Jugami Physics

High School · Physics · Unit 4

Waves and Sound

Ripples, music, sirens and silence: how energy travels without anything traveling with it.

Drop a pebble in a pond and rings spread out, but the water mostly bobs up and down in place. Pluck a guitar string and, a moment later, your ear drum starts to wiggle. In both cases a wave carries energy from one place to another without carrying the material along with it.

On this page you'll learn how to describe any wave with four numbers, how to use v = fλ, what makes a sound high or low and loud or soft, why a passing siren changes pitch, and how two sounds can add up to silence. Then try worked problems, a wave lab you can see and hear, the story of a famous train experiment, and a quiz.

Start with wavesJump to the quiz
Blueprint sketch of a wave. A smooth wave line rises and falls about a dashed rest line. The wavelength, lambda, is marked from one crest to the next, the amplitude, A, is marked from the rest line up to a crest, a trough is labeled, and a blue arrow shows the wave traveling to the right at speed v, with the equation v = f lambda. λ A trough v v = fλ

On this page

Key words first

Wave

A disturbance that travels from place to place, carrying energy but not the material itself.

Medium

The stuff a wave travels through, like water, air, a rope or a steel rail. Sound needs a medium, so it can't cross empty space.

Amplitude (A)

How far the medium moves from its resting position. Bigger amplitude means more energy: a taller water wave, a louder sound.

Wavelength (λ)

The length of one complete wave, for example from one crest to the next. Measured in meters. λ is the Greek letter lambda.

Frequency (f)

How many waves pass a point each second, in hertz (Hz). 1 Hz = 1 wave per second.

Period (T)

The time for one complete wave to pass, in seconds. Period and frequency are flip sides: T = 1/f.

Idea 1: Four numbers describe a wave

Picture a long rope tied to a post. Flick the free end up and down steadily and a train of waves runs along it. Every repeating wave, on a rope, in water or in air, can be described with the same few numbers:

  • Amplitude A: the height of a crest (or depth of a trough) measured from the rest position, not from crest to trough.
  • Wavelength λ: the distance from one crest to the next, or from any point to the matching point on the next wave.
  • Frequency f: how many waves you make each second. Flick the rope twice a second and f = 2 Hz.
  • Period T: how long one wave takes. At 2 Hz, each wave takes half a second.

T = 1 / f   and   f = 1 / T

Notice the two graphs on the right look alike but mean different things. A snapshot of the whole wave at one instant shows the wavelength. Watching one point as time passes shows the period.

Two graphs. Top, a snapshot of a wave along its length, displacement against distance: the wavelength lambda is marked from one crest to the next, and the amplitude A is marked from the middle line down to a trough. Bottom, one point on the wave tracked over time, displacement against time: the period T is marked from one peak to the next. A snapshot along the wave distancedisplacement λ wavelength A amplitude One point on the wave, over time timedisplacement T period
Top: a snapshot in space shows the wavelength λ and amplitude A. Bottom: one point followed over time shows the period T.

Waves carry energy, not stuff. In a stadium "wave," the wave races around the stands but every fan stays in their own seat, standing up and sitting down. A cork on a pond does the same thing: it bobs as ripples pass but doesn't ride away on them.

Idea 2: Transverse and longitudinal waves

Waves come in two basic shapes, depending on which way the medium moves compared with the way the wave travels.

In a transverse wave, the medium moves at right angles to the wave's direction. Shake a rope up and down and the wave travels sideways along it. Waves on a guitar string are transverse, and so are light and radio waves (which need no medium at all).

In a longitudinal wave, the medium moves back and forth along the wave's direction. Push and pull one end of a stretched spring toy and you'll see bunched-up regions, called compressions, and spread-out regions, called rarefactions, travel along it.

Sound is a longitudinal wave. A speaker cone pushes air molecules together, then pulls back and leaves a thin patch, over and over. Those compressions and rarefactions of air pressure travel to your ear.

Two panels. Top, a transverse wave on a rope: the rope forms crests and troughs, and a marked point on the rope moves up and down, shown by an orange double arrow, while the wave travels to the right, shown by a blue arrow. Bottom, a longitudinal wave on a stretched spring: the coils are bunched together in compressions and spread apart in rarefactions, and a marked orange coil moves back and forth along the spring, shown by an orange double arrow, while the wave travels to the right. One wavelength is marked from one compression to the next. Transverse: the rope moves up and down crest trough rope moves wave travels Longitudinal: the coils move back and forth λ one wavelength compression rarefaction wave travels
Top: on a rope, each point moves up and down while the wave moves along. Bottom: on a spring, each coil moves back and forth while compressions and rarefactions move along. Not to scale.

