---
title: "Sound and Acoustics"
book: "High School Physics"
subject: physics
language: en
chapter: 21
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics
---

# Chapter 21 — Sound and Acoustics

Pluck a guitar’s low string: the room fills with a low A, yet nothing traveled to your ear but squeezed and stretched air. This chapter follows that pattern — its speed, how [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) becomes [pitch](#def-g12-sound-acoustics-pitch-loudness) and [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) [loudness](#def-g12-sound-acoustics-pitch-loudness), and why one note differs on every instrument.

## 21.1 Sound is a pressure wave

**Definition 21.1 (Sound wave).**

A *sound wave* is a mechanical [pressure](https://one-course.com/books/physics/2/en/chapter/7-pressure-from-sport-to-diving#def-g10-pressure-pressure) [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) ([Chapter 20](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#ch-g12-mechanical-waves)): a vibrating surface creates traveling *compressions* (slight overpressure) and *rarefactions* (underpressure). Air parcels oscillate *along* the travel direction ([sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) is *longitudinal*), and a medium is needed: a bell under a vacuum jar falls silent ([Chapter 8](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#ch-g10-signals-and-waves)).

![A loudspeaker drives a longitudinal wave: parcels bunch into compressions one wavelength = v/f apart.](https://one-course.com/images/onecourse/chapters/physics-2/g12-sound-acoustics/fig-38131703c916.svg)

*A loudspeaker drives a [longitudinal wave](#def-g12-sound-acoustics-sound-wave): parcels bunch into [compressions](#def-g12-sound-acoustics-sound-wave) one [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) $\lambda = v/f$ apart.*

**Proposition 21.2 (Speed of sound).**

[Sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) travels at about $340\,\mathrm{m}/\mathrm{s}$ in air at $20{}^{\circ}\mathrm{C}$, $1500\,\mathrm{m}/\mathrm{s}$ in water and $5000\,\mathrm{m}/\mathrm{s}$ in steel: the stiffer the medium, the faster.

**Proof.** *Admitted at this level.* ∎

**Remark 21.3 (Warm air is faster).**

These are measured values; predicting them is done in the Year 1 volume. [Sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) speeds up in warmer air ($331\,\mathrm{m}/\mathrm{s}$ at $0{}^{\circ}\mathrm{C}$) — why winds drift out of tune as the hall warms.

**Example 21.4 (The sizes of sounds).**

The relation $\lambda = v/f$ sizes any [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound): in air a $20\,\mathrm{Hz}$ rumble spans $340/20 = 17\,\mathrm{m}$ — a building — and a $20\,\mathrm{kHz}$ hiss fits in $1.7\,\mathrm{cm}$; the metre-sized ones bend easily around doors and corners.

## 21.2 Pitch, loudness and the ear

**Definition 21.5 (Pitch and loudness).**

*Pitch* is the [perception](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-perception) of [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency): the higher $f$, the higher the note; concert A is $440\,\mathrm{Hz}$, and doubling $f$ raises the note one octave. *Loudness* is the [perception](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-perception) of [amplitude](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-amplitude): the stronger the [pressure](https://one-course.com/books/physics/2/en/chapter/7-pressure-from-sport-to-diving#def-g10-pressure-pressure) oscillation, the louder; amplification changes loudness, not pitch.

**Remark 21.6 (The ear’s window).**

Hearing spans about $20\,\mathrm{Hz}$ to $20\,\mathrm{kHz}$ ([Chapter 8](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#ch-g10-signals-and-waves)); below lies [infrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound), above lies [ultrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound) — bats at $50\,\mathrm{kHz}$, medical probes at megahertz. A piano only samples the middle: $27.5\text{ to }4186\,\mathrm{Hz}$.

## 21.3 Sound intensity and the decibel scale

**Definition 21.7 (Sound intensity).**

The *sound intensity* $I$ is the acoustic [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) crossing one square [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) facing the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave): $I = P/S$, in $\mathrm{W}/\mathrm{m}^{2}$. The quietest audible [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) is about $I_0 = 1.0 \times 10^{-12}\,\mathrm{W}/\mathrm{m}^{2}$; pain begins near $1\,\mathrm{W}/\mathrm{m}^{2}$.

**Proposition 21.8 (Inverse-square attenuation).**

A small source radiating a [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) $P$ equally in all directions gives, at distance $r$, $I = P/(4\pi r^2) \propto 1/r^2$: doubling the distance divides the intensity by four.

