Grade by Grade
E. Waves and Sound · Waves and Sound

Standing Waves on Strings

E3.4 E3.3 E2.2 E2.1About 70 minutes
Learning goals
  • I can explain how a standing wave forms from two identical waves travelling in opposite directions.
  • I can identify nodes and antinodes and use the fact that nodes are λ/2 apart.
  • I can state the condition for standing waves on a string fixed at both ends: L = nλ/2.
  • I can calculate the fundamental frequency and harmonics of a string.

Words to know

standing wave
An interference pattern made by two identical waves travelling in opposite directions. Nodes and antinodes stay in fixed places, so the wave seems to "stand still".
node
A point on a standing wave that never moves (continuous destructive interference).
antinode
A point midway between nodes that vibrates with the largest amplitude (continuous constructive interference).
fundamental frequency (f₁)
The lowest frequency at which a standing wave forms; also called the first harmonic.
harmonic
A standing-wave frequency that is a whole-number multiple of the fundamental: f_n = n f₁.
overtone
Any harmonic above the fundamental. The 2nd harmonic is the 1st overtone, the 3rd harmonic is the 2nd overtone, and so on.
loop (segment)
The part of a standing wave between two neighbouring nodes; one loop is half a wavelength long.

Waves that stay put

Pluck a guitar string and look closely: the string becomes a blur that is widest in the middle and does not move at all at the ends. Nothing seems to travel along it. But there are waves on that string, racing back and forth and reflecting from the ends. What you see is their superposition, a pattern called a standing wave. Standing waves are how every string and wind instrument picks its notes.

How a standing wave forms

Send a periodic wave down a string tied at the far end. It reflects (inverted) and travels back, overlapping the waves still coming in. Now two identical waves are moving in opposite directions through each other. At certain points they always cancel; at others they always reinforce.

  • Nodes (N): points that never move. They are spaced λ/2 apart.
  • Antinodes (A): points halfway between nodes that swing with the biggest amplitude (twice the amplitude of each travelling wave).
  • The distance from a node to the next antinode is λ/4.

All the points in one loop move up together and down together. Points in neighbouring loops move in opposite directions.

nodenodenodeantinodeantinodenode to node = λ/2
Snapshots of a standing wave taken at different moments in half a period (solid, dashed, dotted). The nodes never move. The antinodes swing the most. The distance between neighbouring nodes is half a wavelength.

Which frequencies make standing waves?

A string fixed at both ends must have a node at each end. So a whole number of loops must fit exactly on the string. Each loop is λ/2 long, so

L = n (λ/2), which gives λₙ = 2L/n (n = 1, 2, 3, …)

Using v = fλ, the allowed frequencies are

fₙ = n v/(2L) = n f₁

  • n = 1: the fundamental (first harmonic), f₁ = v/(2L). One loop.
  • n = 2: the second harmonic (first overtone), 2f₁. Two loops.
  • n = 3: the third harmonic (second overtone), 3f₁. Three loops.

These are the string's natural frequencies (or resonant frequencies). Shake the string at any other frequency and the reflections do not line up, so no steady pattern forms.

An = 11st harmonicL = λ/2NAAn = 22nd harmonicL = λ = 2(λ/2)NNAAAn = 33rd harmonicL = 3(λ/2)
The first three standing-wave patterns (harmonics) on a string fixed at both ends. N = node, A = antinode. A whole number of half-wavelengths always fits between the fixed ends.

Making a string play a higher note

Since f₁ = v/(2L), a string plays a higher fundamental when:

  • it is shorter (pressing a guitar string against a fret shortens the vibrating part), or
  • the wave speed on it is higher, which happens when the string is tighter (turning a tuning peg) or lighter (thin strings play higher notes than thick ones).

A real plucked string vibrates in several harmonics at once. The mix of harmonics gives each instrument its own sound quality (timbre), which is why a violin and a guitar playing the same note sound different.

Worked example

Example 1: A guitar string

A guitar string is 0.65 m long between its fixed ends. Waves travel along it at 143 m/s. Find (a) the wavelength of the fundamental, (b) the fundamental frequency, and (c) the frequencies of the second and third harmonics.

  1. (a) For n = 1: λ₁ = 2L = 2 × 0.65 m = 1.3 m.
  2. (b) f₁ = v/λ₁ = 143 ÷ 1.3 = 110 Hz.
  3. (c) Harmonics are whole-number multiples: f₂ = 2 × 110 = 220 Hz; f₃ = 3 × 110 = 330 Hz.

λ₁ = 1.3 m; f₁ = 110 Hz; f₂ = 220 Hz; f₃ = 330 Hz.

Worked example

Example 2: From a pattern to a speed

A vibrator shakes a 1.8 m string (fixed at both ends) at 60 Hz, and a standing wave with 4 loops forms. Find the distance between nodes, the wavelength and the wave speed.

