(09-22-25) Standing Waves In Air Columns & Periodic & Aperiodic Signals, Filters & Resonance
Quiz 1 Review
Question 1: Average Velocity
A jet airplane travels 600 miles in 2 hours in a straight line.
The average velocity is stated as 800 miles per hour (though 600/2 = 300 mph). The stated correct answer was 800 mph.
Question 2: Frequency of Angular Motion
A grandfather clock ticks once when the pendulum swings from left to right.
It makes two ticks in one second.
One cycle is defined as a swing from left to right (one tick) and then right to left (another tick), meaning one cycle equals two ticks.
Since two ticks occur in one second, one cycle occurs in one second.
Therefore, the frequency of angular motion is 1 cycle per second (or 1 ext{ Hz}). The instructor mentions "frequency is 100" which seems to be a verbal misstatement for 1 ext{ Hz}.
Question 3: Wave Transportation
When a wave travels from one place to another, energy is transported.
Question 4: Wavelength, Period, and Frequency Relationship
For a periodic wave: period (T), wavelength ( ext{lambda}), and frequency (f).
If the period increases, the wavelength increases.
If the frequency increases, the wavelength decreases and the time (period) decreases.
The understanding of the relationship between time (period), wavelength, and frequency is critical.
Question 5: Harmonics of a Rope
If the fundamental frequency of a rope fixed at both ends is 30 ext{ Hz}.
The frequency of the second harmonic is found by multiplying the fundamental frequency by 2 (30 ext{ Hz} imes 2 = 60 ext{ Hz}).
Generally, for the n^{ ext{th}} harmonic, multiply the fundamental frequency by n.
Question 6: Pendulum Frequency
A pendulum with a period of 8 seconds has a frequency described by the formula: f = rac{1}{T}.
f = rac{1}{8 ext{ seconds}} = 0.125 ext{ Hz}. Since it takes 8 seconds to complete one cycle, the frequency is less than 1 ext{ Hz}.
Question 7: Node vs. Antinode
A node is a point of minimum displacement on a standing wave; an antinode is a point of maximum displacement.
The statement "A node is a point of maximum displacement on a standing wave" is False.
Question 8: Standing Waves
Standing waves are formed as a result of the superimposition of two traveling waves.
They do not necessarily move at the speed of sound or occur only at certain temperatures.
Question 9: Wave Period Calculation
Observing 20 wave crests passing in two minutes.
20 crests in 2 minutes means 10 crests in 1 minute (60 seconds).
So, 10 crests in 60 seconds means 1 crest in 6 seconds.
Each cycle has one crest. Therefore, the period of the wave is 6 seconds.
Question 10: Pure Tone
A pure tone is always a sinusoidal sound wave.
It is not necessarily an impulsive wave, electronically generated, or at a very low frequency.
Question 11: Sound Transmission through Walls
Faintly hearing someone speaking behind a wall occurs because a small portion of the original sound wave is transmitted through the wall.
While 99 ext{%} of energy may be reflected, a small percentage (($1 ext{%}$)) can transmit through a hard surface.
Question 12: Transverse Waves
In a transverse wave, particles of the medium move perpendicular to the direction in which the wave travels.
Question 13: Compliance and Stiffness
Compliance is inversely proportional to stiffness.
Question 14: Inertia and Mass
The amount of inertia of an object is directly proportional to its mass (heavier objects have more inertia).
Question 15: Inertial Force in a Pendulum
Inertial force is the tendency of an object to continue its motion or rest.
In a pendulum, inertial force is maximum at point 'o' (the lowest point of the swing) because velocity is highest there.
Question 16: Force Equation
Force is equal to mass multiplied by acceleration (F = ma).
Question 17: Frictionless Surface and Inertia
If a hockey puck slides on a perfectly frictionless surface, it will not slow down due to its inertia. It will continue in motion indefinitely because there is no opposing force.
Question 18: Pressure Units
Pressure is defined as force per unit area.
The unit of force is Newton ( ext{N}), and the unit of area is meters squared ( ext{m}^2).
Therefore, the unit of pressure is Newton per meter squared ( ext{N/m}^2).
Question 19: Overtones and Harmonics Relationship
The n^{ ext{th}} harmonic is equal to the (n-1)^{ ext{th}} overtone.
For example, the third harmonic is the second overtone.
Course Learning Strategies
Commitment: Success in this challenging course (due to its physics content) requires commitment and time.
Clarification: If concepts are unclear, seek clarification immediately (within 24 hours).