**Proof.** The whole [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) $P$ crosses the sphere of area $4\pi r^2$; divide. ∎

**Definition 21.9 (Sound level).**

The ear spans twelve [powers](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) of ten, so intensities are quoted logarithmically. The *sound level* of $I$ is

$$
L = 10 \log_{10}(I/I_0), \qquad I_0 = 1.0 \times 10^{-12}\,\mathrm{W}/\mathrm{m}^{2},
$$

in *decibels* ($\mathrm{dB}$): hearing threshold $0\,\mathrm{dB}$, pain near $120\,\mathrm{dB}$.

**Proposition 21.10 (Decibel arithmetic).**

Multiplying $I$ by $k$ adds $10\log_{10}k$ [decibels](#def-g12-sound-acoustics-level). In particular $\times 10$ adds $10\,\mathrm{dB}$; $\times 2$ adds $10\log_{10}2 \approx 3.0\,\mathrm{dB}$; $N$ incoherent identical sources add *intensities*, never levels, giving $L_1 + 10\log_{10}N$; doubling the distance to a small source ($I$ divided by four) removes about $6.0\,\mathrm{dB}$.

**Proof.** $10\log_{10}(kI/I_0) = 10\log_{10}(I/I_0) + 10\log_{10}k$; distance: [Proposition 21.8](#prop-g12-sound-acoustics-inverse-square), $k = 1/4$. ∎

**Method 21.11 (Working in decibels).**

1. $L = 10\log_{10}(I/I_0)$ ; back: $I = I_0 \times 10^{L/10}$ .
2. Never add levels: convert to intensities, add, convert back — or $+3\,\mathrm{dB}$ per doubling, $+10\,\mathrm{dB}$ per tenfold.
3. Each doubling of distance to a small source costs $6\,\mathrm{dB}$ .

![A ladder of everyday levels: each 20\, dB step is a hundredfold intensity jump; beyond 85\, dB (red zone) exposure damages hearing.](https://one-course.com/images/onecourse/chapters/physics-2/g12-sound-acoustics/fig-5915489640a7.svg)

*A ladder of everyday levels: each $20\,\mathrm{dB}$ step is a hundredfold intensity jump; beyond $85\,\mathrm{dB}$ (red zone) exposure damages hearing.*

**Remark 21.12 (Hearing damage).**

Hearing loss is a [dose](https://one-course.com/books/physics/2/en/chapter/19-the-nucleus-and-radioactivity#rem-g11-nucleus-radioactivity-dose): $85\,\mathrm{dB}$ is safe for eight hours, and every $3\,\mathrm{dB}$ more — a doubling of intensity — halves the safe time; lost hair cells do not grow back.

## 21.4 Timbre and harmonics

**Definition 21.13 (Harmonics).**

A [periodic](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) of [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) $f_1$ — the *fundamental* — is in general a sum of sines at the frequencies $f_n = n f_1$, $n = 1, 2, 3, \dots$, its *harmonics*. A [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) reduced to its fundamental is a *pure tone*, like a tuning fork’s.

**Definition 21.14 (Spectrum and timbre).**

The *spectrum* of a [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) is the bar chart of its [harmonics](#def-g12-sound-acoustics-harmonics): one bar per $f_n$, the height that [harmonic](#def-g12-sound-acoustics-harmonics)’s strength. The ear hears $f_1$ as the [pitch](#def-g12-sound-acoustics-pitch-loudness) and the *mix* of strengths as the *timbre* — the [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound)’s color: instruments playing the same note share $f_1$, differing only in their spectra.

![The same A (220\, Hz): waveforms (top), spectra (bottom) — same fundamental and pitch; the violin’s harmonic ladder is its timbre.](https://one-course.com/images/onecourse/chapters/physics-2/g12-sound-acoustics/fig-a1a23988157f.svg)

*The same A ($220\,\mathrm{Hz}$): waveforms (top), spectra (bottom) — same [fundamental](#def-g12-sound-acoustics-harmonics) and [pitch](#def-g12-sound-acoustics-pitch-loudness); the violin’s [harmonic](#def-g12-sound-acoustics-harmonics) ladder is its [timbre](#def-g12-sound-acoustics-timbre).*

**Example 21.15 (Why you recognize the caller).**

A flute’s A is nearly a [pure tone](#def-g12-sound-acoustics-harmonics); a violin bowing the same $220\,\mathrm{Hz}$ adds strong [harmonics](#def-g12-sound-acoustics-harmonics) at $440\,\mathrm{Hz}\text{, }660\,\mathrm{Hz}\text{ and }880\,\mathrm{Hz}$. Same [pitch](#def-g12-sound-acoustics-pitch-loudness) and [loudness](#def-g12-sound-acoustics-pitch-loudness), yet no one confuses them: a voice, too, is a [spectrum](#def-g12-sound-acoustics-timbre) you know by heart.

