Waves & Interference

Waves & Interference

Part 1 of 2 — Periodic Motion & Wave Basics

Where this fits. The previous lesson, Fluids in Motion, covered continuity, Bernoulli, viscosity and blood flow. This lesson lays the foundation for waves and sound: periodic motion, simple harmonic motion, the basic properties of waves and $v = f\lambda$. Part 2 of this lesson shows what happens when waves overlap, and the next lesson, Sound, Doppler & Ultrasound, applies both to sound.

Describing Anything That Repeats

A heartbeat, a swinging pendulum, a vibrating vocal fold and a tuning fork all repeat the same motion over and over. Four quantities describe any such periodic motion:

• Period $T$: the time for one complete cycle, in seconds.
• Frequency $f$: the number of cycles per second, in hertz ($1\;\text{Hz} = 1$ cycle per second). Period and frequency are reciprocals: $f = 1/T$. Formulas often use the angular frequency $\omega = 2\pi f$ (in rad/s); passages may write the motion as $x = A\cos\omega t$.
• Amplitude $A$: the MAXIMUM displacement from the equilibrium (rest) position. It is measured from the centre line to a peak, not peak to peak; the peak-to-peak distance is $2A$.
• Phase: where in its cycle an oscillator is at a given instant. One full cycle is $360^\circ$, or $2\pi$ radians. Two oscillators are in phase when they reach their peaks together, and $180^\circ$ out of phase when one is at its peak while the other is at its trough. A quarter-cycle offset is $90^\circ$.

Example: a resting heart at 72 beats per minute has $f = 72/60 = 1.2\;\text{Hz}$ and $T = 1/1.2 \approx 0.83\;\text{s}$. Always convert "per minute" to "per second" before using $f = 1/T$.

Phase is the idea that makes Part 2 of this lesson work: two waves that meet in phase reinforce each other, and two that meet $180^\circ$ out of phase cancel.

Simple Harmonic Motion

Simple harmonic motion (SHM) happens whenever the restoring force is proportional to the displacement and points back toward equilibrium. For a spring this is Hooke's law, $F = -kx$, where $x$ is the displacement and $k$ is the spring constant (stiffness, in N/m). The position then traces a smooth sine curve in time.

The two systems the MCAT uses:

$T_{\text{spring}} = 2\pi\sqrt{\frac{m}{k}} \qquad T_{\text{pendulum}} = 2\pi\sqrt{\frac{L}{g}}$

Here $m$ is the oscillating mass, $L$ the pendulum's length (pivot to the centre of the bob) and $g$ the gravitational acceleration. The pendulum formula holds for small swings (roughly under $15^\circ$).

Two traps hide in that table. Amplitude never changes the period of SHM: a wider swing covers more distance but moves faster. And the pendulum's mass cancels out, because a heavier bob feels a proportionally larger pull of gravity.

Energy sloshes back and forth. At the turning points (maximum displacement) the object is momentarily at rest: all the energy is potential, and the restoring force and acceleration are at their largest. Passing through equilibrium, the force is zero, so acceleration is zero, and all the energy is kinetic, so speed is greatest. (The spring's stored-energy formula belongs to the Work, Energy, and Power lesson.)

Practice: Periodic Motion 🎯

From Oscillation to Wave

A wave is a disturbance that travels through a medium, carrying ENERGY from place to place without carrying the medium along. Each bit of rope or air simply oscillates about its own resting position, often in SHM, while the pattern moves on. A cork on a pond bobs up and down as ripples pass; it does not ride away with them.

Vocabulary for a wave:

• Wavelength $\lambda$: the distance over which the pattern repeats, such as crest to crest, or one compression to the next.
• Amplitude $A$: the maximum displacement of the medium from rest, as for any oscillator.
• Period and frequency: the time for one full wavelength to pass a fixed point, and the number passing per second. A wave's frequency is the frequency of each oscillating particle.

Transverse vs. Longitudinal

Sound in a gas or liquid is purely longitudinal, because a fluid has no restoring force against sideways (shear) displacement: nudge a bit of air sideways and nothing pulls it back. A fluid pushes back only when it is squeezed or stretched along the travel direction.

Solids resist shear, so they can carry both transverse and longitudinal waves.

