Oscillations of a stretched string find application when we hear soothing music coming out of a guitar and a violin. Do you know all these musical instruments work on the principle of standing waves? When we strike the strings of a guitar, the pulse producing it will go to and fro so that the entire string is divided into certain vibrating segments. These vibrations are called standing waves.

Standing (stationary) waves are produced when two identical waves of the same frequency travelling in opposite directions superpose on each other. Due to this, the energy transfer stops completely. Hence energy possessed by individual particles remains constant. How does a change in the position of fingers on the strings of a guitar or a violin result in a change in sound frequency?

The fingers resist the vibration of string which produces nodes that are zero displacement positions. There are several modes of vibration of a string, and each mode of vibration corresponds to a different standing wave. Did you know the properties of standing waves have a direct bearing on musical sounds? Just listen to the varying notes the player is playing! She manages to play a different note merely by pressing on the string at different locations. To know the physics behind it, let’s investigate the vibration modes, the genesis of which lies in standing waves. 

A standing wave is formed by the interaction of two waves of the same amplitude and wavelength, travelling in opposite directions in a linear medium of infinite length. Now the question is whether only one vibrating source can produce a standing wave in a linear medium and what are the different modes of vibration of a stretched string?

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Standing Waves or Stationary Waves

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

Practically, a stationary wave is formed when a wave train is reflected at a rigid boundary. The incident and reflected waves interfere to produce a stationary wave.

Analysis of Standing waves

1. Suppose that the two superimposing waves \({y_1} = a\sin (\omega t – kx)\) are incident wave and reflected wave \({y_2} = a\sin (\omega t + kx)\)

(As \({y_2}\) is the displacement due to a reflected wave from a boundary)

Then by the principle of superposition \(y = {y_1} + {y_2} = a[\sin (\omega t – kx) + \sin (\omega t + kx)]\)

(By using \(\sin C + \sin D = 2\sin \frac{{C + D}}{2}\cos \frac{{C – D}}{2}\)

\( \Rightarrow y = 2a\cos kx\sin \omega t\)(If reflection takes place from the rigid end, then the equation of stationary wave will be \(y = 2a\sin kx\cos \omega t)\)

2. As this equation \(\frac{{{\partial ^2}y}}{{\partial {t^2}}} = {v^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}},\) satisfies the wave equation, It represents a wave.
3. As it is not of form, \(f(ax \pm bt),\) the wave is not progressive.
4. The amplitude of the wave \({A_{sw}} = 2a\cos kx.\)

Amplitude in Two Different Cases

Reflection at the open end	Reflection at the closed-end
\({A_{sw}} = 2a\cos kx\)	\({A_{SW}} = 2a\sin kx\)
Amplitude is maximum when \(\cos kx = \pm 1\) \( \Rightarrow {\rm{kx}} = 0,\pi ,2\pi , \ldots {\rm{n}}\pi .\) where \( \Rightarrow x = 0,\frac{\lambda }{2},\lambda \ldots \ldots ,\frac{{n\lambda }}{2}\) and \(k = \frac{{2\pi }}{\lambda }\)	Amplitude is maximum when \(\sin kx = \pm 1\) \( \Rightarrow kx = \frac{\pi }{2},\frac{{3\pi }}{2} \ldots ..\frac{{(2n – 1)\pi }}{2}\) \( \Rightarrow x = \frac{\lambda }{4},\frac{{3\lambda }}{4} \ldots \ldots \) Where \(k = \frac{{2\pi }}{\lambda }\) and \(n = 1,2,3 \ldots \)
Amplitude is minimum when \(n = 0,1,2,3 \ldots \) \(\cos kx = 0\) \( \Rightarrow kx = \frac{\pi }{2},\frac{{3\pi }}{2}, \ldots ..\frac{{(2n – 1)\pi }}{2}\) \( \Rightarrow x = \frac{\lambda }{4},\frac{{3\lambda }}{4} \ldots \ldots \)	Amplitude is minimum when \({\mathop{\rm sink}\nolimits} x = 0\) \( \Rightarrow kx = \frac{\pi }{2},\frac{{3\pi }}{2} \ldots ..\frac{{(2n – 1)\pi }}{2}\) \( \Rightarrow x = 0,\frac{\lambda }{2},\lambda \ldots \ldots .\frac{{n\lambda }}{2}\)

Parts of a Standing Wave

Nodes (N): 

The points where the amplitude is minimum are called nodes. 