## 21.5 The vibrating string

**Definition 21.16 (Standing wave).**

A [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) confined between the fixed ends of a string can settle into a *standing wave*: an oscillation in place, motionless points — *nodes* — alternating with points of maximal swing — *antinodes*. Nothing travels any more; the string vibrates in a fixed pattern.

**Proposition 21.17 (Modes of a string).**

A string of length $L$, fixed at both ends, carrying [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) of speed $v$, vibrates steadily only in patterns fitting a whole number $n$ of [half-wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) between the imposed end [nodes](#def-g12-sound-acoustics-standing-wave):

$$
\lambda_n = \frac{2L}{n}, \quad f_n = n\,\frac{v}{2L} = n f_1, \quad n = 1, 2, 3, \dots
$$

The lowest mode $f_1 = v/(2L)$ is the [fundamental](#def-g12-sound-acoustics-harmonics); the others are exactly its [harmonics](#def-g12-sound-acoustics-harmonics).

**Proof.** *Admitted at this level.* ∎

**Remark 21.18 (Where the honest proof lives).**

That confined [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) pick exactly these modes follows from superposing the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) with its reflections, a calculus carried out in the Year 1 volume. The geometry convinces: both ends must stay still, so a whole number of [half-wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) must fit.

![The first three modes of a string fixed at both ends: n half-wavelengths fit in L, so _n = 2L/n and f_n = nf_1.](https://one-course.com/images/onecourse/chapters/physics-2/g12-sound-acoustics/fig-a9c0638e47d7.svg)

*The first three modes of a string fixed at both ends: $n$ [half-wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) fit in $L$, so $\lambda_n = 2L/n$ and $f_n = nf_1$.*

**Example 21.19 (A violin’s A string).**

A violin’s A string has $L = 32.8\,\mathrm{cm}$ and [sounds](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) $f_1 = 440\,\mathrm{Hz}$: [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) run along it at $v = 2Lf_1 = 2 \times 0.328 \times 440 \approx 289\,\mathrm{m}/\mathrm{s}$ — set by [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) and mass, tuned by the peg. Pressing a finger shortens $L$ and raises every $f_n$ at once: the [spectrum](#def-g12-sound-acoustics-timbre) slides up.

**Remark 21.20 (Why instruments differ).**

How the string is excited — plucked, bowed, struck — sets how much of each mode is present, and the body amplifies some [harmonics](#def-g12-sound-acoustics-harmonics) more than others; excitation plus body fix the [spectrum](#def-g12-sound-acoustics-timbre), so the same $f_1$ leaves a guitar, a violin and a piano each sounding like itself.

## 21.6 Exercises

**Exercise 21.1 ★.**

In air ($v = 340\,\mathrm{m}/\mathrm{s}$), find the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) of (a) a $55\,\mathrm{Hz}$ bass note; (b) a $1.0\,\mathrm{kHz}$ beep; (c) a $15\,\mathrm{kHz}$ whine. Compare the largest to the smallest.

**Solution of Exercise 21.1.**

$\lambda = v/f$: (a) $340/55 \approx 6.2\,\mathrm{m}$; (b) $0.34\,\mathrm{m}$; (c) $2.3\,\mathrm{cm}$. Ratio $15000/55 \approx 270$.

**Exercise 21.2 ★.**

Classify as [infrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound), audible [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) or [ultrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound): $8\,\mathrm{Hz}$; $100\,\mathrm{Hz}$; $15\,\mathrm{kHz}$; $40\,\mathrm{kHz}$; $3\,\mathrm{MHz}$. Of the audible ones, which [sounds](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) lower in [pitch](#def-g12-sound-acoustics-pitch-loudness)?

**Solution of Exercise 21.2.**

[Infrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound): $8\,\mathrm{Hz}$. Audible: $100\,\mathrm{Hz}$ and $15\,\mathrm{kHz}$. [Ultrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound): $40\,\mathrm{kHz}$ and $3\,\mathrm{MHz}$. Lower [pitch](#def-g12-sound-acoustics-pitch-loudness): $100\,\mathrm{Hz}$ (lower [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency)).