  1. Four loops fit in 1.8 m, so each loop (node to node) is 1.8 ÷ 4 = 0.45 m.
  2. Each loop is λ/2, so λ = 2 × 0.45 = 0.90 m.
  3. v = fλ = 60 × 0.90 = 54 m/s.

Nodes 0.45 m apart, λ = 0.90 m, v = 54 m/s. (This is the fourth harmonic.)

Worked example

Example 3: Next harmonic

A string's standing wave has 3 loops at 270 Hz. What frequency gives 5 loops?

  1. 3 loops is the 3rd harmonic, so f₁ = 270 ÷ 3 = 90 Hz.
  2. 5 loops is the 5th harmonic: f₅ = 5 × 90 = 450 Hz.

450 Hz

Just for fun

Why did the guitar string refuse to go to the party?

It didn't want to be fretted over. (Also, it could only make it at certain frequencies.)

Hands-on activity

Standing waves on a skipping rope

You need: a skipping rope or a 3 m to 4 m rope, a partner (or a door handle to tie one end to), and a metronome app or a stopwatch.

  1. Hold one end each with the rope slightly slack. Your partner holds their end still. You move your hand up and down in small, steady circles or shakes.
  2. Start slowly. Find the frequency where the whole rope swings as one big loop (the fundamental). Count shakes in 10 s and calculate f₁.
  3. Shake about twice as fast. Can you make two loops with a still point (node) in the middle? Count again. Is f₂ about 2f₁?
  4. Try for three loops. (It is hard work!)
  5. Measure the rope length and use λ = 2L/n and v = fλ to find the wave speed for each pattern. Do you get roughly the same speed each time?

What to look for: the frequencies should be close to 1 : 2 : 3. A floor rope typically gives f₁ around 1 Hz to 2 Hz. The calculated speeds should roughly agree, because the rope (the medium) did not change. Your hand end is not quite a node, which causes some error.

Your turn

1

What is the distance between two neighbouring nodes in terms of λ?

2

What is the distance from a node to the next antinode?

3

A string fixed at both ends is 1.2 m long. Find the wavelength of the fundamental.

4

For the same string, find the wavelength of the 3rd harmonic.

5

A string has a fundamental of 196 Hz. Find the frequency of its 2nd and 4th harmonics.

6

A standing wave has nodes 25 cm apart and the frequency is 120 Hz. Find v.

7

Name two ways a musician can raise the pitch of a string.

More practice: warm-up

1

What two things must overlap to make a standing wave?

2

What is the fundamental frequency also called?

3

Which harmonic is the first overtone?

4

How many loops does the 3rd harmonic have on a string fixed at both ends?

5

If f₁ = 100 Hz, what is f₅?

More practice: on your own

1

A 0.75 m string has a wave speed of 300 m/s. Find its first three harmonic frequencies.

2

A violin string plays 440 Hz as its fundamental and is 0.33 m long. Find the wave speed on it.

3

A guitarist presses a 0.64 m string so only 0.48 m vibrates. If the open string's fundamental is 330 Hz, what is the new fundamental? (Same tension.)

4

A 2.0 m rope fixed at both ends shows a standing wave with 5 nodes, counting both ends. Which harmonic is this and what is λ?

5

Explain why only certain frequencies make standing waves on a string with fixed ends.

Challenge

1

Two neighbouring harmonics of a string fixed at both ends are 420 Hz and 490 Hz. Find the fundamental and identify which harmonics these are.

2

A wave speed on a string is given by v = √(F_T/μ), where F_T is tension and μ is mass per metre. If the tension is increased by a factor of 4, by what factor does the fundamental frequency change? Explain using f₁ = v/(2L).

3

A 1.50 m string vibrates in its 3rd harmonic. Where are the nodes and antinodes (measured from one end)?

Check your understanding

Quiz

1.

A standing wave is produced by…

2.

The distance between neighbouring nodes is…

3.

For a string fixed at both ends, the fundamental wavelength is…

4.

A string's fundamental is 150 Hz. Its third harmonic is…

5.

Which change would lower the fundamental frequency of a guitar string?

6.

The point on a standing wave that vibrates with the largest amplitude is a(n)…

For parents and teachers

Big idea: expectations E3.3 and E3.4 ask students to explain standing waves using superposition and to state the conditions needed for them. The key result is L = nλ/2 for a string fixed at both ends.

How to help: Making standing waves on a skipping rope is fun and convincing. If you have a guitar or ukulele, gently touch a string at its exact midpoint while plucking: you will hear the higher second harmonic (a "harmonic" note).

Common difficulties:

  • Thinking the node spacing is λ (it is λ/2).
  • Counting nodes instead of loops: the number of loops is the harmonic number.
  • Confusing harmonics and overtones: the nth harmonic is the (n − 1)th overtone.