(i) Distance between two successive nodes is \(\frac{\lambda }{2}.\)
(ii) Nodes are at permanent rest. 
(iii) At nodes, air pressure and density both are high.

Antinodes (A): 

The points of maximum amplitudes are called antinodes. 

(i) The distance between two successive antinodes is \(\frac{\lambda }{2}\)
(ii) At nodes, air pressure and density both are low.
(iii) The distance between a node (N) and adjoining antinode (A) is \(\frac{\lambda }{4}.\)

Characteristics of Standing Waves

1. Standing waves can be transverse or longitudinal.
2. The wave disturbance is confined to a particular area between the starting point and the reflecting point of the wave.
3. There is no forward motion of the disturbance from one particle to the adjoining particle beyond this particular region. 
4. The total energy associated with a standing wave is double the energy of each incident and reflected wave. The nodes are permanently at rest in stationary waves. So, no energy can be transmitted across them, i.e. the energy of one region (segment or loop) is confined in that region. However, this energy oscillates between elastic potential energy and kinetic energy of particles of the medium. 
5. The medium splits up into several loops or segments. Each loop is vibrating up and down as a whole.
6. All the particles in one particular loop vibrate in the same phase. Particles in two consecutive segments differ in phase by \({180^ \circ }\) or \({\pi ^c}\).
7. All the particles execute simple harmonic motion about their mean position with the same time period except those at nodes,.
8. The amplitude of vibration of particles varies from zero at nodes to the maximum at antinodes \(\left( {2a} \right).\)
9. All points (except nodes) of a stationary wave pass their mean position twice in one time period. 
10. The velocity of particles while crossing the mean position varies from a maximum \(\left( {\omega {A_{sW}} = \omega .2a} \right)\) at antinodes to zero at nodes.
11. The resultant amplitude at nodes will be minimum (but not zero) in standing waves, if the amplitude of component waves are not equal. Therefore, some energy will pass across nodes, and waves will be partially standing.
12. Application of stationary waves: 
    a. Vibration in a stretched string
    b. Vibration in organ pipes (closed and open)
    c. Kundt’s tube

Progressive vs Stationary Wave

Progressive wave	Stationary wave
These waves transfer energy	This wave does not transfer energy
All particles have the same amplitude	Between a node and an antinode, all particles have different amplitudes
Over one wavelength span, all particles have different phase	Between a node and an antinode, all particles have the same phase.
No point is at rest	Nodes are always at rest
All particles do not cross the mean position simultaneously.	All particles cross the mean position simultaneously.

Standing Waves on a String

1. Consider a string of length l, stretched under tension T between two fixed points. 
2. If the string is plucked and released, a transverse harmonic wave propagates along its length and is reflected at the end. 
3. The incident and reflected waves will superimpose or interfere to produce transverse stationary waves in a string.
4. Nodes (N) are formed at a rigid end, and antinodes (A) are formed between them. 
5. Number of antinodes\(=\)number of nodes\(-1\)
6. The velocity of wave (incident or reflected wave) is given by \(v = \sqrt {\frac{T}{m}} ;\) \(m = \)mass per unit length of the wire
7. Frequency of vibration \((n) = \)frequency of the wave
   \( = \frac{v}{\lambda } = \frac{1}{\lambda }\sqrt {\frac{T}{m}} \)
8. For obtaining \(p\) loops (\(p\)-segments) in a string, it has to be plucked at a distance \(\frac{l}{{2p}}\) from one fixed end.