**Exercise 21.3 ★.**

A worker strikes a rail $1.0\,\mathrm{km}$ away. Compute the travel times in the steel and in the air. What does a listener with one ear pressed to the rail hear?

**Solution of Exercise 21.3.**

Steel: $1000/5000 = 0.20\,\mathrm{s}$; air: $1000/340 \approx
2.9\,\mathrm{s}$. Two clicks, $2.7\,\mathrm{s}$ apart — the rail first.

**Exercise 21.4 ★.**

(a) The [sound level](#def-g12-sound-acoustics-level) for $I = 1.0 \times 10^{-5}\,\mathrm{W}/\mathrm{m}^{2}$? (b) The intensity at $60\,\mathrm{dB}$? (c) Name an everyday [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) near each level.

**Solution of Exercise 21.4.**

(a) $L = 10\log_{10}10^{7} = 70\,\mathrm{dB}$. (b) $I = 10^{-12} \times
10^{6} = 1.0 \times 10^{-6}\,\mathrm{W}/\mathrm{m}^{2}$. (c) $70\,\mathrm{dB}$: a vacuum cleaner; $60\,\mathrm{dB}$: a conversation.

**Exercise 21.5 ★.**

A violin G string [sounds](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) $f_1 = 196\,\mathrm{Hz}$: give its next three [harmonics](#def-g12-sound-acoustics-harmonics). Why does a $12\,\mathrm{kHz}$ whistle, however rich at the source, reach the ear as a [pure tone](#def-g12-sound-acoustics-harmonics)?

**Solution of Exercise 21.5.**

$392\,\mathrm{Hz}\text{, }588\,\mathrm{Hz}\text{ and }784\,\mathrm{Hz}$. Its higher [harmonics](#def-g12-sound-acoustics-harmonics) ($n \ge 2$) start at $24\,\mathrm{kHz}$, above the audible ceiling: only the $12\,\mathrm{kHz}$ [fundamental](#def-g12-sound-acoustics-harmonics) is heard.

**Exercise 21.6 ★★.**

A siren radiates $P = 0.50\,\mathrm{W}$ equally in all directions. Compute the intensity and the [sound level](#def-g12-sound-acoustics-level) $5.0\,\mathrm{m}$ away.

**Solution of Exercise 21.6.**

$I = 0.50/(4\pi \times 25) \approx 1.6 \times 10^{-3}\,\mathrm{W}/\mathrm{m}^{2}$; $L = 10\log_{10}(1.6 \times 10^{9}) \approx 92\,\mathrm{dB}$.

**Exercise 21.7 ★★.**

One violin gives $68\,\mathrm{dB}$ at your seat. What level do two give? Four? How many violins would it take to reach $78\,\mathrm{dB}$?

**Solution of Exercise 21.7.**

Two: $+3\,\mathrm{dB} \to 71\,\mathrm{dB}$; four: $74\,\mathrm{dB}$. $+10\,\mathrm{dB}$ needs $\times 10$ in intensity: $10$ violins.

**Exercise 21.8 ★★.**

The level is $95\,\mathrm{dB}$ at $2.0\,\mathrm{m}$ from a small loudspeaker. Predict the level at $8.0\,\mathrm{m}$: below the $85\,\mathrm{dB}$ mark?

**Solution of Exercise 21.8.**

$r \times 4$: $I \div 16$, so $-12\,\mathrm{dB}$: $83\,\mathrm{dB}$. Yes — just below the $85\,\mathrm{dB}$ mark.

**Exercise 21.9 ★★.**

A $440\,\mathrm{Hz}$ note passes from air into water. Compute its [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) in each medium. The diver hears the same [pitch](#def-g12-sound-acoustics-pitch-loudness): which quantity is fixed by the source, and which one adjusts?

**Solution of Exercise 21.9.**

Air: $340/440 \approx 0.77\,\mathrm{m}$; water: $1500/440 \approx
3.4\,\mathrm{m}$. The source fixes the [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency); the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) $\lambda = v/f$ adjusts to the medium.

