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A wave's speed is its frequency times its wavelength; in one medium the speed is fixed, so higher frequency means shorter waves.
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By the end of this lesson you will be able to find a wave's speed, frequency or wavelength from the other two.
You can find a speed as distance divided by time, and you know that sound and light travel from place to place. This lesson describes waves with numbers and gives the one equation that links a wave's speed, frequency and wavelength.
| Term | What it means |
|---|---|
| Wave | A disturbance that carries energy from place to place without carrying material. |
| Frequency | The number of waves passing a point each second, in hertz, Hz. |
| Wavelength | The distance from one crest to the next, $\lambda$, in meters. |
| Amplitude | The height of a wave from its middle to a crest. |
| Period | The time for one wave to pass, $T = 1/f$. |
| Wave speed | How fast the wave travels, $v = f\lambda$. |
A wave's speed, frequency and wavelength are linked by the wave equation:
$$v = f\lambda.$$
In one medium, such as air, the speed stays the same. A higher frequency then means a shorter wavelength.
Another way: picture
Picture a line of train cars rolling past a crossing. If three cars pass every second and each car is twenty meters long, the train moves sixty meters every second. A wave works the same way: count the waves passing each second and multiply by each wave's length.
Another way: steps
Drop a stone in a pond and ripples spread outward. The water itself mostly bobs up and down in place, but the ripples travel across the pond, carrying energy that can rock a toy boat on the far side.
Every wave works this way: it carries energy, not material. Sound carries energy through air, light through space, and earthquake waves through rock.
Frequency counts how many waves pass a point each second. Its unit, the hertz, means one wave per second. A wave with a frequency of five hertz sends five crests past you every second.
For sound, frequency is pitch. A low note on a tuba might be a hundred hertz; a high note on a flute, a couple of thousand. Humans can hear from about twenty to twenty thousand hertz.
Wavelength is the length of one wave: the distance from one crest to the next, or from one trough to the next. It is written with the Greek letter lambda.
Ocean swells may have wavelengths of a hundred meters or more; ripples on a puddle, a few centimeters; the light your eyes see, less than a millionth of a meter.
In one second, a certain number of waves pass, the frequency. Each is one wavelength long. So in one second the front of the wave moves the frequency times the wavelength, which is the distance traveled per second: the speed.
A wave with frequency $5$ Hz and wavelength $2$ m moves $5 \times 2 = 10$ meters every second. The units multiply too: hertz, per second, times meters gives meters per second.
The wave equation can find any one of its three quantities from the other two. Wavelength is speed divided by frequency; frequency is speed divided by wavelength.
Sound at $343$ m/s with a frequency of $686$ Hz has a wavelength of $343 \div 686 = 0.5$ m. A water wave at $3$ m/s with crests $6$ m apart arrives $3 \div 6 = 0.5$ times per second.
The period is the time for one wave to pass, the reverse of frequency: $T = 1/f$. A wave with a frequency of four hertz has a period of a quarter of a second.
Since one wavelength passes in one period, the speed is also wavelength divided by period. This gives a quick check on any wave calculation.
A wave's speed is set by what it travels through, not by its frequency. Sound travels at about $343$ m/s in room-temperature air, about fifteen hundred meters per second in water, and about five thousand in steel.
Every note from a piano reaches your ear at the same speed. If high notes traveled faster, music heard from a distance would arrive jumbled.
Because sound travels at one speed in air, a higher frequency must have a shorter wavelength. Doubling the frequency halves the wavelength.
This is why small instruments, like a piccolo, make high notes and large instruments, like a tuba, make low ones. The size of the instrument matches the length of the waves it makes.
Checking an answer. Sound in air must come out near $343$ m/s. In one medium, a higher frequency must give a shorter wavelength. Radio waves must travel at the speed of light.
The wave equation is really just speed equals distance over time, applied to one wave. In one period, the wave travels one wavelength, so its speed is wavelength divided by period, which is wavelength times frequency.
Because it comes from the meaning of speed itself, the equation works for every kind of wave: water, sound, light and radio.
Light and radio waves are electromagnetic waves, which can travel through empty space. They all move at the speed of light, three hundred million meters per second.
