PHY131H1/Unit 10

Unit 10 · Introduction to Physics I

Waves & Sound

An oscillation that travels. Nothing material moves along with a wave — only the disturbance does, and with it the energy. The last unit of the course, and the one that sets up all of the optics in PHY132H1.

By the end you can

• Use $v = f\lambda$ and say what sets each quantity.
• Distinguish transverse from longitudinal waves.
• Apply superposition to get interference and standing waves.
• Compute beat frequencies and Doppler shifts.

Why it matters & where it comes from

Why it exists

Waves carry energy and information without carrying matter. That is how you hear, see, and receive every signal ever transmitted. The mathematics is identical whether the wave is on a string, in air, or electromagnetic.

In the real world

Every musical instrument is a standing wave in a string or air column. Noise-cancelling headphones use destructive interference. Ultrasound imaging, radar and sonar are all reflection and timing.

Story & open questions

Whether light was a wave or a stream of particles was argued for two centuries. Young's double-slit experiment of 1801 settled it for the wave side — until quantum mechanics showed the answer is both.

Describing a wave

Definition

Wavelength $\lambda$: distance between successive identical points.

Frequency $f$: cycles per second, in hertz.

Period $T = 1/f$.

Amplitude $A$: maximum displacement. Intensity goes as $A^2$.

Speed $v = f\lambda$.

Rule The medium sets the speed

This is the most misunderstood point in the unit. Wave speed is fixed by the medium, not by the source. The source sets the frequency. Wavelength then follows from $\lambda = v/f$.

On a string, $v = \sqrt{F/\mu}$ — tension over mass per unit length. Tightening a guitar string raises its speed and so its pitch.

In air, sound travels at about $343$ m/s at room temperature, faster when warmer.

When a wave crosses into a new medium its frequency stays the same and its wavelength changes. Frequency is set at the source and cannot change en route.

Definition

Transverse: the medium oscillates perpendicular to the travel direction. Waves on a string, light.

Longitudinal: the medium oscillates along the travel direction, making compressions and rarefactions. Sound.

Sound cannot be transverse in a gas, because gases have no resistance to shear. This is why seismic S-waves, which are transverse, do not pass through the Earth's liquid outer core — the observation that revealed the core is liquid.

Superposition and interference

Rule Displacements add

When two waves overlap, the total displacement at each point is the sum of the individual displacements. Afterwards both waves continue unchanged — they pass through each other.

Example Constructive and destructive

In phase (path difference a whole number of wavelengths): crests align, amplitudes add. Constructive.

Out of phase by half a wavelength: crest meets trough and they cancel. Destructive.

With equal amplitudes the cancellation is total. The energy is not destroyed — it is redistributed to the constructive regions.

Definition

Beats arise when two close frequencies overlap. The amplitude rises and falls at

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

Musicians tune by this: adjust until the beating slows and stops, meaning the frequencies match. A $440$ Hz and a $443$ Hz tone give $3$ beats per second.

Standing waves

Definition

A wave reflecting off a boundary interferes with itself. At the right frequencies the pattern stands still: nodes never move, antinodes oscillate maximally.

Rule Boundaries select the frequencies

For a string fixed at both ends, of length $L$:

$\lambda_n = \dfrac{2L}{n}$ and $f_n = \dfrac{nv}{2L}$ for $n = 1, 2, 3, \dots$

$f_1$ is the fundamental; the rest are harmonics. Only these discrete frequencies survive — everything else interferes with itself destructively and dies out.

For a pipe open at both ends the formula is the same. For a pipe closed at one end, only odd harmonics exist, and $f_n = nv/(4L)$ with $n$ odd — which is why a clarinet sounds different from a flute of the same length.

Example Guitar string

A $0.65$ m string carries waves at $400$ m/s.

$f_1 = 400/(2 \times 0.65) = 308$ Hz.

Harmonics at $616$, $924$ Hz and so on. The mix of harmonics is what makes a guitar sound like a guitar rather than a piano at the same pitch.

Pressing a fret shortens $L$ and raises every frequency. Tightening the string raises $v$ and does the same.

Standing waves transport no energy

The two travelling waves carry energy in opposite directions and the net flow is zero. Energy sloshes between nodes but does not move along the string.

