Back to the on-screen lesson ·
Sound as a pressure wave, intensity and decibels, resonance in open and closed pipes, beats, and the Doppler effect.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to compute sound levels, resonant frequencies and Doppler shifts.
From the last lesson you know that waves obey $v = f\lambda$, that longitudinal waves compress and stretch their medium, and that standing waves fit a whole number of half-wavelengths between fixed ends. This lesson applies those ideas to sound: what we hear as pitch, loudness and tone.
| Term | What it means |
|---|---|
| Sound | A longitudinal pressure wave, about $343$ m/s in air at $20$ °C. |
| Pitch | How high a sound seems, set by its frequency. |
| Intensity | Power per area, $I = P/(4\pi r^2)$ from a point source, in W/m². |
| Decibel level | $\beta = 10\log(I/I_0)$, with $I_0 = 10^{-12}$ W/m². |
| Resonance | Large vibrations when a driving frequency matches a natural one. |
| Beats | A throbbing loudness at the difference of two close frequencies. |
| Doppler effect | A change in heard frequency when source or listener moves. |
A vibrating object pushes and pulls on the air, sending out compressions and rarefactions: a longitudinal wave traveling at about $343$ m/s at room temperature. Its frequency is what we hear as pitch, from about $20$ Hz to $20$ kHz. Its intensity, the power per area, falls as $1/r^2$ from a small source. Because the ear responds to an enormous range, intensity is measured on the decibel scale:
$$\beta = 10\log\frac{I}{I_0}, \qquad I_0 = 10^{-12}\ \text{W/m}^2.$$
Air columns resonate at frequencies set by their length, and a moving source or listener hears a shifted frequency, the Doppler effect.
Another way: picture
Picture a crowd doing the wave, but front to back: each person bumps the one in front and springs back. The bump travels forward as a pulse of crowding. Sound in air is exactly that, with molecules jostling their neighbors, and the ear detects the tiny rise and fall of pressure as the crowded and thinned regions pass.
Another way: steps
Sound travels faster in warmer air, about $331$ m/s at $0$ °C plus $0.6$ m/s for each degree: $343$ m/s at room temperature. It travels about four times faster in water and fifteen times faster in steel, since stiffer media pass on pushes more quickly.
Count the seconds between a lightning flash and its thunder: light arrives almost instantly, sound takes about three seconds per kilometer, or five seconds per mile. The National Weather Service's advice, when thunder roars go indoors, reflects that any storm close enough to hear is close enough to strike.
The ear can hear sounds whose intensities range over a factor of a trillion, from $10^{-12}$ W/m² at the threshold of hearing to about $1$ W/m² at the threshold of pain. A logarithmic scale tames that range: each factor of ten in intensity adds $10$ dB. Whispering is about $30$ dB, conversation $60$, a lawn mower $90$, a rock concert $110$.
Because the scale is logarithmic, decibels do not simply add. Two $60$ dB sources together give $63$ dB, not $120$: doubling the intensity adds $10\log 2 \approx 3$ dB. The ear judges a $10$ dB rise as roughly twice as loud.
Blow across a bottle or play a flute and the air inside resonates. Sound waves reflect from the ends of the column and form standing waves. An open end is a pressure node, where the air moves freely; a closed end is where the air cannot move.
A pipe open at both ends fits a whole number of half-wavelengths: $f_n = nv/(2L)$ for every $n$, like a flute. A pipe closed at one end fits an odd number of quarter-wavelengths: $f_n = nv/(4L)$ for odd $n$ only, like a clarinet, which is why it sounds an octave lower than a flute of the same length and has a distinctive hollow tone.
When a siren approaches, each crest is emitted a little closer to you than the last, so the crests arrive bunched together: you hear a higher frequency. As it recedes, the crests are stretched out and the pitch drops. For a source moving at $v_s$, $f' = f\,v/(v - v_s)$ approaching and $f\,v/(v + v_s)$ receding.
The effect works for light too. Police radar and weather radar measure speeds from the Doppler shift of reflected radio waves, and astronomers measure the speeds of stars and galaxies from the shift of their spectral lines, which is how Edwin Hubble found the universe expanding.
Checking an answer. Doubling intensity must add about $3$ dB. An approaching source must sound higher. A closed pipe's second resonance must be three times its first.
The inverse-square law holds for a small source radiating evenly into open air, since the same power spreads over a sphere of area $4\pi r^2$. Indoors, reflections add to the direct sound, and the level falls more slowly with distance.
The pipe formulas come from the boundary conditions on standing waves, as for strings, with a small end correction in real pipes. The Doppler formula follows from counting how crests are spaced when the source moves between emitting them; it assumes speeds well below the speed of sound.