Where you see each kind

Idea 3: Wave speed, v = fλ

Every period T, the wave moves forward exactly one wavelength λ. Speed is distance over time, so v = λ / T. Since 1/T is the frequency, that gives the most useful equation in this unit:

v = f × λ

Wave speed (m/s) = frequency (Hz) × wavelength (m). Rearranged: f = v / λ and λ = v / f.

Here's the key idea: the medium sets the speed. All sounds in the same air travel at the same speed, whether they're high or low, loud or soft. So if the frequency goes up, the wavelength must get shorter to keep f × λ the same. Double the frequency, halve the wavelength.

Speed of sound in some materials. Air at 0 °C and the liquids and solids are from OpenStax Physics Table 14.1; air at 20 °C is from OpenStax College Physics 2e. Sound travels fastest in stiff solids.
MediumSpeed of sound
Air at 0 °C331 m/s
Air at 20 °C (room temperature)343 m/s
Helium at 0 °C965 m/s
Fresh water at 20 °C1,480 m/s
Sea water at 20 °C1,540 m/s
Steel (longitudinal or bulk)5,960 m/s

For comparison, light and radio waves travel at about 300,000,000 m/s (3.00 × 10⁸ m/s), nearly a million times faster than sound in air. That's why you see lightning before you hear the thunder.

Idea 4: Sound, pitch and loudness

Sound starts with something vibrating: a guitar string, a speaker cone, your vocal folds. The vibration squeezes and stretches the air next to it, and that pattern of pressure travels outward as a longitudinal wave.

Pitch is frequency. A high note is a high-frequency wave and a low note is a low-frequency wave. Humans can normally hear from about 20 Hz to 20,000 Hz. Below 20 Hz is infrasound; above 20,000 Hz is ultrasound. Dogs can hear up to about 45,000 Hz, and bats and dolphins up to about 110,000 Hz.

Loudness is amplitude. A bigger pressure swing carries more energy each second, and we hear it as louder. Turning up the volume makes the waves taller, not more frequent, so the pitch stays the same.

A two-by-two grid of sound waves drawn as graphs. Left column, small amplitude: quieter. Right column, large amplitude: louder. Top row, waves spread out with a low frequency: lower pitch. Bottom row, waves packed close together with a high frequency: higher pitch. quietersmall amplitude louderlarge amplitude lower pitchlow frequency higher pitchhigh frequency
Pitch depends on frequency (how closely the waves are packed). Loudness depends on amplitude (how tall they are). Not to scale.

Decibels, briefly

Our ears handle an enormous range: the loudest sound you can stand carries about a trillion times as much energy each second as the faintest one you can hear. To keep the numbers manageable, sound level is measured on a scale in decibels (dB). The faintest sound a person with normal hearing can detect is set at 0 dB, and every 10 dB step means 10 times the intensity. So 20 dB more is 10 × 10 = 100 times the intensity, and 30 dB more is 1,000 times.

Examples from OpenStax College Physics 2e, Table 17.2.
Sound levelExampleCompared with 0 dB
0 dBThreshold of hearing1 time
20 dBWhisper at 1 m100 times
60 dBNormal conversation1 million times
90 dBInside a heavy truck; hearing damage from long exposure1 billion times
120 dBLoud rock concert; threshold of pain1 trillion times

Look after your ears. Hearing damage builds up over time and doesn't heal. Keep earbuds at a level where you can still hear someone talking to you, and take breaks from loud music.

Idea 5: The Doppler effect

Stand by a road as an ambulance races past with its siren on. As it comes toward you the siren sounds higher; the moment it passes, the pitch drops. The siren itself never changes. What changes is how the waves reach you. This is the Doppler effect.

The siren sends out each wave crest from a slightly different spot, because it keeps moving forward. Ahead of the ambulance, the crests are squeezed closer together: a shorter wavelength, so more crests reach you each second, a higher frequency. Behind it, the crests are stretched out: a longer wavelength and lower frequency.