• Earthquakes show this: the fast longitudinal P-waves pass through Earth's liquid outer core, while the slower transverse S-waves cannot.

The Wave Equation and Who Controls What

In one period, a wave advances exactly one wavelength, so:

$v = f\lambda$

Three quantities, but each is controlled by something different. This is the single most-tested wave idea.

• The medium sets the speed. On a string, $v = \sqrt{F_T/\mu}$, where $F_T$ is the tension and $\mu$ is the linear mass density (mass per length, in kg/m). A tighter string carries waves faster; a heavier string carries them slower. For sound, the speed depends on the medium's stiffness against compression (its bulk modulus $B$, the pressure needed to squeeze it by a given fraction of its volume) and its density: $v = \sqrt{B/\rho}$.
• The source sets the frequency. A tuning fork, a vocal fold or an ultrasound crystal vibrates at its own rate, and the wave inherits that rate.
• Wavelength is the result: $\lambda = v/f$.
• Energy grows as amplitude squared: triple the amplitude and the wave carries nine times the power.

Sound is fastest in solids, then liquids, then gases. Solids are denser, which on its own would slow sound, but they are enormously stiffer, and stiffness wins. Typical values: air about $343\;\text{m/s}$ at room temperature (often rounded to $340$), water about $1500\;\text{m/s}$, soft tissue about $1540\;\text{m/s}$, bone roughly $3000$ to $4000\;\text{m/s}$, steel about $5000\;\text{m/s}$ or more. In a gas, warmer means faster.

Frequency never changes at a boundary. When a wave crosses into a new medium, every crest that arrives at the boundary launches exactly one crest on the far side, so crests leave at the rate they arrive. Speed changes, so wavelength changes in proportion: sound passing from air into water speeds up about $4.4$ times, and its wavelength stretches $4.4$ times.

Changing the source in the same medium changes only the wavelength. Double a speaker's frequency and the sound still travels at $343\;\text{m/s}$; its wavelength halves. Thinking "higher pitch travels faster" is the classic error.

• Exact figure: the speed of sound in air rises by about $0.6\;\text{m/s}$ for each degree Celsius of warming.

Worked Example — Who Controls What, Step by Step

Problem 1 — One tone, three media. A $500\;\text{Hz}$ tone travels through air ($340\;\text{m/s}$), then water ($1500\;\text{m/s}$), then bone (take $4000\;\text{m/s}$).

• (a) What is the tone's frequency in each medium?
• (b) What is its wavelength in each medium?
• (c) From air to bone, which quantities change and which stays fixed?

(a) Frequency. The source sets it, and boundaries cannot change it: $500\;\text{Hz}$ in all three media.

(b) Wavelengths. With $f = 500\;\text{Hz}$ from (a), $\lambda = v/f$ in each:

(c) Sanity check. Speed and wavelength rise together (from $0.68$ to $8.0\;\text{m}$) while frequency holds still at $500\;\text{Hz}$. If your answer shows frequency changing at a boundary, start again.

Problem 2 — A string under tension. A string with $\mu = 0.010\;\text{kg/m}$ is held at tension $F_T = 100\;\text{N}$ and shaken by a vibrator at $250\;\text{Hz}$.

• (a) How fast do waves travel along the string?
• (b) What is their wavelength?
• (c) If the tension is quadrupled to $400\;\text{N}$, what are the new speed and wavelength?

(a) Speed from the medium. $v = \sqrt{F_T/\mu} = \sqrt{100/0.010} = \sqrt{10^4} = 100\;\text{m/s}$

(b) Wavelength. With $v = 100\;\text{m/s}$ from (a): $\lambda = v/f = 100/250 = 0.40\;\text{m}$

(c) Quadruple the tension. The speed scales as $\sqrt{F_T}$, so it doubles to $200\;\text{m/s}$. The vibrator still sets $250\;\text{Hz}$, so the wavelength doubles to $0.80\;\text{m}$. Notice the square root: four times the tension gives only twice the speed.

Problem 3 — A pendulum clock. A pendulum is $1.0\;\text{m}$ long ($g = 10\;\text{m/s}^2$).