Play two notes of nearly the same frequency together and their waves drift in and out of step. The loudness swells and fades at the difference frequency: $440$ Hz and $443$ Hz give three beats per second.
Musicians tune by listening for beats: as two strings approach the same pitch, the beats slow down and vanish. Piano tuners set some intervals slightly off to a specific number of beats per second, because equal temperament, the tuning system that lets a piano play in every key, deliberately makes most intervals slightly impure.
Loud sound damages the delicate hair cells of the inner ear, and they do not grow back. The Occupational Safety and Health Administration allows workers eight hours a day at $90$ dBA, halving the time for every $5$ dB more. The National Institute for Occupational Safety and Health recommends a stricter limit, $85$ dBA for eight hours with halving every $3$ dB.
Earbuds at full volume can exceed $100$ dB. Hearing loss from noise is gradual and painless, which is why audiologists urge the sixty-sixty rule: no more than sixty percent volume for sixty minutes at a time.
Sound above $20$ kHz, beyond human hearing, is ultrasound. Medical ultrasound at $2$ to $15$ MHz sends pulses into the body and times their echoes from tissue boundaries, building images of babies, hearts and organs without radiation. Doppler ultrasound measures blood flow speed from the frequency shift of echoes off moving blood cells.
Ships and submarines use sonar the same way underwater, where sound travels far. NOAA maps the seafloor with multibeam sonar, and bats and dolphins navigate by echolocation, emitting clicks and listening for returns, the natural version of the same physics.
When a source moves faster than sound, the crests it emits cannot get ahead of it. They pile up along a cone trailing behind, a shock wave, heard on the ground as a sonic boom when the cone sweeps past. Chuck Yeager first flew faster than sound in the Bell X-1 over California's Mojave Desert in 1947.
Because of booms, supersonic flight over land has been banned in the United States since 1973. NASA's X-59 research aircraft, built by Lockheed Martin, is designed to reshape the shock waves so that the boom becomes a quiet thump, to test whether people on the ground would accept it.
In a room, sound reflects from walls, floor and ceiling, and the reflections persist after the source stops: reverberation. Wallace Sabine, a Harvard physicist, founded architectural acoustics in 1895 by measuring how long sound took to fade in lecture halls, and he designed Boston's Symphony Hall, still admired for its sound.
Concert halls aim for about two seconds of reverberation, rich for orchestras; lecture halls and classrooms aim for well under a second, so speech stays clear. Absorbing panels, carpets and even the audience's clothing shorten the time, which is why a hall sounds different empty than full.
The Occupational Safety and Health Administration limits workplace noise under its standard 29 CFR 1910.95. A worker may spend eight hours a day at $90$ dBA; for every $5$ dB more, the allowed time halves: four hours at $95$, two at $100$, one at $105$, and half an hour at $110$ dBA, the level of a chainsaw.
Above $85$ dBA averaged over a day, employers must run a hearing conservation program with annual hearing tests and free hearing protection. Because $5$ dB is more than a tripling of intensity, OSHA's rule is more lenient than the equal-energy rule of $3$ dB that NIOSH recommends, and many companies now follow the stricter guideline.
American emergency vehicles use sirens that sweep between about $500$ and $1500$ Hz. As an ambulance drives past at $30$ m/s, a $700$ Hz tone is heard at about $767$ Hz as it approaches and about $644$ Hz as it recedes, a drop of over $120$ Hz, nearly three semitones.
That sudden fall in pitch tells listeners the vehicle has passed, which is part of why sirens remain effective. Police radar guns work on the same principle with radio waves: a car moving toward the gun shifts the reflected frequency up by an amount proportional to its speed, and the gun reads the shift. Doppler weather radar at the National Weather Service's stations measures wind speeds inside storms the same way, spotting the rotation that warns of tornadoes.
It is natural to think two $60$ dB sounds together make $120$ dB. The decibel scale is logarithmic: combining two equal sounds doubles the intensity, which adds only about $3$ dB, giving $63$ dB. Reaching $120$ dB would take a million of them.
A related error is to think the Doppler effect changes the source's own frequency. The siren always sounds the same to the driver; only a listener moving relative to it hears the shift.
One speaker gives $75$ dB at your seat. Find its intensity.
$I = 10^{-12} \times 10^{7.5} = 3.16 \times 10^{-5}\ \text{W/m}^2$
Undo the logarithm.
A second identical speaker is added. Find the total intensity.
$I = 6.32 \times 10^{-5}\ \text{W/m}^2$
Intensities add.
Find the new level.
$\beta = 75 + 10\log 2 = 78\ \text{dB}$
Not $150$ dB.
Find how many speakers make $85$ dB.