For a source moving straight toward or away from a listener who is standing still:

fheard = fsource × v / (v ∓ vs)

Here v is the speed of sound and vs is the speed of the source. Use minus when the source is coming toward you (smaller bottom, higher frequency) and plus when it's moving away.

The Doppler effect. A siren moving to the right gives out a series of circular wavefronts. Each circle was sent out from a point where the siren used to be, so the circles are bunched close together in front of the siren and spread far apart behind it. A listener in front hears a shorter wavelength and a higher pitch; a listener behind hears a longer wavelength and a lower pitch. siren moving right Behind: waves stretched, longer λ, lower pitch Ahead: waves bunched, shorter λ, higher pitch
Each circle is a wave crest sent out from where the siren was at that moment (gray dots). Ahead of the siren the crests bunch up; behind it they spread out. Not to scale: the siren here moves at half the speed of sound, far faster than any real ambulance.

Where the Doppler effect is used

Idea 6: Interference, beats and noise-cancelling

Two waves can pass right through each other. Where they overlap, their displacements simply add. This is called superposition, and it leads to interference.

Constructive interference: crest meets crest and trough meets trough. The waves are "in step," and they add up to a bigger wave, a louder sound.

Destructive interference: crest meets trough. The waves are exactly out of step. If they have the same amplitude, they cancel completely and the result is silence at that spot.

Noise-cancelling headphones use destructive interference on purpose. A tiny microphone picks up the outside noise, electronics flip it so every crest becomes a trough, and the speaker plays that "anti-noise" into your ear along with your music. The noise and anti-noise cancel, especially for steady low sounds like an airplane engine.

Interference. Left, constructive: two identical waves line up crest to crest, and they add to make one wave twice as tall, a louder sound. Right, destructive: two identical waves line up crest to trough, and they add to make a flat line, so they cancel out. Constructive Destructive crest meets crest crest meets trough +=+= twice the amplitude they cancel: silence
Adding two identical waves. In step, they make a wave twice as tall. Exactly out of step, they cancel.

Beats

What if two notes are almost the same frequency, like 440 Hz and 442 Hz? They drift in and out of step. When in step they add (loud); a moment later they're out of step and cancel (quiet). You hear one note that throbs, "wah-wah-wah." These throbs are called beats, and the number per second is the difference between the two frequencies:

fbeat = | f1 − f2 |

440 Hz and 442 Hz make 2 beats per second. Musicians use this to tune: they adjust a string until the beats slow down and disappear, which means the two frequencies match.

Beats. Two sound waves with slightly different frequencies, one blue and one orange, are drawn one above the other. Added together they make a wave whose loudness swells and fades: the dashed outline grows large where the two waves are in step and shrinks to nearly nothing where they are out of step. The number of swells each second, the beat frequency, equals the difference between the two frequencies. f₁ f₂ add them loudloudloud quietquiet
Two waves with slightly different frequencies (blue and orange) add up to a wave that swells and fades. The dashed outline shows the beats. Not to scale.

Worked examples

Try each one on paper first, then open the solution. Use 343 m/s for the speed of sound in air.

1. Waves on a rope

You shake a rope 4 times a second, and the waves are 0.5 m long. How fast do they travel along the rope?

Show solution
v = f × λ = 4 Hz × 0.5 m = 2 m/s

2. Concert A

Orchestras tune to the note A at 440 Hz. What is its wavelength in air, and what is its period?

Show solution
λ = v ÷ f = 343 ÷ 440 ≈ 0.78 m
T = 1 ÷ f = 1 ÷ 440 ≈ 0.00227 s (2.27 milliseconds)

3. The edges of hearing

Find the wavelengths of the lowest (20 Hz) and highest (20,000 Hz) sounds a person can hear.

Show solution
20 Hz: λ = 343 ÷ 20 ≈ 17 m
20,000 Hz: λ = 343 ÷ 20,000 ≈ 0.017 m (1.7 cm)

Low notes are longer than a bus; high notes are shorter than your thumb.

4. An FM radio station

A radio station broadcasts at 100 MHz (100,000,000 Hz). Radio waves travel at 3.00 × 10⁸ m/s. What is the wavelength?

Show solution
λ = v ÷ f = 300,000,000 ÷ 100,000,000 = 3 m

Same equation, totally different kind of wave.