The fundamental mode of vibration

1. Number of loops  \(p = 1\)
2. Plucking at \(\frac{l}{2}\) (from one fixed end)
3. \(l = \frac{{{\lambda _1}}}{2} \Rightarrow {\lambda _1} = 2l\)
4. Fundamental frequency or first harmonic \({n_1} = \frac{1}{{{\lambda _1}}}\sqrt {\frac{T}{m}} = \frac{1}{{2l}}\sqrt {\frac{T}{m}} \)

The second mode of vibration (First overtone or second harmonic)

1.  Number of loops  \(p = 2\)
2. Plucking at \(\frac{l}{{2 \times 2}} = \frac{l}{4}\) (from one fixed end)
3. \(l = {\lambda _2}\)
4. Second harmonic or first overtone \({n_1} = \frac{1}{{{\lambda _1}}}\sqrt {\frac{T}{m}} = \frac{1}{l}\sqrt {\frac{T}{m}} = 2{n_1}\)

The third normal mode of vibration (Second overtone or third harmonic)

1. Number of loops \(p = 3\)
2. Plucking at \(\frac{l}{{2 \times 3}} = \frac{l}{6}\)(from one fixed end)
3. \(l = \frac{{3{\lambda _3}}}{2} \Rightarrow {\lambda _3} = \frac{{2l}}{3}\)
4. Third harmonic or second overtone\({n_3} = \frac{1}{{{\lambda _3}}}\sqrt {\frac{T}{m}} = \frac{3}{{2l}}\sqrt {\frac{T}{m}} = 3{n_1}\)

String Vibration

1. In general, if the string is plucked at length, \(\frac{l}{{2p}},\) then it vibrates in \(p\) segments (loops), and we have the \({p^{th}}\) harmonic is give \({f_p} = \frac{p}{{2l}}\sqrt {\frac{T}{m}} \)
2. All even and odd harmonics are present. Ratio of harmonic\(= 1: 2: 3…..\)
3. The ratio of overtones\(= 2: 3: 4…\)
4. The general formula for wavelength \(\lambda = \frac{{2l}}{N},\) where \(N=1,2,3,…\) correspond to \({1^{{\rm{st}}}},{2^{{\rm{nd}}}},{3^{{\rm{rd}}}}\)  modes of vibration of the string.
5. The general formula for frequency \(n = N \times \frac{v}{{2l}}\)
6. Position of nodes:\(x = 0,\frac{l}{N},\frac{2}{N},\frac{{3l}}{N} \ldots ..l\)
7. Position of antinodes:\(X = \frac{l}{{2N}},\frac{{3l}}{{2N}},\frac{{5l}}{{2N}}, \ldots .\frac{{(2N – 1)l}}{{2N}}\)

Melde’s Experiment

Melde’s experiment is an experimental representation of a transverse stationary wave. In Melde’s experiment, one end of a flexible string is tied to the end of a tuning fork. The other end passes over a smooth pulley carrying a suitable load.  If \(P\) is the number of the loop formed in stretched string and \(T\) is the tension in the string, then Melde’s law is \(p\sqrt T = {\rm{constant}} \Rightarrow \frac{{{p_1}}}{{{p_2}}} = \sqrt {\frac{{{T_2}}}{{{T_1}}}} \) (For comparing two cases)

Two arrangements of connecting a string to turning fork

Arrangement 1	Arrangement 2
Prongs of tuning fork vibrate at right angles to the thread.	Prongs vibrated along the length of the thread.
Frequency of vibration of tuning fork =  frequency of vibration of the thread.	Frequency of turning fork =2 × (Frequency of vibration of thread)
If the number of loops in the string is p, then \(l = \frac{{p\lambda }}{2} \Rightarrow \lambda = \frac{{2l}}{p}\) Frequency of string \( = \frac{v}{\lambda } = \frac{p}{2}\sqrt {\frac{T}{m}} \left( {v = \sqrt {\frac{T}{m}} } \right)\) \( \Rightarrow \) Frequency of tuning fork \( = \frac{p}{{2l}}\sqrt {\frac{T}{m}} \) \( \Rightarrow \) If \(l,m,n \to \) constant then \(p\sqrt T = \)constant	If the number of the loops  in the string is \(p,\) then \(l = \frac{{p\lambda }}{2} \Rightarrow \lambda = \frac{{2l}}{p}\) \( \Rightarrow \) Frequency of string \( = \frac{v}{\lambda } = \frac{p}{{2l}}\sqrt {\frac{T}{m}} \) Frequency of turning fork \(l = \frac{P}{l}\sqrt {\frac{T}{m}} \) If \(l,m,n \to \) constant then \(p\sqrt T = \)constant