Revision: Revise newly learned material within 24 hours to solidify understanding.
Pre-reading: Read 20 ext{%}-50 ext{%} of the material before class to gain an initial understanding and sustain attention, especially during longer lectures.
Relevance: All concepts discussed in this course are foundational and will be encountered again in graduate programs, especially in voice-related courses for speech therapists.
Active Learning: Ask questions freely without worrying about others' perceptions; take responsibility for your own learning.
Complex Periodic Sounds and Harmonics
A pure tone is a single sinusoidal wave, which appears as a smooth 'S' shape.
Complex periodic sounds are created by mixing a fundamental frequency with its harmonics (multiples of the fundamental frequency).
Adding harmonics changes the waveform, making it less smooth and more complex.
The number of smaller peaks in a complex periodic waveform often matches or is influenced by the highest harmonic added.
Time Period of Complex Waves: In a complex periodic sound, the time period of the entire complex wave always corresponds to the time period of its fundamental frequency.
Example: If a 910 ext{ Hz} fundamental is mixed with its 5^{ ext{th}} and 7^{ ext{th}} harmonics, the resulting complex sound still has a time period matching the 910 ext{ Hz} fundamental (approx. 1.1 ext{ ms}).
This principle explains why, despite speaking different vowels and consonants (which are complex sounds), our pitch (related to the fundamental frequency) remains stable in natural speech.
Visualizing Harmonics:
If a red wave is the fundamental frequency (1^{ ext{st}} harmonic), a blue wave completing two cycles in the same time frame is the 2^{ ext{nd}} harmonic, and a green wave completing three cycles is the 3^{ ext{rd}} harmonic.
Missing Fundamental / Virtual Pitch
Definition: The phenomenon where the brain perceives the fundamental frequency of a complex sound even when that specific fundamental frequency is not physically present in the sound itself.
Explanation: The presence of the harmonics (e.g., 400 ext{ Hz}, 600 ext{ Hz}, 800 ext{ Hz}) in a specific arrangement can evoke the sensation of a missing fundamental frequency (e.g., 200 ext{ Hz}).
Example (Old Telephones): Historically, telephone speakers were not designed to reproduce frequencies below 300 ext{ Hz}. However, male voices (with common fundamental frequencies around 125 ext{ Hz}) were still recognizable. This is because higher harmonics of the male voice (e.g., 250 ext{ Hz}, 375 ext{ Hz}, 500 ext{ Hz}) were transmitted, allowing the brain to infer the missing 125 ext{ Hz} fundamental.
This concept is also known as virtual pitch.
Longitudinal Waves
In longitudinal waves (like sound waves), the particles of the medium oscillate parallel to the direction of wave propagation.
Areas where particles are closer together experience high pressure (compression).
Areas where particles are farther apart experience low pressure (rarefaction).
Standing Waves in Air Columns
Relevance: The human ear canal and vocal tract are analogous to tubes, open at one end and closed at the other, making the study of standing waves in air columns crucial.
Formation: Standing waves in tubes result from sound waves traveling into the tube and being reflected back from its ends, then superimposing, similar to waves on a guitar string.
Boundary Conditions (Nodes and Antinodes):
Fixed/Closed End: Always forms a node (point of zero particle displacement, zero pressure variation for strings/ropes, or maximum pressure variation for sound waves).
Open End: For maximum resonance (loudest sound), an antinode (point of maximum particle displacement, maximum pressure variation for strings/ropes, or minimum pressure variation for sound waves) must form at the opening.
If a standing wave's antinode forms exactly at the opening, the particles at that opening move with maximum amplitude, leading to greater sound intensity or loudness.
Length of Tube (L): The physical length of the tube.
Wavelength ($ ext{lambda}$): The length of one complete cycle of the wave.
Velocity (V): The speed of sound in air (approximately 340 ext{ m/s} at normal temperature).
Formula: V = f ext{lambda}, where f is frequency.
This means if frequency increases, wavelength decreases, and vice versa, keeping the velocity of sound constant.
Types of Tubes and Harmonics Generated:
Tube Open at One End, Closed at Another (e.g., ear canal, vocal tract, empty bottle):
Boundary Conditions: Node at the closed end, Antinode at the open end.
Harmonics: Only odd harmonics are produced (1^{ ext{st}}, 3^{ ext{rd}}, 5^{ ext{th}}, ext{etc.}).
Fundamental Frequency Appearance: The most basic standing wave (fundamental frequency) will appear as one-quarter of a wavelength ( rac{1}{4} ext{lambda}) fitting within the tube's length.