$10^{(85 - 75)/10} = 10$
Ten times the intensity.
Move twice as far from one speaker. Find its level.
$75 - 10\log 4 = 69\ \text{dB}$
Inverse square.
A train's horn sounds $400$ Hz as it approaches at $35$ m/s. Find the heard frequency.
$f' = 400 \times \dfrac{343}{343 - 35} = 445\ \text{Hz}$
Approaching: higher.
Find the frequency after it passes.
$f' = 400 \times \dfrac{343}{343 + 35} = 363\ \text{Hz}$
Receding: lower.
Find the drop.
$445 - 363 = 82\ \text{Hz}$
About three semitones.
Find the wavelength ahead of the train.
$\lambda = \dfrac{343}{445} = 0.771\ \text{m}$
Squeezed.
Find the wavelength behind.
$\lambda = \dfrac{343}{363} = 0.945\ \text{m}$
Stretched.
Explain what the engineer hears.
$400\ \text{Hz}$
No relative motion between horn and engineer.
A flute behaves as a pipe open at both ends. To play $262$ Hz at $343$ m/s, find the needed length.
$L = \dfrac{v}{2f} = \dfrac{343}{2 \times 262} = 0.655\ \text{m}$
Fundamental of an open pipe.
Find its second harmonic.
$f_2 = 524\ \text{Hz}$
An octave up.
Close one end of the same tube. Find the new fundamental.
$f_1 = \dfrac{343}{4 \times 0.655} = 131\ \text{Hz}$
An octave lower.
Find the closed tube's next resonance.
$f_3 = 393\ \text{Hz}$
Three times, not two.
The room warms to $30$ °C. Find the new speed.
$v = 331 + 0.6 \times 30 = 349\ \text{m/s}$
Warmer air, faster sound.
Find the open flute's new pitch.
$f = \dfrac{349}{2 \times 0.655} = 266\ \text{Hz}$
Wind instruments go sharp as they warm.
Write the change in level.
$\Delta\beta = 10\log k$
For an intensity factor $k$.
Substitute the factor.
$\Delta\beta = 10\log 100$
Two powers of ten.
Evaluate the change.
A sound has a level of $50$ dB. Its intensity is then multiplied by $2$. What is the new sound level, in dB?
Complete the worked solution: a pipe $0.45$ m long is closed at one end. With sound at $343$ m/s, find its fundamental frequency in Hz, its next resonance, the third harmonic, in Hz, and the fundamental's wavelength in m.
Find the fundamental.
$f_1 = \dfrac{v}{4L} =$ f
A quarter wavelength fits.
Find the next resonance.
$f_3 = 3f_1 =$ t
Only odd harmonics exist.
Find the fundamental's wavelength.
$\lambda_1 = 4L =$ w
Four times the pipe.
Explain the missing even harmonics.
$\text{a node must sit at the closed end}$
Even patterns would put an antinode there.
Match each sound situation to its rule.
| resonates at all whole-number harmonics of v/2L | resonates only at odd harmonics of v/4L | makes beats at the difference of the frequencies | sounds higher than it really is | |
|---|---|---|---|---|
| a pipe open at both ends | ||||
| a pipe closed at one end | ||||
| two notes of slightly different pitch | ||||
| a siren approaching a listener |
An ambulance siren sounds $500$ Hz as the ambulance drives at $25$ m/s past a pedestrian. With sound at $343$ m/s, fill in the frequency the pedestrian hears as it approaches, as it recedes, and the drop between the two, all in Hz.
| value | |
|---|---|
| frequency approaching (Hz) | |
| frequency receding (Hz) | |
| drop in frequency (Hz) |
An organ pipe open at both ends is $1.0$ m long, and sound travels at $343$ m/s. Write the frequency of its $n$th harmonic, in Hz, as a function of $n$.
Answer:
A speaker radiates $5$ W of sound evenly in all directions. What is the sound level $20$ m away, in dB? Use $I_0 = 10^{-12}$ W/m².
Answer: dB
Federal OSHA rules allow $8$ hours a day of exposure to $90$ dBA and halve the allowed time for every $5$ dB louder. How many hours a day may a worker spend in $110$ dBA of noise without hearing protection?
Answer: h
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
An organ pipe open at both ends is $0.7$ m long, and sound travels at $343$ m/s. Write the frequency of its $n$th harmonic, in Hz, as a function of $n$.
Answer:
You can analyze sound. Explain to someone why two equally loud speakers are not twice as many decibels as one.
22. Your turn: a sound's intensity is multiplied by $100$. By how many decibels does its level rise?, step 3
$\Delta\beta = 20\ \text{dB}$
Ten per factor of ten.