5. How far is the storm?

You see a lightning flash and hear the thunder 3 s later. About how far away was the strike?

Show solution
The light arrives almost instantly, so the delay is the sound's travel time.
d = v × t = 343 × 3 ≈ 1,030 m, about 1 km

Rule of thumb: about 3 seconds per kilometer.

6. Sonar

A ship sends a sound pulse straight down. The echo returns 2.0 s later. Sound travels at 1,540 m/s in sea water. How deep is the sea?

Show solution
Distance there and back = 1,540 × 2.0 = 3,080 m
Depth = 3,080 ÷ 2 = 1,540 m

Don't forget to halve it: the sound makes a round trip.

7. Decibels

Normal conversation is about 60 dB. Inside a heavy truck it's about 90 dB. How many times more intense is the truck?

Show solution
90 − 60 = 30 dB = three 10 dB steps
10 × 10 × 10 = 1,000 times the intensity

8. A passing ambulance

An ambulance siren sounds at 700 Hz and the ambulance drives at 25 m/s. What frequency do you hear as it comes toward you, and as it moves away?

Show solution
Toward: f = 700 × 343 ÷ (343 − 25) = 700 × 343 ÷ 318 ≈ 755 Hz
Away: f = 700 × 343 ÷ (343 + 25) = 700 × 343 ÷ 368 ≈ 652 Hz

As it passes, the pitch drops by about 100 Hz.

9. Tuning a guitar

A tuning fork plays 440 Hz. A guitar string played with it makes 3 beats per second. What could the string's frequency be?

Show solution
fbeat = |f1 − f2| = 3 Hz
So the string is 437 Hz or 443 Hz

Tighten the string a little: if the beats speed up, it was already too high.

Try it: the wave lab

See it, hear it

v = fλ calculator

Things to try

  • In the wave lab, double the frequency from 220 Hz to 440 Hz. What happens to the number of waves on the graph, and to the wavelength?
  • Change only the amplitude. Does the pitch change when you play it?
  • In the calculator, find the wavelength of a 440 Hz note in air, then in water. Why is it longer in water?
  • Find the frequency of green light, with a wavelength of 0.00000055 m (550 nanometers).

A short history: Doppler and the train test

Christian Doppler (1803–1853) was an Austrian mathematician and physicist, born in Salzburg. On 25 May 1842, in Prague, he presented a paper to the Royal Bohemian Society of Sciences with a long title: On the Colored Light of Double Stars and Some Other Stars of the Heavens. In it he argued that if a source of waves and an observer move toward each other, the observer meets the waves more often and so measures a higher frequency; if they move apart, a lower one.

Doppler hoped this would explain why some stars look red or blue. That part didn't hold up: stars move far too slowly compared with light to change color noticeably. But the principle itself was right, and it worked for sound as well as light.

Not everyone believed it. The Dutch scientist C. H. D. Buys Ballot (1817–1890) of Utrecht was skeptical and decided to test it with sound, using the new railway. On 3 June 1845, and again on 5 June, a steam locomotive pulled musicians, including a horn player, along the line between Utrecht and Maarssen, while listeners with a good ear for pitch stood by the track and rode on the train. With no instruments to measure pitch, the human ear was the measuring device.

The listeners heard the note rise as the train approached and fall as it moved away, just as Doppler predicted. Buys Ballot published his results in 1845 in the German journal Annalen der Physik, with remarks on Doppler's theory in the same paper. He later founded the Netherlands' national weather institute, the KNMI, which now uses Doppler radar to track storms.

A simplified drawing of Buys Ballot's 1845 test. A small steam locomotive pulls an open wagon to the right along a railway. A musician on the wagon plays a horn. Listeners stand by the track ahead of the train and behind it. Curved sound waves spread from the horn, bunched up in front of the train and stretched out behind it. a musician plays one steady note on a moving train hears higher hears lower Utrecht–Maarssen railway, June 1845
A simplified, original drawing of the idea behind Buys Ballot's 1845 test, not his exact setup. Not to scale.

Why it's called a decibel

The decibel came from telephone engineering. In 1923 the Bell System began measuring signal strength in a "transmission unit," where each unit meant a fixed power ratio. In talks with a European telephone committee formed in 1924, engineers suggested naming a unit ten times bigger the bel, after Alexander Graham Bell. The Bell System then renamed its unit the decibel, one tenth of a bel, and announced the new name in the Bell System Technical Journal in January 1929.