• (a) What is its period?
• (b) What is the period with a bob twice as heavy, or with a slightly wider (still small) swing?
• (c) What is the period if the length is increased to $4.0\;\text{m}$?

(a) Period. $T = 2\pi\sqrt{L/g} = 2\pi\sqrt{0.10} \approx 2\pi(0.32) \approx 2.0\;\text{s}$, one second for each swing across.

(b) Mass and amplitude. Neither appears in $T = 2\pi\sqrt{L/g}$, so both leave the period at $2.0\;\text{s}$.

(c) Four times the length. $T \propto \sqrt{L}$ and $\sqrt{4} = 2$, so the $2.0\;\text{s}$ from (a) doubles to $4.0\;\text{s}$.

Practice: Waves, Media & the Wave Equation 🎯

Key Takeaways

• Periods in SHM. $f = 1/T$; $T = 2\pi\sqrt{m/k}$ for a spring, $2\pi\sqrt{L/g}$ for a pendulum; amplitude never changes $T$.
• Energy sloshes back and forth. Turning points: speed zero, acceleration maximum; equilibrium: speed maximum, acceleration zero.
• Who controls what. In $v = f\lambda$ the medium sets $v$, the source sets $f$, and $\lambda$ follows.
• Sound in different media. Sound in fluids is longitudinal, fastest in solids (stiffness beats density); air is about $343\;\text{m/s}$.

Common traps

• Leaving it per minute. Convert to hertz first: 72 beats per minute is $1.2\;\text{Hz}$, so $T \approx 0.83\;\text{s}$.
• Heavier bob, wider swing. Neither changes a pendulum's period; only the length and $g$ do.
• Frequency changing at a boundary. A new medium changes $v$ and $\lambda$ together; $f$ stays what the source set.
• Forgetting the square root. $v = \sqrt{F_T/\mu}$, so four times the tension only doubles the wave speed.

• Low-yield extras: P-waves vs S-waves in earthquakes; the exact temperature coefficient of the speed of sound in air.

Waves & Interference

Part 2 of 2 — Superposition, Interference & Standing Waves

Building on Part 1. Part 1 of this lesson covered oscillations, simple harmonic motion and the basics of waves. This part shows what happens when waves overlap: superposition, path difference, beats, standing waves and resonance.

Superposition: Overlapping Waves Simply Add

When two waves pass through the same point at the same time, the medium's displacement there is the SUM of the displacements each wave would cause alone. This is the principle of superposition. Displacements carry signs: a crest is positive, a trough negative. After the overlap, each wave continues on exactly as before, unchanged by the meeting.

• Constructive interference: the waves arrive in phase (crest on crest). The amplitudes add: $A_1 + A_2$.
• Destructive interference: the waves arrive $180^\circ$ out of phase (crest on trough). The amplitudes subtract: $|A_1 - A_2|$. Equal amplitudes cancel completely at that point.
• Any other phase difference gives an amplitude somewhere in between.

Energy is not destroyed where waves cancel. It is redistributed: the quiet spots are matched by loud spots elsewhere, where the same waves reinforce.

Path Difference Decides the Phase

Two speakers driven by the same signal start out in phase. A listener hears them in or out of phase depending on how much farther one sound has travelled. Call the path difference $\Delta d = |d_1 - d_2|$, where $d_1$ and $d_2$ are the listener's distances from the two speakers.

Always convert the path difference into WAVELENGTHS before deciding. A path difference of $0.5\;\text{m}$ is destructive if $\lambda = 1.0\;\text{m}$ (half a wavelength); if $\lambda = 0.5\;\text{m}$ it is one full wavelength, and the sound is loud.

Noise-cancelling headphones put this to work. A microphone samples the incoming noise, and the speaker plays a copy with the same frequency and amplitude, shifted by half a cycle. The two cancel at the ear. The same rules govern light's interference patterns, which belong to the Optics lesson.

Beats

Play two tones of slightly different frequencies, $f_1$ and $f_2$, together. They drift in and out of phase, so the loudness swells and fades in a regular throb. The number of throbs per second is the beat frequency:

$f_{\text{beat}} = |f_1 - f_2|$

The pitch you hear is the average of the two frequencies, and it pulses in loudness.

