identical waves that have different phase shifts.
Phase rules:
Whole λ → constructive
Odd ½λ → destructive
15. Standing Waves
Standing wave: an interference pattern produced when incoming and reflected waves interfere with each other.
The wave pattern appears to be stationary.
Node: no motion
Antinode: max motion (2× amplitude)
Fixed–Fixed Ends
Nodes at both ends
Fundamental frequency (f₁) = lowest frequency that can produce standing wave in medium
Harmonics: Whole number multiples that will cause second standing wave
f₂ = 2f₁
f₃ = 3f₁
Overtones: All natural frequencies higher than fundamental frequency
1st overtone = 2nd harmonic
Free–Free Ends
Antinodes at ends
16. Beats
Acoustical Beat: loudness variation due to close frequencies heard as change in loudness
Beat Frequency Formula: the frequency of beats produced by the interference of two waves with slightly different frequencies
fbeat = |f₂ − f₁|
Used for:
Tuning instruments
Finding unknown frequencies
17. Resonance
Objects vibrate best at specific frequencies
Resonant frequency = easiest vibration (usually f₁)
When driving frequency = resonant frequency → large amplitude
Reducing resonance:
Add damping (important in buildings)
18. Doppler Effect
Doppler Effect:
Change in observed frequency due to relative motion
Formula:
fobs = [(v_sound ± v_detector) / (v_sound ± v_source)] f₀
Approaching → higher frequency (compressing the waves)
Moving away → lower frequency (stretching the waves)
Sonic Boom
If the wave source moves at the same speed as the waves, the compressions and rarefactions add together.
This creates a very large compression (high pressure) followed by a very large rarefaction (low pressure).
Occurs when waves pile up at Mach ≥ 1
Creates shock wave
Homework Topics to Review
1. Production and Characteristics of Waves
How can we create a wave? (ex. Water waves or waves on a spring)
Does anyone see the relationship between frequency and period?
Does anyone see the relationship between frequency and period? (pendulum example)
What do you notice about the wavelength of the waves with changing frequency? (oscilloscope observation)
Are waves transmitted more effectively in liquids or gases? Why do you think so?
Homework:
Pg. 380 #1–6
2. Types of Mechanical Waves
Describe a transverse wave in terms of particle motion vs wave motion
Describe a longitudinal wave in terms of particle motion vs wave motion
Are sound waves transmitted more effectively in liquids or gases? Why?
What do you hear when you use a frequency generator or audio illusion?
Homework:
Pg. 384 #1–9
3. Wave Speed
If a wave has a frequency of 230 Hz and a wavelength of 2.3 m, what is its speed?
If a wave has a speed of 1500 m/s and a frequency of 11 Hz, what is the wavelength?
If a wave has a speed of 405 m/s and a wavelength of 2.0 m, what is its frequency?
If a wave machine string has a linear density of 0.2 kg/m and a wave speed of 200 m/s, what tension is required?
If a string on a wave machine has a linear density of 0.011 kg/m and a tension of 250 N, what is the wave speed?
If the temperature of the air is 32°C, what is the speed of sound?
If the speed of sound near you is 333 m/s, what is the temperature of the air?
If the local speed of sound is 344 m/s and an aircraft is flying at 910 km/h, what is the Mach number?
If the Mach number is 0.93 and the local speed of sound is 320 m/s, what is the speed of an airplane?
If the Mach number is 0.81 and the airplane speed is 850 km/h, what is the local speed of sound?
Homework:
Pg. 391 #1–7
Pg. 397 #2–4
4. Waves at Media Boundaries
What will happen to the speed of the reflected wave and the transmitted wave?
What happens when a wave hits a fixed-end boundary?
What happens when a wave hits a free-end boundary?
5. Interference of Waves
What type of phase shifts cause constructive interference?
What type of phase shifts cause destructive interference?
Homework:
Pg. 419 #1–31
6. Standing Waves
What are the locations of nodes and antinodes in a standing wave?
Why are there always two nodes at the ends of a fixed–fixed standing wave?
What is the fundamental frequency (first harmonic)?
What is the relationship between harmonics and overtones?
In a free–free standing wave, where are the antinodes located?
Predict the standing wave frequencies in a Ruben’s tube (tube length 70 cm, two fixed ends)
7. Beats
How many beats will be heard if a 437 Hz tuning fork is sounded with a 425 Hz fork for 3.0 s?
A tuning fork of unknown frequency is sounded with a 440 Hz fork. Over 15 s, 46 beats are produced. What are the possible frequencies of the unknown fork?
A trumpet player plays a note while middle C (256 Hz) is sounded on a piano. She hears 10 beats in 2.0 s. What are the possible frequencies of the note?
Homework:
Pg. 429 #1–4
Worksheet: “More Practice Problems – Beats”
8. Resonance
What happens when two tuning forks of the same frequency are placed near each other?
What happens when two tuning forks of different frequencies are placed near each other?
How can resonance make a pendulum start swinging?
Is it possible to shatter a glass using only sound waves? Why or why not?
Homework:
Pg. 432 #1–3, 5
9. Doppler Effect
What happens to the pitch of a fire truck’s siren as it approaches and then passes?
A police car is approaching at 20.0 m/s with a 1.0 kHz siren. What frequency does a stationary observer detect? (v_sound = 330 m/s)
An ambulance has passed you. You detect 900 Hz; the siren’s frequency is 950 Hz. How fast is the ambulance moving?
Homework:
Pg. 435 #1–7