The Doppler effect

Rule Relative motion shifts frequency

Motion between source and observer changes the observed frequency:

$f' = f\dfrac{v \pm v_o}{v \mp v_s}$

with the upper signs when the motion is toward. Rather than memorise the signs, remember the direction of the effect: approaching raises the pitch, receding lowers it, and choose signs to match.

Example The ambulance

A siren at $600$ Hz approaches at $30$ m/s, with sound at $343$ m/s.

Approaching: $f' = 600 \times 343/(343 - 30) = 658$ Hz — higher, as expected.

Receding: $f' = 600 \times 343/(343 + 30) = 552$ Hz.

The drop as it passes is $106$ Hz, and it happens at the moment of passing, not gradually.

Example Why it matters elsewhere

Light from distant galaxies is shifted toward the red, meaning they are receding — the observation behind the expanding universe.

Doppler radar measures the speed of vehicles and storms by the shift in reflected waves.

Medical ultrasound images blood flow by the shift off moving cells.

Common pitfalls

• Thinking the source sets the wave speed. The medium does.
• Expecting frequency to change when a wave enters a new medium. The wavelength changes instead.
• Thinking matter travels with the wave. Only the disturbance and its energy do.
• Using the open-pipe formula for a closed pipe. Closed pipes have only odd harmonics.
• Believing destructive interference destroys energy. It redistributes it.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Wavelength

Sound at $440$ Hz travels at $343$ m/s. Find the wavelength.

Answer

$\lambda = 343/440 = 0.78$ m.

Practice 2 · Crossing media

A sound wave passes from air into water, where it travels faster. What changes?

Answer

Speed and wavelength both increase; frequency is unchanged, being set at the source.

Practice 3 · Beats

Two tuning forks at $256$ and $260$ Hz sound together. What do you hear?

Answer

A tone near $258$ Hz pulsing $4$ times per second.

Practice 4 · Fundamental

A $0.5$ m string carries waves at $200$ m/s. Find the first two harmonics.

Answer

$f_1 = 200 = 200$ Hz and $f_2 = 400$ Hz.

Practice 5 · Closed pipe

Why does a pipe closed at one end lack even harmonics?

Solution

The closed end must be a node and the open end an antinode. Only quarter-wavelength odd multiples satisfy both boundary conditions, so $f_n = nv/(4L)$ with $n$ odd.

Practice 6 · Doppler

A train horn at $500$ Hz recedes at $25$ m/s. What do you hear?

Solution

$f' = 500 \times 343/(343 + 25) = 466$ Hz — lower, as receding requires.

Practice 7 · Tightening a string

What happens to the pitch if tension is quadrupled?

Solution

$v = \sqrt{F/\mu}$ doubles, so every $f_n = nv/(2L)$ doubles — the pitch rises by an octave.

Practice 8 · Where the energy goes

Two speakers cancel at a point. Is energy destroyed?

Answer

No. Energy is redistributed to the constructive regions elsewhere, and the total over all space is unchanged. Interference rearranges energy; it never removes it.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY131H1 topic map. It should cover the following items before you move on:

• Wave quantities. $v = f\lambda$, and which of the three is set by the medium and which by the source.
• Wave types. Transverse versus longitudinal, and why sound in a gas must be longitudinal.
• Superposition. Interference, beats, and standing waves.
• Boundaries and Doppler. Harmonic series for open and closed pipes, and the direction of the frequency shift.

Textbook/source alignment

The calendar closes its topic list with "waves, sound". Check the syllabus on Quercus for assigned problems; this material is the direct foundation for the optics half of PHY132H1.

Core idea map

A wave is Unit 9's oscillation set travelling. The medium fixes the speed, the source fixes the frequency, superposition explains everything that happens when waves meet, and boundaries select which frequencies can persist.

Study routine. For any wave question, write down which of $v$, $f$ and $\lambda$ is fixed by the medium and which by the source. The third follows, and confusing the two causes most errors here.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · What sets speed

Does a louder sound travel faster?

Guide

No. Amplitude does not affect speed; the medium alone determines it.

Extension 2 · Crossing a boundary

Which quantity is unchanged?

Guide

Frequency. It is fixed at the source, so speed and wavelength change together.

Extension 3 · Standing wave energy

Does a standing wave transport energy?

Guide

No. The two component travelling waves carry equal energy in opposite directions and the net flow is zero.

Extension 4 · Doppler direction

How do you avoid memorising the sign convention?

Guide

Recall that approaching raises pitch and receding lowers it, then pick the signs that produce the right direction of shift.

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