Standing Sound Waves

Edited By Vishal kumar | Updated on Jul 02, 2025 06:19 PM IST

Standing sound waves are a fascinating phenomenon that occurs when two identical sound waves travelling in opposite directions interfere with each other, creating a pattern of nodes and antinodes. This creates the impression that the wave is "standing still" rather than moving through the medium. These waves are not just a theoretical concept but have practical applications in real life. For example, musical instruments like guitars and flutes rely on standing waves to produce harmonious sounds. When a guitar string is plucked, it vibrates, creating standing waves that determine the pitch of the note. Similarly, standing sound waves are crucial in acoustics, influencing the design of concert halls to enhance sound quality by controlling how sound waves interact within the space. In this article, we will understand how standing sound waves help us grasp how sound is produced, manipulated, and optimized in various everyday scenarios.

This Story also Contains

Standing Waves
Standing Wave in a Closed Organ Pipe
Standing Waves in Open Organ Pipes
Solved Examples Based on Standing Sound Waves
Example 1: The phase difference between the two particles situated on both sides of a node is
Summary

Standing Sound Waves

Standing Waves

When two sets of progressive waves of the same type (both longitudinal or both transverse) having the same amplitude and the same time period or frequency or wavelength travelling along the same straight line with the same speed in opposite directions superimpose, a new set of waves are formed. These are called stationary waves.

Some of the Characteristics of Standing Waves

(1) In this the disturbance is confined to a particular region between the starting point and the reflecting point of the wave.
(2) In this there is no forward motion of the disturbance from one particle to the adjoining particle and so on, beyond this particular region.
(3) The total energy in the stationary wave is twice the energy of each incident and reflected wave. But there is no flow or transference of energy along the stationary wave.
(4) Points in a standing wave, which are permanently at rest. These are called nodes. The distance between two consecutive nodes is λ2

(5) The Points on the standing wave having maximum amplitude are known as antinodes. The distance between two consecutive antinodes is also λ2
(6) All the particles execute simple harmonic motion about their mean position (except those at nodes) within the same time period.

Note - In standing waves, if the amplitude of component waves is not equal. The resultant amplitude at nodes will not be zero. It will be a minimum. Because of this, some energy will pass across nodes and waves will be partially standing.

Let us take an example to understand and derive the equation of standing wave

Let us take a string and when a string is under tension and set into vibration, transverse harmonic waves propagate along its length. If the length of the string is fixed, reflected waves will also exist. These incident and reflected waves will superimpose to produce transverse stationary waves in a string

Incident wave y1=asin⁡2πλ(vt+x)
Reflected wave y2=asin⁡2πλ[(vt−x)+π]=−asin⁡2πλ(vt−x)

Now we can apply the principle of superposition to this and get

y=y1+y2=2acos⁡2πvtλsin⁡2πxλ

Standing Wave in a Closed Organ Pipe

Organ pipes are musical instruments which are used for producing musical sound by blowing air into the pipe. In this longitudinal stationary waves are formed due to superimposition of incident and reflected longitudinal waves.

A closed organ pipe is a cylindrical tube having an air column with one end closed. Sound waves enter from a source vibrating near the open end. An ongoing pressure wave gets reflected from the fixed end. This inverted wave is again reflected at the open end. After two reflections, it moves towards the fixed end and interferes with the new wave sent by the source in that direction. The twice-reflected wave has travelled an extra distance of 2l causing a phase advance of 2πλ⋅2l=4πlλ
Similarly at open ends, the twice reflected wave suffered a phase change of π at the open end.

So the phase difference is δ=4πlλ+π. Also, the waves interfere constructively if the phase difference is 2nπ

4πlλ+π=2nπl=(2n−1)λ4

Here n = 1,2,3... But if we take n = 0,1,2,.... then the above equation can also be written as l=(2n−1)λ4

So, the frequency can be written as ν=vλ=v⋅(2n−1)4l

Equation of standing wave is given by and explained earlier =y=2acos⁡2πtλsin⁡2πxλ
As, general formula for wavelength defined earlier =λ=4L(2n−1)
The minimum allowed frequency is obtained by putting n=1

(1) First normal mode of vibration : n1=v4L

This is called fundamental frequency. The note so produced is called the fundamental note or first harmonic.
(2) Second normal mode of vibration : n2=vλ2=3v4L=3n1

This is called the third harmonic or first overtone.
(3) Third normal mode of vibration : n3=5v4L=5n1

This is called the fifth harmonic or second overtone.

