Standing Waves Examples in Physics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Standing Waves.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Standing waves are wave patterns that stay in place, formed when two waves of the same frequency and amplitude travel in opposite directions and interfere.

The pattern looks frozen, with points that never move and others that vibrate the most.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Standing Waves asks what oscillates, what travels, and which wave quantity is being measured.

Common stuck point: Students often know a formula related to standing waves but skip the recognition step: Am I describing a repeating disturbance using wavelength, frequency, amplitude, speed, medium, or superposition? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I describing a repeating disturbance using wavelength, frequency, amplitude, speed, medium, or superposition?

Worked Examples

Example 1

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A guitar string of length 0.650.65 m has wave speed 400400 m/s. Find the fundamental frequency.

Answer

f1≈307.7 Hzf_1 \approx 307.7 \text{ Hz}

First step

1
λ1=2L=1.30\lambda_1 = 2L = 1.30 m.

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Example 2

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A string 1.21.2 m long vibrates at 150150 Hz with wavelength 0.60.6 m. (a) What mode is this? (b) What is the wave speed?

Example 3

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A string under tension T=80T = 80 N has linear density μ=0.005\mu = 0.005 kg/m, length 1.01.0 m. Find the fundamental frequency. (Use v=T/μv = \sqrt{T/\mu}.)

Example 4

hard
Two identical waves y1=0.02sin⁡(4πx−100πt)y_1 = 0.02\sin(4\pi x - 100\pi t) and y2=0.02sin⁡(4πx+100πt)y_2 = 0.02\sin(4\pi x + 100\pi t) (SI units) superpose. Write the resulting standing wave and find the antinode amplitude.

Example 5

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A 1.51.5 m string has linear density μ=0.004\mu = 0.004 kg/m and tension T=100T = 100 N. Find the frequency of the 3rd harmonic.

Example 6

hard
A standing wave on a string is y(x,t)=0.05sin⁡(2πx)cos⁡(40πt)y(x,t) = 0.05\sin(2\pi x)\cos(40\pi t) (SI). Find (a) the wavelength, (b) the frequency, and (c) the spacing between consecutive nodes.

Example 7

challenge
A string fixed at both ends has length L=1.0L = 1.0 m and supports a standing wave at f=240f = 240 Hz with 3 internal nodes. Find the wave speed.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A standing wave on a string forms with n=1n=1. Using L=nλ2L = n\frac{\lambda}{2}, find λ\lambda if L=0.6L = 0.6 m.

Example 2

easy
What is a node in a standing wave?

Example 3

easy
What is an antinode in a standing wave?

Example 4

easy
A string of length 1 m vibrates in its second mode (n=2n=2). Find the wavelength using L=nλ2L = n\frac{\lambda}{2}.

Example 5

easy
Are standing waves a new kind of wave or an interference pattern?

Example 6

easy
How many half-wavelengths fit on a fixed-fixed string in its 3rd mode (n=3n=3)?

Example 7

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In the fundamental mode of a fixed-fixed string, how many antinodes are there?

Example 8

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On a fixed-fixed string, adjacent nodes are how far apart in terms of wavelength?

Example 9

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A fixed-fixed string of length 0.8 m supports a standing wave with 3 half-wavelengths (n=3n=3). Find the wavelength.

Example 10

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A standing wave on a 1.5 m string has wavelength 1.0 m. Which mode number nn is this? (L=nλ2L = n\frac{\lambda}{2}.)

Example 11

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A string fixed at both ends shows 4 nodes (including the ends). How many half-wavelengths fit, and what mode is it?

Example 12

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A string (fixed-fixed, length 0.5 m) has wave speed 100 m/s. Find the fundamental frequency.

Example 13

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A standing wave has nodes 0.4 m apart. What is the wavelength?

Example 14

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A 2 m string carries a standing wave at 60 Hz with wavelength 0.8 m. Find the wave speed.

Example 15

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A standing wave is set up so that the string length holds exactly 2.5 wavelengths is impossible for fixed-fixed ends. What is the smallest mode number above 2.5 half-waves... Actually: how many half-wavelengths are in 2 full wavelengths?

Example 16

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A fixed-fixed string has fundamental wavelength 1.6 m. What is the wavelength of its 4th harmonic?

Example 17

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A standing wave on a fixed-fixed string shows 5 nodes counting both ends. What mode number nn is this?

Example 18

challenge
A wire fixed at both ends (length 0.75 m) has wave speed 300 m/s. Find the frequency of its 3rd harmonic.

Example 19

challenge
Two waves y1=Asin⁡(kx−ωt)y_1 = A\sin(kx - \omega t) and y2=Asin⁡(kx+ωt)y_2 = A\sin(kx + \omega t) superpose. The resulting standing wave has the form 2Asin⁡(kx)cos⁡(ωt)2A\sin(kx)\cos(\omega t). At what positions xx (smallest positive) is the first node, given k=πk = \pi per meter?

Example 20

challenge
A string fixed at both ends has its 2nd harmonic at 220 Hz. What is the frequency of the 5th harmonic? (fn=nf1f_n = n f_1.)

Example 21

easy
A standing wave on a fixed-fixed string has fundamental wavelength λ1=2.0\lambda_1 = 2.0 m. What is the string length LL?

Example 22

easy
A fixed-fixed string of length 0.90.9 m vibrates in mode n=3n=3. Find the wavelength.

Example 23

easy
A standing wave has nodes spaced 0.250.25 m apart. Find the wavelength.

Example 24

easy
An open-open pipe of length 0.50.5 m supports a standing wave with n=2n=2. Find the wavelength using L=nλ2L = n\frac{\lambda}{2}.

Example 25

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A fixed-fixed string has fundamental frequency 100100 Hz. What is the frequency of its 4th harmonic?

Example 26

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A string fixed at both ends shows 6 antinodes. What mode nn is this?

Example 27

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A pipe closed at one end and open at the other (length 0.850.85 m) has vsound=340v_{\text{sound}} = 340 m/s. Find the fundamental frequency.

Example 28

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A closed-open pipe only supports odd harmonics. If f1=200f_1 = 200 Hz, what is the next allowed frequency?

Example 29

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On a fixed-fixed string, the 2nd harmonic has frequency 180180 Hz. What is the fundamental?

Example 30

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If the tension on a string is quadrupled, by what factor does the fundamental frequency change?

Example 31

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A standing wave on a 1.0 m fixed-fixed string has wavelength 0.40.4 m. What is the next-longer wavelength that the same string supports?

Example 32

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A string vibrates in its 3rd harmonic at 300300 Hz. The wave speed is 200200 m/s. Find the string length.

Example 33

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A fixed-fixed string at n=2n=2 has wavelength 0.50.5 m. Find the string length.

Example 34

hard
In a closed-open pipe of length LL, what is the wavelength of the 3rd allowed mode? (Modes are n=1,3,5,…n = 1, 3, 5, \ldots.)

Example 35

hard
A guitar string sounds A4 (440440 Hz) as its fundamental. The player presses a fret so the vibrating length is 80%80\% of the original. What is the new fundamental frequency?

Example 36

hard
An open-open organ pipe has fundamental 256256 Hz. The room temperature rises and the sound speed increases by 2%2\%. Estimate the new fundamental.

Example 37

hard
A standing wave on a fixed-fixed string shows 4 antinodes. The string is 0.80.8 m long. Find the wavelength.

Example 38

challenge
A closed-open pipe and an open-open pipe both have fundamental frequency 200200 Hz. The closed-open pipe is how many times the length of the open-open pipe?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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