  1. 1842 Doppler presents his paper in Prague (25 May), predicting the frequency shift from motion.
  2. 1845 Buys Ballot's railway tests between Utrecht and Maarssen (3 and 5 June) confirm the effect for sound.
  3. 1845 Buys Ballot's results appear in Annalen der Physik.
  4. 1923 The Bell System introduces the "transmission unit" for measuring signal levels.
  5. 1929 The Bell System Technical Journal announces the new name: the decibel.
  6. Today The Doppler effect is used in weather radar, medical ultrasound and astronomy.

Test yourself

The Wave Check

Talk about it

  1. In space movies, explosions make a loud boom. Would a real explosion in space make any sound you could hear? Why not?
  2. When you breathe helium, your voice sounds squeaky. Sound travels faster in helium. How might that change the wavelengths and frequencies coming out of your mouth?
  3. Noise-cancelling headphones work best on steady low hums and worst on sudden sounds like a door slam. Why might that be?
  4. Buys Ballot used human ears as his measuring instruments. What could go wrong with that? How would you test the Doppler effect today?

Sources

  1. OpenStax, Physics (high school), Chapter 13 "Waves and Their Properties": 13.1 Types of Waves, 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period, 13.3 Wave Interaction: Superposition and Interference.
  2. OpenStax, Physics (high school), Chapter 14 "Sound": 14.1 Speed of Sound, Frequency, and Wavelength (Table 14.1, speeds of sound), 14.2 Sound Intensity and Sound Level (hearing ranges, decibels), 14.3 Doppler Effect and Sonic Booms, 14.4 Sound Interference and Resonance (beats, noise-cancelling headphones).
  3. OpenStax, College Physics 2e: 17.2 Speed of Sound, Frequency, and Wavelength (343 m/s at 20 °C), 17.3 Sound Intensity and Sound Level (Table 17.2, sound levels).
  4. Christian Doppler, "Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels," Abhandlungen der Königlichen Böhmischen Gesellschaft der Wissenschaften, 5th series, vol. 2, pp. 465–482 (presented 25 May 1842). Full text on Wikisource (German): https://de.wikisource.org/wiki/Über_das_farbige_Licht_der_Doppelsterne_und_einiger_anderer_Gestirne_des_Himmels
  5. Kurt Toman, "Christian Doppler and the Doppler effect," Eos, Transactions American Geophysical Union 65, no. 48 (1984): 1193–1194, doi:10.1029/EO065i048p01193 (Doppler's life and the 1842 paper). https://doi.org/10.1029/EO065i048p01193
  6. C. H. D. Buys Ballot, "Akustische Versuche auf der Niederländischen Eisenbahn, nebst gelegentlichen Bemerkungen zur Theorie des Hrn. Prof. Doppler," Annalen der Physik 142, no. 11 (1845): 321–351, doi:10.1002/andp.18451421102. https://doi.org/10.1002/andp.18451421102
  7. KNMI (Royal Netherlands Meteorological Institute), "Doppler experiment Buys Ballot 160 jaar geleden" (in Dutch: 3 June 1845, Utrecht–Maarssen, locomotive, horn player and listeners; Buys Ballot as KNMI's founder; Doppler radar and ultrasound today). https://www.knmi.nl/over-het-knmi/nieuws/doppler-experiment-buys-ballot-160-jaar-geleden
  8. Utrecht University, "Professor Buys Ballot investigates the Doppler Effect," Utrechtse muurformules (Buys Ballot's dates, his doubts about Doppler's theory, musicians on a flat wagon). https://muurformules.sites.uu.nl/mural-paintings/christophorus-buys-ballot/?lang=en
  9. W. H. Martin, "DeciBel—The Name for the Transmission Unit," Bell System Technical Journal 8, no. 1 (January 1929): 1–2, doi:10.1002/j.1538-7305.1929.tb02302.x (the 1923 transmission unit, the bel named for Alexander Graham Bell, and the decibel). https://doi.org/10.1002/j.1538-7305.1929.tb02302.x

Credits: all diagrams on this page, including the drawing of Buys Ballot's train test and the share image, are original SVG made for Jugami. The wave graphs are drawn from exact sine curves.