Standing Waves in Open Organ Pipes

The general formula for wavelength λ=2Ln where n=1,2,3………

Then the first normal mode of vibration is n1=vλ1=v2L

This is called fundamental frequency and the node so produced is called fundamental node or first harmonic.

(2) Second normal mode of vibration n2=vλ2=vL=2(v2L)=2n1⇒n2=2n1

This is called the second harmonic or first overtone.
(3) Third normal mode of vibration n3=vλ3=3v2L,n3=3n1

This is called the third harmonic or second overtone.

Solved Examples Based on Standing Sound Waves

Example 1: The phase difference between the two particles situated on both sides of a node is

1) 0∘
2) 90∘
3) 180∘
4) 360∘

Solution:

Standing wave

Two identical waves travel in opposite directions in the same medium and combine to form a stationary wave.

All the particles vibrating in a given loop are in the same phase and particles in two consecutive loops are in opposite phase to each other.

So two particles situated on both the sides of node will be in opposite phases of vibration since they correspond to different but consecutive loops.

Hence, the answer is the option (3).

Example 2: y=5cos⁡πx25sin⁡100πt. A node will not occur at a distance x is equal to in a stationary wave ( answer in cms)

1) 25

2) 62.5

3) 12.5

4) 37.5

Solution:

The node in the standing wave

The positions at which the amplitude of oscillation has zero value are called nodes.

wherein

At nodes,

sin⁡kx=0kx=0,π,2π……nπ
nodes will occur, where
πX25=π2,3π2,5π2 etc. X=12.5,37.5,62.5etc

Hence, the answer is the option (1).

Example 3: Tube A has both ends open while tube B has one end closed, otherwise, they are identical. The ratio of the fundamental frequency of tubes A and B

1) 1:2
2) 1:4
3) 2:1
4) 4:1

Solution:

Closed organ pipe

A closed organ pipe is a cylindrical tube having an air column with one end closed.

wherein

Condition of constructive interference

l=(2n+1)λ4

In tube A,λA=2l
In tube BλB=4l

∴vA=νλA=ν2l∴vB=νλB=ν4lvAvB=21=2:1

Hence, the answer is the option (3).

Example 4: A closed organ pipe has a length ‘l'. The air in it is vibrating in 3rd overtone with maximum amplitude 'a'. Find the amplitude at a distance of l /7 from the closed end of the pipe.

1) a

2) a2

3) 3a2

4) a2

Solution:

The frequency in a closed organ pipe

ν=(2n+1)V4ln=0,1,2,3……
wherein

V= velocity of sound wave l= length of pipe n= number of overtones

The figure shows the variation of displacement of particles in a closed organ pipe for 3rd overtone.

For the third overtone l=7λ4 or λ=4l7 or λ4=l7

Hence the amplitude at P at a distance l7 from the closed end is ‘a’ because there is an antinode at that point

Alternate

Because there is a node at x = 0 the displacement amplitude as a function of x can be written as

A=asin⁡kx=asin⁡2πλX

For the third overtone l=7λ4 or λ=4l7
∴A=asin⁡7π2ll7=asin⁡π2=a at x=l7⇒A=a

Hence, the answer is the option (1).

Summary

If two waves with the same frequency and amplitude are moving in opposite directions and interfere with each other, then standing waves will be formed. It causes an interference pattern to form, wherein some points stay fixed and others oscillate with maximum amplitude, called antinodes. Standing sound waves are very important for musical instruments because they determine the pitch and timbre of the sound produced. They are equally important in many engineering applications for the design of acoustic devices or in searching for an understanding of resonance in structures.

Frequently Asked Questions (FAQs)

1. What determines the frequency of a standing sound wave in a closed pipe?

The frequency of a standing sound wave in a closed pipe is determined by the length of the pipe and the speed of sound in the medium inside the pipe. The fundamental frequency is given by f = v/4L, where v is the speed of sound and L is the length of the pipe.