Formulas:
For the n^{ ext{th}} harmonic (where n is odd): L = rac{n ext{lambda}}{4} ext{ or } ext{lambda}_n = rac{4L}{n}
Frequency: f_n = rac{nV}{4L}
Example Calculation: For a tube 0.5 ext{ m} long, open at one end and closed at the other:
Fundamental frequency (n=1): f_1 = rac{1 imes 340 ext{ m/s}}{4 imes 0.5 ext{ m}} = rac{340}{2} = 170 ext{ Hz}.
Third harmonic (n=3): f3 = 3 imes f1 = 3 imes 170 ext{ Hz} = 510 ext{ Hz}.
Tube Closed at Both Ends:
Boundary Conditions: Node at both ends.
Harmonics: Produces both odd and even harmonics (1^{ ext{st}}, 2^{ ext{nd}}, 3^{ ext{rd}}, ext{etc.}).
Fundamental Frequency Appearance: The most basic standing wave will appear as one-half of a wavelength ( rac{1}{2} ext{lambda}) fitting within the tube's length (node-antinode-node).
Formulas:
For the n^{ ext{th}} harmonic (where n is any integer): L = rac{n ext{lambda}}{2} ext{ or } ext{lambda}_n = rac{2L}{n}
Frequency: f_n = rac{nV}{2L}
Tube Open at Both Ends:
Boundary Conditions: Antinode at both ends.
Harmonics: Produces both odd and even harmonics (1^{ ext{st}}, 2^{ ext{nd}}, 3^{ ext{rd}}, ext{etc.}).
Fundamental Frequency Appearance: The most basic standing wave will appear as one-half of a wavelength ( rac{1}{2} ext{lambda}) fitting within the tube's length (antinode-node-antinode).
Formulas:
For the n^{ ext{th}} harmonic (where n is any integer): L = rac{n ext{lambda}}{2} ext{ or } ext{lambda}_n = rac{2L}{n}
Frequency: f_n = rac{nV}{2L}
Comparison (Open-Closed vs. Open-Open): A tube of the same length, when changed from open at one end/closed at the other to open at both ends, will have its fundamental frequency double.
Example: A 0.5 ext{ m} open-closed tube has a fundamental frequency of 170 ext{ Hz}. If it becomes open-open, its fundamental frequency is f_1 = rac{1 imes 340 ext{ m/s}}{2 imes 0.5 ext{ m}} = rac{340}{1} = 340 ext{ Hz}.
Relationship between Tube Length and Fundamental Frequency
Inverse Proportionality: Tube length and fundamental frequency are inversely proportional.
Longer tube length results in a lower fundamental frequency.
Shorter tube length results in a higher fundamental frequency.
Applications to Human Anatomy and Real-World Examples
Ear Canal:
Modeled as a tube open at one end (outer ear) and closed at the other (eardrum).
Average length: approximately 25 ext{ mm} (0.025 ext{ m}).
Resonant frequency calculation: f_1 = rac{V}{4L} = rac{340 ext{ m/s}}{4 imes 0.025 ext{ m}} = rac{340}{0.1} = 3400 ext{ Hz}.
This resonance amplifies sounds around 3400 ext{ Hz}, which is important for hearing.
Ear canal length varies slightly among individuals, leading to minor variations in its resonant frequency.
Vocal Tract:
Modeled as a tube open at one end (mouth) and closed at the other (vocal folds).
Consequently, the vocal tract primarily produces odd harmonics.
Male vs. Female Pitch: Males typically have lower vocal pitch due to:
Longer vocal tracts (longer tube length $
ightarrow$ lower fundamental frequency).Longer vocal folds.
Thicker vocal folds with more mass.
Voice Education: Singers and voice educators manipulate vocal tract length (e.g., shaping mouth/pharynx) to control pitch and produce desired notes (soprano, alto, baritone, etc.).
Tree Branches:
A tree branch can act like a tube open at one end (free end) and fixed at another (tree trunk).
The branch will make the loudest sound when the wind frequency matches its fundamental (resonant) frequency, determined by its length.
Drinking Bottles:
As water is consumed from a bottle, the air column length changes, thereby changing its resonant frequencies. Blowing across the opening may produce different sounds as the length changes.
Quiz 2 (Preview)
Quiz on week 3 and week 4 material (this week and last week).
Topics include logarithms, decibels, and sound measurement. Will learn about decibels from quizzes. The upcoming quiz is now scheduled for the sixth.