**Exercise 21.10 ★★.**

A [spectrum](#def-g12-sound-acoustics-timbre) shows peaks at $130\,\mathrm{Hz}\text{, }260\,\mathrm{Hz}\text{, }390\,\mathrm{Hz}\text{ and }520\,\mathrm{Hz}$ of decreasing heights: what is the [fundamental](#def-g12-sound-acoustics-harmonics)? Another instrument plays the same note at the same [loudness](#def-g12-sound-acoustics-pitch-loudness): what is the same in its [spectrum](#def-g12-sound-acoustics-timbre), what differs?

**Solution of Exercise 21.10.**

$f_1 = 130\,\mathrm{Hz}$ (the peak spacing). Same: the bar positions $nf_1$ (same [pitch](#def-g12-sound-acoustics-pitch-loudness)) and the total intensity; different: the bar heights — the mix that makes the [timbre](#def-g12-sound-acoustics-timbre).

**Exercise 21.11 ★★.**

A guitar string of length $65.0\,\mathrm{cm}$ [sounds](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) $f_1 = 110\,\mathrm{Hz}$. Find the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) speed, then the vibrating lengths giving the octave ($220\,\mathrm{Hz}$) and the fifth ($165\,\mathrm{Hz}$).

**Solution of Exercise 21.11.**

$v = 2Lf_1 = 2 \times 0.650 \times 110 = 143\,\mathrm{m}/\mathrm{s}$. Octave: $L' = v/(2 \times 220) = L/2 = 32.5\,\mathrm{cm}$; fifth: $L' = 143/330 \approx 43.3\,\mathrm{cm} = 2L/3$.

**Exercise 21.12 ★★★.**

A medical probe sends $5.0\,\mathrm{MHz}$ [ultrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound) into soft tissue ($v = 1540\,\mathrm{m}/\mathrm{s}$): the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength)? [Echoes](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) return after $26\,\text{µ}\mathrm{s}$ and $58\,\text{µ}\mathrm{s}$ from an organ’s two walls: find their depths and the organ’s thickness.

**Solution of Exercise 21.12.**

$\lambda = 1540/5.0 \times 10^{6} \approx 0.31\,\mathrm{mm}$. Depth $= vt/2$: $1540 \times 26 \times 10^{-6}/2 \approx 2.0\,\mathrm{cm}$ and $1540 \times 58 \times 10^{-6}/2 \approx 4.5\,\mathrm{cm}$; thickness $\approx 2.5\,\mathrm{cm}$.

**Exercise 21.13 ★★★.**

An outdoor stage produces $110\,\mathrm{dB}$ at $3.0\,\mathrm{m}$. Treating it as a small source, at what distance does the level fall to $85\,\mathrm{dB}$? Give two reasons the real distance differs.

**Solution of Exercise 21.13.**

$-25\,\mathrm{dB}$ means $I \div 10^{2.5} \approx 316$, so $r \times \sqrt{316} \approx 17.8$: $r \approx 3.0 \times 17.8
\approx 53\,\mathrm{m}$. Real halls differ: reflections from ground and walls add intensity, while air and the crowd absorb it (and the stage is not an isotropic point source).

**Exercise 21.14 ★★★.**

Find the ratio of the pain threshold ($1\,\mathrm{W}/\mathrm{m}^{2}$) to $I_0$. Taking the eardrum as $0.5\,\mathrm{cm}^{2}$, compute the [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) received at each threshold; comment on the ear as a detector.

**Solution of Exercise 21.14.**

$1/10^{-12} = 10^{12}$. Eardrum $S = 5 \times 10^{-5}\,\mathrm{m}^{2}$: received [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) $5 \times 10^{-17}\,\mathrm{W}$ at threshold, $5 \times 10^{-5}\,\mathrm{W}$ at pain — the ear detects tens of attowatts and still works at a trillion times more: a detector with a twelve-decade dynamic [range](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-strong).

**Exercise 21.15 ★★★.**

Each identical festival loudspeaker alone gives $80\,\mathrm{dB}$ at the mixing desk. What level do $2$ give? $10$? $100$? What does each extra $10\,\mathrm{dB}$ cost — what does this say of [loudness](#def-g12-sound-acoustics-pitch-loudness) races?

**Solution of Exercise 21.15.**

$2$: $83\,\mathrm{dB}$; $10$: $90\,\mathrm{dB}$; $100$: $100\,\mathrm{dB}$. Each extra $10\,\mathrm{dB}$ costs a tenfold in loudspeakers: [loudness](#def-g12-sound-acoustics-pitch-loudness) races are exponentially expensive (and dangerous long before they are won).