Radio stations are labeled by frequency. An FM station at a hundred megahertz sends waves about three meters long; an AM station at one thousand kilohertz sends waves about three hundred meters long.
The amplitude of a wave is how tall it is, from the middle to a crest. For sound, amplitude is loudness; for water waves, it is height; for light, it is brightness.
Amplitude does not appear in the wave equation. A loud note and a quiet note of the same pitch have the same frequency, wavelength and speed. Only the energy they carry is different.
To measure a water wave, count how many crests pass a post in a set time, and divide to get the frequency. Measure the distance between neighboring crests for the wavelength. Multiplying gives the speed.
Oceanographers measure ocean waves the same way, using buoys that bob with each passing crest and radio the data back to shore.
The most common error is dividing when you should multiply, or the other way around. Speed is the product of frequency and wavelength; wavelength is speed over frequency.
Another is forgetting prefixes. A frequency in kilohertz must be multiplied by a thousand, and in megahertz by a million, before using the equation.
Light travels almost instantly compared with sound, which covers about a third of a kilometer each second. During a storm, the flash of lightning reaches you first and the thunder follows.
Counting the seconds between flash and thunder and dividing by three gives the distance to the lightning in kilometers. Five seconds means the storm is less than two kilometers away.
Doctors use ultrasound, sound with frequencies of millions of hertz, far above what humans can hear, to see inside the body. At those frequencies the wavelength in body tissue is less than a millimeter, short enough to show fine detail.
The machine sends sound pulses into the body and times their echoes. Knowing the speed of sound in tissue, it works out how deep each echo came from.
A few benchmarks help. Speech has frequencies of a few hundred hertz and wavelengths of about a meter. Ocean waves arrive every several seconds and are tens of meters long. FM radio waves are a few meters long.
If a calculation gives a sound wave a kilometer long, or an FM wave a millimeter long, check the prefixes and whether you multiplied or divided.
When sound hits a wall or a cliff, part of it bounces back as an echo. Timing the echo gives the distance: the sound travels there and back, so the distance is half the speed times the time. Shout at a canyon wall and hear the echo a second later, and the wall is about one hundred seventy meters away.
Ships and submarines use the same idea underwater, with sonar. The U.S. Navy and fishing boats send sound pulses down and time the echoes from the seafloor or schools of fish, using the speed of sound in seawater, about fifteen hundred meters per second.
In some waves, like ripples and waves on a rope, the material moves up and down while the wave travels sideways; these are called transverse waves. In others, like sound, the air moves back and forth along the direction the wave travels, squeezing and stretching; these are called longitudinal waves. The wave equation works the same way for both.
Every radio station in the United States is assigned a frequency by the Federal Communications Commission. FM stations broadcast between about eighty-eight and one hundred eight megahertz; AM stations between about five hundred thirty and seventeen hundred kilohertz.
Radio waves travel at the speed of light, so the wave equation gives their wavelengths. An FM station at ninety megahertz sends waves about three and a third meters long, while an AM station near nine hundred kilohertz sends waves over three hundred meters long. That is why AM stations need tall broadcast towers, often over a hundred meters, while FM antennas can be much smaller. AM waves also bend around hills and travel farther at night.
Surfers in Hawaii and California check wave forecasts from the National Weather Service and ocean buoys. The forecasts list wave height and period, the time between crests, often ten to twenty seconds for big swells from distant storms.
In deep water, longer-period waves travel faster and carry more energy. A swell with a sixteen-second period may travel over twenty meters per second across the open ocean, arriving days after the storm that made it. Surfers use the period, together with the wave equation, to judge how powerful the waves will be when they finally reach the beach.
It seems natural that a wave with more crests per second must be moving faster. But in one medium, every frequency travels at the same speed. A higher frequency just means shorter waves packed more closely together.
A related error is to think a louder sound travels faster. Loudness is amplitude, which does not appear in the wave equation. A whisper and a shout reach you at the same speed.
A student shakes a rope $4$ times each second, making waves $1.5$ m long. Find the frequency.
$f = 4\ \text{Hz}$
Waves per second.
Write the wave equation.
$v = f\lambda$
Frequency times wavelength.
Find the speed.
$v = 4 \times 1.5 = 6\ \text{m/s}$
Meters per second.