Worked Example — Tuning by Beats

Problem. A tuner sounds a $440\;\text{Hz}$ tuning fork together with a string and hears $3$ beats per second. Tightening a string raises its frequency.

• (a) What frequency could the string have?
• (b) She tightens the string slightly. What does it mean if the beats get SLOWER, and what if they get FASTER?
• (c) How does she know when the string is in tune?

(a) Two candidates. $f_{\text{beat}} = |f_1 - f_2| = 3\;\text{Hz}$, so the string is at either $437$ or $443\;\text{Hz}$; the beats alone cannot say which.

(b) Change it and listen. Tightening raises the string's frequency. If the beats slow down, the string is approaching $440$: it was flat ($437\;\text{Hz}$). If they speed up, it is moving away: it was sharp ($443\;\text{Hz}$).

(c) In tune. When the beats vanish, the frequencies match.

Practice: Superposition, Interference & Beats 🎯

Superposition and Standing Waves

Send a wave down a string fixed at the far end and it reflects back. The outgoing and returning waves, identical but travelling in opposite directions, superpose into a standing wave: a pattern that oscillates in place instead of travelling.

• Nodes are points that never move; the two waves always cancel there.
• Antinodes are points that oscillate with the greatest amplitude, midway between nodes.
• Adjacent nodes are $\lambda/2$ apart; a node and the next antinode are $\lambda/4$ apart.

Only certain wavelengths fit a given length, because the ends force a node or an antinode. Each allowed pattern is a harmonic. The lowest-frequency one is the fundamental, or first harmonic ($f_1$), and the $n$th harmonic has frequency $nf_1$.

• String fixed at both ends: both ends are nodes, so the string must hold a whole number of half-wavelengths: $L = n\lambda/2$.
• Pipe open at both ends: air at an open end moves freely, so both ends are displacement antinodes. Antinode to antinode also fits whole half-wavelengths: $L = n\lambda/2$.
• Pipe closed at one end: the closed end is a displacement node (air cannot move into a wall), and the open end is an antinode. Node to antinode fits an ODD number of quarter-wavelengths: $L = n\lambda/4$ with $n = 1, 3, 5, \ldots$

Four things to remember:

• The closed pipe has no even harmonics. Its resonances are $f_1, 3f_1, 5f_1, \ldots$, so consecutive resonances differ by $2f_1$. For strings and open pipes, consecutive resonances differ by $f_1$.
• Overtones are the harmonics above the fundamental. The first overtone is the 2nd harmonic for strings and open pipes, but the 3rd harmonic for a closed pipe.
• A closed pipe's fundamental is half that of an open pipe of the same length: its fundamental wavelength is $4L$ rather than $2L$.
• $v$ is the speed in the vibrating medium. For a string, it is the wave speed on the string, $\sqrt{F_T/\mu}$. The sound the string sends into the air has the SAME frequency but a different wavelength, because air carries it at $343\;\text{m/s}$ (Part 1 of this lesson: the source sets $f$, the medium sets $v$).

A passage may describe pipes by pressure instead of displacement. The two are swapped: an open end is a pressure node (it stays at room pressure), and a closed end is a pressure antinode. The harmonic formulas are the same.

Resonance, Natural Frequency & Damping

The harmonics of an object are its natural frequencies. Push an object periodically at one of them and each push arrives in phase with the motion, so the amplitude builds up dramatically. That build-up is resonance.

• A child on a swing goes higher when pushed once per swing, at the swing's natural frequency.
• A singer holding the natural frequency of a wine glass can make it vibrate hard enough to shatter.
• The ear canal is a pipe closed at one end (open to the air, closed by the eardrum). About $2.5\;\text{cm}$ long, it resonates near $v/4L = 340/0.10 = 3400\;\text{Hz}$, which helps explain why human hearing is most sensitive at roughly $2$ to $5\;\text{kHz}$.

Damping is the gradual loss of a vibration's energy to friction, viscosity or absorption, so its amplitude dies away. A heavily damped system resonates less sharply: its amplitude at resonance is smaller. Damping of sound travelling through a medium is called attenuation, covered in the Sound, Doppler & Ultrasound lesson.