Sonometer

Sonometer is an apparatus that works on the principle of resonance (matching frequency) of a tuning fork (or any source of sound) with stretched vibrating string. It consists of a hollow rectangular box of light wood. The experimental setup fitted on the box is as shown in the figure:

The box increases the loudness of the sound produced by the vibrating wire. If the length of the wire between the two bridges is \(l,\) then the frequency of vibration is \(n = \frac{1}{{2l}}\sqrt {\frac{T}{m}} = \sqrt {\frac{T}{{\pi {r^2}d}}} \)

(\(r=\)Radius of the wire, \(d=\)density of material of wire) \(m=\)mass per unit length of the wire)
Resonance: When a vibrating tuning fork is placed on the box, and if the length between the bridges is properly adjusted, the \({(n)_{{\rm{Fork}}}} = {(n)_{{\rm{String}}}} \to \) rider is thrown off the wire.

Laws of String

If the length of the wire between the two bridges is \(l,\) then the frequency of vibration is \(n = \frac{1}{{2l}}\sqrt {\frac{T}{m}} = \sqrt {\frac{T}{{\pi {r^2}d}}} \)

1. Law of length: If \(T\) and \(m\) are constant, then \(n \propto \frac{1}{1}\)
   \( \Rightarrow nl = {\rm{constant}} \Rightarrow {n_1}{l_1} = {n_2}{l_2}\)

2. Law of mass: If (T) and \(m\) are constant, then \(n \propto \frac{1}{{\sqrt m }}\)
\( \Rightarrow n\sqrt m = {\rm{constant}} \Rightarrow \frac{{{n_1}}}{{{n_2}}} = \sqrt {\frac{{{m_2}}}{{{m_1}}}} \)

3. Law of density: If \(T,l\) and \(r\) are constant, then \(n \propto \frac{1}{{\sqrt d }}\)
\( \Rightarrow n\sqrt d = {\rm{constant}} \Rightarrow \frac{{{n_1}}}{{{n_2}}} = \sqrt {\frac{{{d_2}}}{{{d_1}}}} \)

4. Law of tension: If \(l\) and \(m\) are constant, then \(n \propto \sqrt T \)
\( \Rightarrow \frac{n}{{\sqrt T }} = {\rm{constant}} \Rightarrow \quad \frac{{{n_1}}}{{{n_2}}} = \sqrt {\frac{{{T_2}}}{{{T_1}}}} \)

Vibration of Composite Strings

Consider two strings of different material and lengths tied between clamps and joined end to end as shown in the figure. After plucking the string, stationary waves are established only at those frequencies which match with any one harmonic of both the independent strings \({S_1}\) and \({S_1}.\)

As the frequency of the wave in both strings must be the same so:

\(\frac{p}{{2{l_1}}} = \sqrt {\frac{T}{{{m_1}}}} = \frac{q}{{2{l_2}}}\sqrt {\frac{T}{{{m_2}}}} \Rightarrow \frac{p}{q} = \frac{{{l_1}}}{{{l_2}}}\sqrt {\frac{{{m_1}}}{{{m_2}}}} = \frac{{{l_1}}}{{{l_2}}}\sqrt {\frac{{{\rho _1}}}{{{\rho _2}}}} \)