2. What is the role of boundary conditions in forming standing sound waves?

Boundary conditions determine how waves reflect at the ends of a medium, influencing the formation of standing waves. They dictate which frequencies can resonate and where nodes and antinodes occur, shaping the possible standing wave patterns.

3. How do standing sound waves contribute to the formation of sound in wind instruments?

In wind instruments, the player's breath creates a vibrating air column, forming standing waves within the instrument's body. The length and shape of the instrument determine which frequencies (and thus which notes) can form as standing waves, producing the instrument's characteristic sound.

4. How do standing sound waves differ in open and closed pipes?

In an open pipe, both ends are antinodes (pressure nodes), allowing all harmonics to form. In a closed pipe, one end is a node and the other an antinode, permitting only odd harmonics. This results in different resonant frequencies and tonal qualities.

5. What is the significance of the Doppler effect in standing sound waves?

The Doppler effect typically doesn't apply to standing sound waves because they don't propagate through space. However, if the source or reflector of the waves is moving, it can affect the formation and characteristics of the standing wave pattern.

6. What is a standing sound wave?

A standing sound wave is a pattern of vibration that occurs when two waves of the same frequency traveling in opposite directions interfere with each other. This results in fixed points of no vibration (nodes) and maximum vibration (antinodes) along the wave.

7. How do standing sound waves differ from traveling sound waves?

Standing sound waves have fixed nodes and antinodes, while traveling sound waves move through a medium. In standing waves, energy doesn't propagate; it oscillates between potential and kinetic forms at fixed locations. Traveling waves, however, transfer energy through the medium.

8. What are nodes and antinodes in a standing sound wave?

Nodes are points of minimum displacement where the wave appears stationary. Antinodes are points of maximum displacement where the wave oscillates with the largest amplitude. In a standing sound wave, these points remain fixed in space.

9. How are standing sound waves created in musical instruments?

Standing sound waves in musical instruments are created by reflecting sound waves at the ends of the instrument (like the ends of a flute or guitar string). The interference between the original and reflected waves forms a standing wave pattern, producing specific musical notes.

10. What is the relationship between standing sound waves and sound intensity?

The intensity of sound in a standing wave varies with position. At nodes, the intensity is minimum (ideally zero), while at antinodes, the intensity is maximum. This creates a pattern of loud and quiet spots in the medium.

11. How does the wavelength of a standing sound wave relate to the length of an open pipe?

In an open pipe, the length of the pipe is equal to half the wavelength of the fundamental standing wave. For higher harmonics, the pipe length is equal to whole number multiples of half-wavelengths.

12. How do standing sound waves contribute to resonance?

Standing sound waves create resonance when the frequency of the wave matches the natural frequency of the system. This leads to constructive interference and amplification of the wave, resulting in a much larger amplitude of vibration.

13. What is the fundamental frequency in a standing sound wave?

The fundamental frequency is the lowest frequency at which a standing wave can form in a given system. It corresponds to the longest wavelength that fits the boundary conditions of the system, such as the length of a pipe or string.

14. What is the difference between harmonics and overtones in standing sound waves?

Harmonics are integer multiples of the fundamental frequency, including the fundamental itself. Overtones are higher frequencies above the fundamental. In standing sound waves, all harmonics are overtones, but not all overtones are harmonics (in some systems).

15. How do temperature changes affect standing sound waves?

Temperature changes affect the speed of sound in a medium. As temperature increases, the speed of sound typically increases. This can alter the wavelength and frequency of standing sound waves, potentially changing the resonant frequencies of an instrument or cavity.

16. How do standing sound waves contribute to the phenomenon of acoustic streaming?

While acoustic streaming primarily involves fluid motion induced by traveling waves, standing waves can also generate streaming effects. The time-averaged acoustic energy in a standing wave field can drive steady fluid currents, particularly near boundaries or in the presence of obstacles.

17. How do standing sound waves relate to the concept of acoustic holography?

Acoustic holography often involves analyzing complex sound fields, which can include standing wave components. Understanding standing wave patterns can be crucial in reconstructing and visualizing sound fields in acoustic holography applications.

18. How do standing sound waves relate to the concept of phononic crystals?

Phononic crystals are designed to control sound wave propagation through periodic structures. The formation and manipulation of standing waves within these structures are key to their function, allowing them to create band gaps and control the flow of acoustic energy.