Find the period.
$T = \dfrac{1}{4} = 0.25\ \text{s}$
Seconds per wave.
Shaking faster at $8$ Hz, find the new wavelength.
$\lambda = \dfrac{6}{8} = 0.75\ \text{m}$
Same rope, same speed.
Middle C has a frequency of about $262$ Hz. Sound travels at $343$ m/s. Write the formula for wavelength.
$\lambda = \dfrac{v}{f}$
Rearranged.
Substitute the values.
$\lambda = \dfrac{343}{262}$
Meters per second over hertz.
Evaluate the wavelength.
$\lambda = 1.31\ \text{m}$
About a meter.
Find the wavelength one octave up, at $524$ Hz.
$\lambda = \dfrac{343}{524} = 0.655\ \text{m}$
Half as long.
Find the period of middle C.
$T = \dfrac{1}{262} = 0.0038\ \text{s}$
Under four thousandths.
Explain why both notes arrive together.
$\text{same speed in air}$
Speed set by the medium.
At a beach in California, $12$ waves reach the shore in $96$ s. Find the frequency.
$f = \dfrac{12}{96} = 0.125\ \text{Hz}$
Waves per second.
Find the period.
$T = \dfrac{96}{12} = 8\ \text{s}$
Seconds between waves.
The crests are $100$ m apart. Find the speed.
$v = 0.125 \times 100 = 12.5\ \text{m/s}$
Frequency times wavelength.
Check with wavelength over period.
$\dfrac{100}{8} = 12.5\ \text{m/s}$
The same.
Convert the speed to miles per hour.
$12.5 \times 2.237 = 28\ \text{mph}$
Faster than a bicycle.
Find how long a wave takes to cross $1000$ m.
$t = \dfrac{1000}{12.5} = 80\ \text{s}$
Distance over speed.
Explain what the water itself does.
$\text{bobs up and down in place}$
The energy moves on.
Write the wave equation.
$v = f\lambda$
Frequency times wavelength.
Substitute the values.
$v = 500 \times 0.68$
Hertz times meters.
Evaluate the speed.
A wave has a frequency of $256$ Hz and a wavelength of $1.34$ m. How fast does it travel, in m/s?
Complete the worked solution: standing on a pier, a student counts $10$ waves passing a post in $40$ s, and sees that neighboring crests are $8$ m apart. Find the waves' frequency in Hz, their period in s, and their speed in m/s.
Find the frequency.
$f = \dfrac{n}{t} =$ f
Waves per second.
Find the period.
$T = \dfrac{t}{n} =$ p
Seconds per wave.
Find the speed.
$v = f\lambda =$ v
The wave equation.
Check with one wave.
$v = \dfrac{\lambda}{T}$
One wavelength per period.
Match each wave word to its meaning.
| the number of waves passing each second | the distance from one crest to the next | the height from the middle to a crest | frequency times wavelength | |
|---|---|---|---|---|
| frequency | ||||
| wavelength | ||||
| amplitude | ||||
| wave speed |
A musical note of $440$ Hz travels through air at $343$ m/s. Fill in its wavelength in m, the wavelength in m of a note with twice the frequency, and the note's period in s.
| value | |
|---|---|
| wavelength (m) | |
| wavelength at twice the frequency (m) | |
| period (s) |
Sound travels through fresh water at $1480$ m/s. Write the wavelength of a sound, in m, as a function of its frequency $f$ in hertz.
Answer:
waves on a lake travel at $12$ m/s with crests $30$ m apart. What is their frequency, in Hz?
Answer: Hz
the FM station WNYC in New York broadcasts at $93.9$ MHz, which is $93900000$ Hz. Radio waves travel at $3 \times 10^8$ m/s. What is the wavelength of its signal, in m?
Answer: m
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Sound travels through seawater at $1500$ m/s. Write the wavelength of a sound, in m, as a function of its frequency $f$ in hertz.
Answer:
You can use the wave equation. Explain to someone why a high note and a low note from the same piano reach your ear at the same moment.
30. Your turn: a sound wave has a frequency of $500$ Hz and a wavelength of $0.68$ m. How fast does it travel?, step 3
$v = 340\ \text{m/s}$
Sound in air.