Solved Examples on Oscillations of a Stretched String

Q.1. The stationary wave produced on a string is represented by the equation\(y = 5\cos \left( {\frac{{\pi x}}{3}} \right)\sin (40\pi t)\) where \(x\) and \(y\) are in \({\rm{cm}}\) and \(t\) is in seconds. The distance between consecutive nodes is
(a) \({\rm{5}}\,{\rm{cm}}\)
(b) \(\pi \,{\rm{cm}}\)
(c) \({\rm{3}}\,{\rm{cm}}\)
(d) \(40\,{\rm{cm}}\)
Ans: (c) By comparing with the standard equation of stationary wave,
\({\rm{y}} = {\rm{a}}\cos \frac{{2\pi {\rm{x}}}}{\lambda }\sin \frac{{2\pi vt}}{\lambda },\)
We get\(\frac{{2\pi x}}{\lambda } = \frac{{\pi x}}{3} \Rightarrow \lambda = 6;\) Distance between two consecutive nodes\( = \frac{\lambda }{2} = 3\;{\rm{cm}}\)

Q.2.The equation \(y = 0.15\sin 5x\cos 300t,\) describes a stationary wave. The wavelength of the stationary wave is
(a) Zero meter
(b) \(1.256\) meter
(c) \(2.512\) meter
(d) \(0.628\) meter
Ans: (b) By comparing with standard equation \(\therefore \frac{{2\pi X}}{\lambda } = 5X \Rightarrow \lambda = \frac{2}{5} \times \pi = 1.256\) meter

Q.3. The tuning fork and sonometer wire were sounded together and produced \(4\) beats/second when the length of the sonometer wire was \({\rm{95}}\,{\rm{cm}}\) or \({\rm{100}}\,{\rm{cm}}{\rm{.}}\) The frequency of the tuning fork is
(a) \(156\;{\rm{Hz}}\)
(b) \(152\;{\rm{Hz}}\)
(c) \(148\;{\rm{Hz}}\)
(d) \(160\;{\rm{Hz}}\)
Ans: (a) Frequency \(n \propto \frac{1}{l}\therefore {\rm{As}}\,n = \frac{1}{{2l}}\sqrt {\frac{T}{m}} \) If \(n\) is the frequency of tuning fork.
\(n + {4^{ \propto \frac{1}{{95}}}} \Rightarrow n – {4^{ \propto \frac{1}{{100}}}} \Rightarrow (n + 4)95 = (n – 4)100 \Rightarrow n = 156\;{\rm{Hz}}{\rm{.}}\)

Summary

1. When two waves of the same amplitude and wavelength travelling in opposite directions meet, they give rise to a distinct pattern that appears to be standing still. Such a wave is called a standing wave.
2. There are points on the string where the particles never move. These points are called nodes.
3. The points where the displacement of the particles is the maximum are called antinodes.
4. Unlike a progressive wave, a standing wave does not transfer energy. The energy is only redistributed such that there is less energy near the nodes and more of it near the antinodes.
5. In a stretched string, only a few waves with a specific frequency will form standing waves.
6. The lowest frequency at which a standing wave occurs is called the fundamental frequency or the first harmonic. It occurs when the wavelength of the wave is twice the length of the string.
7. Higher harmonics are integral multiples of the first harmonic.

FAQ’s on Oscillations of a Stretched String

Q.1. On what factors frequency of stretched string depends?
Ans: The frequency of vibration of a stretched string \(\left( f \right)\) depends on length \(\left( l \right)\) tension \(\left( T \right)\) and mass per unit length \(\left( m \right)\) of the string.

Q.2. What kind of motion does a stretched string have?
Ans: A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch.

Q.3. Which are the laws of stretched string?
Ans: There are three laws in the case of the vibrating string. The laws of transverse vibrations of stretched strings are (i) the law of length, (ii) the law of tension and (iii) the law of mass.

Q.4. How does the frequency of sound given by a stretched string depends on its length and tension?
Ans: (a) The frequency of sound is inversely proportional to the length of the string.
(b) The sound frequency is directly proportional to the square root of the tension in the string.

Q.5. What type of wave is produced in a stretched string?
Ans: The type of wave that occurs in a string is called a transverse wave.

Q.6. Is it possible to have a longitudinal wave on a stretched string?
Ans: No, it is not possible to produce to longitudinal wave in a stretched string. That is because it is almost impossible to compress the string along its length. It will bend and produce the transverse wave.

Q.7. How many natural frequencies can a string have?
Ans: A string can have only one natural frequency with which it vibrates naturally when disturbed.

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