19. How do standing sound waves relate to the concept of acoustic diodes?

Acoustic diodes are devices that allow sound to pass in one direction but not the other. Some designs of acoustic diodes utilize standing wave patterns in conjunction with nonlinear materials to achieve this directional behavior.

20. How do standing sound waves relate to the concept of phonon confinement?

Phonon confinement occurs when sound waves (phonons) are restricted in nanostructures. The confined space creates standing wave patterns that quantize the allowed phonon energies, affecting thermal and acoustic properties of the material.

21. What is the significance of standing sound waves in non-destructive testing?

In non-destructive testing, understanding standing wave patterns can be crucial for interpreting results, especially in resonance-based techniques. Standing waves can reveal information about material properties and structural integrity.

22. How do standing sound waves relate to the concept of acoustic black holes?

Acoustic black holes are structures designed to trap and absorb sound waves. They often utilize carefully designed geometries that support specific standing wave patterns to gradually slow down and eventually absorb incoming sound waves.

23. How do standing sound waves relate to the concept of beats?

While standing waves and beats are different phenomena, they both involve wave interference. Beats occur when two waves of slightly different frequencies interfere, creating a pulsating sound. Standing waves, however, result from waves of the same frequency interfering in opposite directions.

24. What is the relationship between standing sound waves and sound absorption?

Sound absorption can affect the formation and quality of standing waves. Materials that absorb sound energy can reduce the amplitude of reflected waves, potentially weakening or preventing the formation of standing waves. This principle is used in acoustic treatments to control room resonances.

25. How do standing sound waves relate to the concept of acoustic impedance?

Acoustic impedance affects how sound waves reflect at boundaries, which is crucial for forming standing waves. Changes in impedance can alter the reflection coefficient, affecting the amplitude and phase of reflected waves and thus the standing wave pattern.

26. What is the importance of standing sound waves in acoustic levitation?

Acoustic levitation uses standing sound waves to create pressure nodes where small objects can be suspended. By carefully controlling the sound field, objects can be trapped at these nodes, counteracting gravity and allowing them to "float" in air.

27. How do standing sound waves contribute to the phenomenon of sonoluminescence?

Sonoluminescence occurs when standing sound waves in a liquid create and collapse tiny bubbles, producing brief flashes of light. The standing wave pattern concentrates acoustic energy, driving the extreme conditions necessary for this phenomenon.

28. What is the relationship between standing sound waves and sound diffraction?

While diffraction is more relevant to traveling waves, it can affect the formation of standing waves in certain scenarios. Diffraction around obstacles or through apertures can alter wave patterns, potentially influencing the standing wave formation in complex geometries.

29. How do standing sound waves relate to the concept of acoustic radiation pressure?

Standing sound waves can create regions of varying acoustic radiation pressure. This pressure is typically strongest at the antinodes and weakest at the nodes. In intense standing wave fields, this pressure can be used to manipulate small particles or droplets.

30. What is the significance of standing sound waves in architectural acoustics?

In architectural acoustics, understanding standing waves is crucial for designing spaces with good sound quality. Room modes (standing waves in rooms) can cause uneven sound distribution and frequency response, which architects and acousticians must address in their designs.

31. What is the relationship between standing sound waves and sound scattering?

Sound scattering can affect the formation and stability of standing waves. When sound waves encounter objects or irregularities in a medium, scattering can disrupt the clean interference pattern needed for standing waves, potentially altering or weakening the standing wave field.

32. What is the importance of standing sound waves in ultrasonic imaging?

While ultrasonic imaging primarily uses pulsed waves, understanding standing wave phenomena is important for interpreting results and avoiding artifacts. Standing waves can form between the transducer and strong reflectors, potentially affecting image quality.

33. How do standing sound waves contribute to the design of acoustic metamaterials?

Acoustic metamaterials often exploit standing wave patterns within their structure to achieve unusual acoustic properties. By carefully designing the geometry and materials, researchers can create metamaterials that manipulate standing waves to control sound in unprecedented ways.