Worked Example — Pipes, Strings and Tuning

Problem 1 — A closed organ pipe. A pipe $0.85\;\text{m}$ long is closed at one end ($v = 340\;\text{m/s}$).

• (a) What is its fundamental frequency?
• (b) What are the next two resonances above it?
• (c) What are the fundamental and the resonance series of an open pipe of the same length?

(a) Fundamental. A closed pipe holds a quarter-wavelength: $\lambda_1 = 4L = 3.4\;\text{m}$, so $f_1 = v/4L = 340/3.4 = 100\;\text{Hz}$.

(b) Next resonances. Even harmonics are missing, so the next is the THIRD: $f_3 = 300\;\text{Hz}$, then $f_5 = 500\;\text{Hz}$.

(c) Compare an open pipe. $f_1 = v/2L = 340/1.7 = 200\;\text{Hz}$, with the full series $200, 400, 600\;\text{Hz}$. Against the closed pipe's $100, 300, 500\;\text{Hz}$: capping one end halves the fundamental and deletes every even harmonic.

Problem 2 — A guitar string. A string $0.50\;\text{m}$ long carries waves at $200\;\text{m/s}$. Sound travels at $340\;\text{m/s}$ in the surrounding air.

• (a) What is the string's fundamental frequency?
• (b) What are the third harmonic's frequency and wavelength, and how many antinodes and nodes does it show?
• (c) When the fundamental reaches your ear, what are its frequency and wavelength in the air?

(a) Fundamental. $\lambda_1 = 2L = 1.0\;\text{m}$, so $f_1 = 200/1.0 = 200\;\text{Hz}$.

(b) Third harmonic. $f_3 = 600\;\text{Hz}$ and $\lambda_3 = 2L/3 \approx 0.33\;\text{m}$. The string shows three antinodes ("loops") and four nodes, counting both fixed ends.

(c) The sound in the air. The fundamental reaches your ear at $200\;\text{Hz}$, but its wavelength in air is $340/200 = 1.7\;\text{m}$, not the string's $1.0\;\text{m}$. Frequency carries over; wavelength does not.

Problem 3 — Identify a mystery tube. A tube's two consecutive resonances are $150\;\text{Hz}$ and $250\;\text{Hz}$.

• (a) What is the gap between consecutive resonances?
• (b) Could the tube be open at both ends?
• (c) Could it be closed at one end, and if so, what is its fundamental?

(a) The gap. Consecutive resonances differ by $250 - 150 = 100\;\text{Hz}$.

(b) Test an open tube. If open, the $100\;\text{Hz}$ gap from (a) equals $f_1$, so every resonance would be a multiple of $100$; but $150$ is not. It is not open at both ends.

(c) Test a closed tube. If closed, the gap is $2f_1$, so $f_1 = 50\;\text{Hz}$, and $150 = 3f_1$, $250 = 5f_1$. Both odd: the tube is closed at one end.

Practice: Standing Waves & Resonance 🎯

Key Takeaways

• Interference. Displacements add with signs; whole wavelengths of path difference are loud, odd half-wavelengths quiet.
• Beats. Two close tones throb at $f_{\text{beat}} = |f_1 - f_2|$; tighten or loosen and listen to tell which side you are on.
• Standing waves. Strings and open pipes: $f_n = nv/2L$, every $n$; a pipe closed at one end: $f_n = nv/4L$, odd $n$ only.
• Resonance. Driving at a natural frequency builds large amplitude; the ear canal, a closed pipe, resonates near $3.4\;\text{kHz}$.

Common traps

• Metres instead of wavelengths. Convert the path difference first: $0.5\;\text{m}$ is quiet if $\lambda = 1.0\;\text{m}$, loud if $\lambda = 0.5\;\text{m}$.
• Counting loops as nodes. The $n$th harmonic on a string has $n$ loops but $n + 1$ nodes, counting both fixed ends.
• Even harmonics in a closed pipe. A closed pipe has only odd harmonics, and its fundamental wavelength is $4L$, not $2L$.
• The string's wavelength in air. The sound keeps the string's frequency, but air carries it at its own speed, so $\lambda$ changes.

Next up: the Sound, Doppler & Ultrasound lesson, which covers sound intensity, decibels and hearing, then the Doppler effect, ultrasound and shock waves.