## 21.7 Problem: The Luthier’s Workshop

**Problem 21.1.**

Weekend problem — the luthier’s workshop: a guitar string laid out fret by fret, its harmonics tuned into a timbre, a recital kept below the danger line, and an ultrasound hunt for a flaw in the wood

A luthier is finishing a guitar. Its A string has vibrating length $L = 65.0\,\mathrm{cm}$ and must [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) $f_1 = 110\,\mathrm{Hz}$. Take $340\,\mathrm{m}/\mathrm{s}$ for [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) in air, and admit $f_1 = v/(2L)$ ([Proposition 21.17](#prop-g12-sound-acoustics-string-modes)), $v$ being the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) speed on the string.

**Part I — Laying out the fretboard.**

1. Compute the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) speed $v$ on the string.
2. Compute the [fundamental](#def-g12-sound-acoustics-harmonics) ’s [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) *on the string* , then the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) of the $110\,\mathrm{Hz}$ [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) *in the air* . Why do they differ at the same [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) ?
3. The octave ( $220\,\mathrm{Hz}$ ): what vibrating length gives it? Where must the twelfth fret sit, as a fraction of $L$ ?
4. The fifth ( $165\,\mathrm{Hz}$ ): what vibrating length gives it?
5. Each fret raises the note one semitone, a factor $2^{1/12}$ . Check with logarithms how many semitones separate $110\,\mathrm{Hz}$ from $220\,\mathrm{Hz}$ ; length at the first fret?

**Part II — [Harmonics](#def-g12-sound-acoustics-harmonics) and [timbre](#def-g12-sound-acoustics-timbre).**

6. List the first five [harmonics](#def-g12-sound-acoustics-harmonics) of the open A string.
7. A violinist’s A [sounds](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) $440\,\mathrm{Hz}$ . Which [harmonic](#def-g12-sound-acoustics-harmonics) of the guitar string is that, and what musical interval separates it from the [fundamental](#def-g12-sound-acoustics-harmonics) ? And $330\,\mathrm{Hz}$ ?
8. The luthier plucks near the bridge, then over the fingerboard: same [pitch](#def-g12-sound-acoustics-pitch-loudness) , different color. Explain with spectra.
9. A light touch at the midpoint while plucking [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) a [node](#def-g12-sound-acoustics-standing-wave) there. Which [harmonics](#def-g12-sound-acoustics-harmonics) survive, and what [pitch](#def-g12-sound-acoustics-pitch-loudness) is heard?
10. How many [harmonics](#def-g12-sound-acoustics-harmonics) of the open string lie, in principle, within the audible [range](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-strong) ?

**Part III — The recital.** At the guitar’s first concert, each instrument radiates an acoustic [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) $P = 6.0 \times 10^{-3}\,\mathrm{W}$ as a small source, equally in all directions.

11. Compute the intensity at $r = 3.0\,\mathrm{m}$ from one instrument.
12. Deduce the [sound level](#def-g12-sound-acoustics-level) there.
13. A quartet plays: what level do four such instruments give at the same spot? Why is it not four times the level?
14. Twelve instruments play: what is the level at $3.0\,\mathrm{m}$ ?
15. How far back must a listener sit for the ensemble to fall to $85\,\mathrm{dB}$ ?
16. Safety rule: $85\,\mathrm{dB}$ is safe for eight hours, and every $3\,\mathrm{dB}$ halves the safe time. How long may the front-row listener of question 14 safely stay?

**Part IV — The flaw in the wood.** Before varnishing, the luthier probes a $45\,\mathrm{mm}$ thick neck blank with a $2.0\,\mathrm{MHz}$ tester; in this wood take $v = 5.0 \times 10^{3}\,\mathrm{m}/\mathrm{s}$.

17. Compute the [ultrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound) [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) in the wood. Why must a flaw-hunting [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) be ultrasonic rather than audible?
18. Compute the [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) [delay](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-delay) from the back face of the blank.
19. An [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) arrives after $7.2\,\text{µ}\mathrm{s}$ : how deep is the flaw?
20. Write the workshop card: [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) speed on the string, twelfth- and first-fret lengths, safe listening distance, flaw depth.

**Solution of Problem 21.1.**

**1.** $v = 2Lf_1 = 2 \times 0.650 \times 110 = 143\,\mathrm{m}/\mathrm{s}$.

**2.** On the string $\lambda_1 = 2L = 1.30\,\mathrm{m}$; in air $\lambda = 340/110 \approx 3.1\,\mathrm{m}$. Same $f$, but $\lambda = v/f$ and the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) speeds differ.