34. What is the relationship between standing sound waves and acoustic cloaking?

Acoustic cloaking techniques often involve manipulating the propagation and scattering of sound waves. Understanding standing wave patterns is crucial in designing structures that can guide waves around objects, making them "invisible" to sound.

35. What is the significance of standing sound waves in musical acoustics?

Standing sound waves are fundamental to musical acoustics. They determine the resonant frequencies of instruments, affect timbre, and influence how instruments project sound. Understanding standing waves is crucial for instrument design and performance techniques.

36. How do standing sound waves contribute to the phenomenon of acoustic cavitation?

Acoustic cavitation occurs when high-intensity sound waves create and collapse bubbles in a liquid. Standing wave patterns can concentrate acoustic energy at antinodes, promoting cavitation at these locations. This is important in applications like ultrasonic cleaning and sonochemistry.

37. What is the relationship between standing sound waves and acoustic levitation in fluids?

Acoustic levitation in fluids, like in air, relies on standing wave patterns. In fluids, these patterns can create stable positions where buoyancy forces and acoustic radiation forces balance, allowing objects to be suspended or manipulated within the fluid.

38. How do standing sound waves relate to the concept of acoustic metamaterials?

Acoustic metamaterials often utilize subwavelength structures to manipulate sound waves. The interaction between these structures and standing wave patterns can lead to exotic acoustic properties, such as negative refractive index or superlensing effects.

39. What is the importance of standing sound waves in noise control engineering?

In noise control, understanding standing waves is crucial for designing effective sound absorption and isolation systems. Standing waves can create resonances in enclosed spaces, which need to be addressed to achieve optimal noise reduction.

40. How do standing sound waves contribute to the phenomenon of thermoacoustics?

Thermoacoustic devices use standing sound waves to pump heat or generate sound from heat. The pressure and velocity oscillations in the standing wave drive the thermodynamic cycle that makes these devices work.

41. What is the relationship between standing sound waves and acoustic emission testing?

While acoustic emission testing primarily deals with transient elastic waves, understanding standing wave patterns can be important in interpreting results, especially in enclosed structures where reflections and resonances can affect the detected signals.

42. What is the significance of standing sound waves in seismology?

While seismic waves are typically studied as traveling waves, standing wave patterns can form in certain geological structures. Understanding these patterns can be important for interpreting seismic data and studying Earth's structure.

43. How do standing sound waves contribute to the design of acoustic lenses?

Acoustic lenses often manipulate wave propagation to focus or redirect sound. Understanding standing wave patterns within lens structures can be crucial for designing lenses that effectively control sound waves at specific frequencies.

44. What is the relationship between standing sound waves and acoustic cloaking devices?

Acoustic cloaking devices aim to guide sound waves around objects, making them "invisible" to sound. Designing these devices often involves manipulating standing wave patterns to achieve the desired wave guidance.

45. What is the importance of standing sound waves in ultrasonic welding?

Ultrasonic welding uses high-frequency sound waves to join materials. Standing wave patterns in the welding tool and materials being joined are crucial for concentrating energy at the weld interface, enabling effective bonding.

46. How do standing sound waves contribute to the phenomenon of sonochemistry?

In sonochemistry, chemical reactions are driven by ultrasound. Standing wave patterns in reaction vessels can create regions of high acoustic intensity (at antinodes) where cavitation occurs, driving sonochemical reactions.

47. What is the relationship between standing sound waves and acoustic holograms?

Acoustic holograms use interference patterns to create three-dimensional sound fields. Understanding and manipulating standing wave patterns is crucial for designing acoustic holograms that can shape sound in complex ways.

48. How do standing sound waves relate to the concept of phononic waveguides?

Phononic waveguides control the propagation of sound waves along specific paths. The design of these waveguides often involves creating structures that support specific standing wave patterns to guide sound in desired directions or with particular properties.

49. How do standing sound waves contribute to the design of acoustic filters?

Acoustic filters often utilize resonant cavities or structures that support specific standing wave patterns. By designing these patterns, filters can be created to selectively transmit or attenuate certain frequencies of sound.

50. What is the relationship between standing sound waves and acoustic tweezers?

Acoustic tweezers use sound waves to manipulate small particles. Many designs rely on creating standing wave patterns where particles can be trapped at pressure nodes or antinodes, allowing precise control over their position.