**3.** $L' = v/(2 \times 220) = L/2 = 32.5\,\mathrm{cm}$: the twelfth fret sits at the midpoint, $L/2$ from the bridge.

**4.** $L' = 143/(2 \times 165) \approx 43.3\,\mathrm{cm} = 2L/3$.

**5.** $\log(220/110)/\log 2^{1/12} = 12$ semitones — twelve frets to the octave. First fret: $L_1 = L/2^{1/12} = 65.0/1.059
\approx 61.4\,\mathrm{cm}$.

**6.** $110\,\mathrm{Hz}\text{, }220\,\mathrm{Hz}\text{, }330\,\mathrm{Hz}\text{, }440\,\mathrm{Hz}\text{ and }550\,\mathrm{Hz}$.

**7.** $440\,\mathrm{Hz} = 4f_1$: the fourth [harmonic](#def-g12-sound-acoustics-harmonics), two octaves up. $330\,\mathrm{Hz} = 3f_1$: the third [harmonic](#def-g12-sound-acoustics-harmonics), an octave plus a fifth.

**8.** The pluck position sets the mode mix: near the bridge the high [harmonics](#def-g12-sound-acoustics-harmonics) are strong (bright [spectrum](#def-g12-sound-acoustics-timbre)), over the fingerboard the [fundamental](#def-g12-sound-acoustics-harmonics) dominates (mellow). Same $f_1$, so same [pitch](#def-g12-sound-acoustics-pitch-loudness) — different [spectrum](#def-g12-sound-acoustics-timbre), different [timbre](#def-g12-sound-acoustics-timbre).

**9.** Only [harmonics](#def-g12-sound-acoustics-harmonics) with a [node](#def-g12-sound-acoustics-standing-wave) at the midpoint survive: the even ones, $220\,\mathrm{Hz}\text{, }440\,\mathrm{Hz}\text{ and }660\,\mathrm{Hz}$, …The [pitch](#def-g12-sound-acoustics-pitch-loudness) jumps an octave, to $220\,\mathrm{Hz}$.

**10.** $nf_1 \le 20\,\mathrm{kHz}$: $n \le 20000/110 = 181.8$, so $181$ [harmonics](#def-g12-sound-acoustics-harmonics).

**11.** $I = P/(4\pi r^2) = 6.0 \times 10^{-3}/(4\pi \times 9.0)
\approx 5.3 \times 10^{-5}\,\mathrm{W}/\mathrm{m}^{2}$.

**12.** $L = 10\log_{10}(5.3 \times 10^{7}) \approx 77\,\mathrm{dB}$.

**13.** $I \times 4$: $+6\,\mathrm{dB} \to 83\,\mathrm{dB}$. Levels are logarithms: intensities add, levels do not.

**14.** $+10\log_{10}12 \approx +10.8\,\mathrm{dB} \to
88\,\mathrm{dB}$.

**15.** Target $I = I_0 \times 10^{8.5} \approx
3.2 \times 10^{-4}\,\mathrm{W}/\mathrm{m}^{2}$ with $12P = 7.2 \times 10^{-2}\,\mathrm{W}$: $r = \sqrt{12P/(4\pi I)} \approx \sqrt{18.1} \approx 4.3\,\mathrm{m}$.

**16.** $88\,\mathrm{dB}$ is $+3\,\mathrm{dB}$ over the eight-hour limit: the safe time halves once — four hours. The recital fits.

**17.** $\lambda = 5000/2.0 \times 10^{6} = 2.5\,\mathrm{mm}$. A [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) only reflects off flaws at least about a [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) across: audible [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) in wood has $\lambda$ of tens of centimetres and diffracts around any small defect.

**18.** $t = 2d/v = 2 \times 0.045/5000 = 18\,\text{µ}\mathrm{s}$.

**19.** $d = vt/2 = 5000 \times 7.2 \times 10^{-6}/2 = 18\,\mathrm{mm}$.

**20.** The card: $v = 143\,\mathrm{m}/\mathrm{s}$ on the string; frets at $32.5\,\mathrm{cm}$ (twelfth) and $61.4\,\mathrm{cm}$ (first); ensemble safe beyond $\approx 4.3\,\mathrm{m}$; flaw $18\,\mathrm{mm}$ deep — four numbers, one chapter.