(1) It is an apparatus, used to produce resonance (matching frequency) of tuning fork (or any source of sound) with stretched vibrating string.
(2) It consists of a hollow rectangular box of light wood. The experimental fitted on the box as shown.
(3) The box serves the purpose of increasing the loudness of the sound produced by the vibrating wire.
(4) 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)
(5) Resonance : When a vibrating tuning fork is placed on the box, and if the length between the bridges is properly adjusted then if \[{{(n)}_{Fork}}={{(n)}_{String}}\to \] rider is thrown off the wire.
(6) Laws of string
(i) Law of length : If T and m are constant then \[n\propto \frac{1}{l}\]
\[\Rightarrow \] nl = constant \[\Rightarrow \] \[{{n}_{1}}{{l}_{1}}={{n}_{2}}{{l}_{2}}\]
(ii) Law of mass : If T and l are constant then \[n\propto \frac{1}{\sqrt{m}}\]
\[\Rightarrow \] \[n\sqrt{m}=\]constant \[\Rightarrow \] \[\frac{{{n}_{1}}}{{{n}_{2}}}=\sqrt{\frac{{{m}_{2}}}{{{m}_{1}}}}\]
(iii) Law of density : If T, l and r are constant then \[n\propto \frac{1}{\sqrt{d}}\]
\[\Rightarrow \] \[n\sqrt{d}=\] constant \[\Rightarrow \] \[\frac{{{n}_{1}}}{{{n}_{2}}}=\sqrt{\frac{{{d}_{2}}}{{{d}_{1}}}}\]
(iv) Law of tension : If l and m are constant then \[n\propto \sqrt{T}\] \[\Rightarrow \] \[\frac{n}{\sqrt{T}}=\]constant
\[\Rightarrow \] \[\frac{{{n}_{1}}}{{{n}_{2}}}=\sqrt{\frac{{{T}_{2}}}{{{T}_{1}}}}\]
(1) It is an experimental representation of transverse stationary wave.
(2) In Melde's experiment, one end of a flexible piece of string is tied to the end of a tuning fork. The other end passes over a smooth pulley carries a suitable load.
(3) If p is the number of loop's formed in stretched string and T is the tension in the string then Melde's law is \[p\sqrt{T}=\]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
Transversely
Example
Prongs of tuning fork vibrates at right angles to the thread.
Prongs vibrated along the length of the thread.
Frequency of vibration of turning fork: frequency of vibration of the thread.
Frequency of turning fork \[=2\times \] (Frequency of vibration of thread)
If number of loops in string is P then
\[l=\frac{p\lambda }{2}\Rightarrow \lambda =\frac{2l}{p}\]
\[\Rightarrow \] Frequency of string
\[=\frac{v}{\lambda }=\frac{p}{2l}\sqrt{\frac{T}{m}}\]\[\left( \because \,v=\sqrt{\frac{T}{m}} \right)\]
\[\Rightarrow \] Frequency of tuning fork
\[=\frac{p}{2l}\sqrt{\frac{T}{m}}\]
\[\Rightarrow \] If l, m, n \[\to \] constant then \[p\sqrt{T}=\] constant
It number of loop so in string is p then
\[l=\frac{p\lambda }{2}\Rightarrow \,\lambda =\frac{2l}{p}\]
\[\Rightarrow \]Frequency of string
\[=\frac{v}{\lambda }=\frac{p}{2l}\sqrt{\frac{T}{m}}\]
\[\Rightarrow \]Frequency of turning fork
\[I=\frac{P}{l}\sqrt{\frac{T}{m}}\] \[\Rightarrow \]If l, m, n \[\to \] constant then \[p\sqrt{T}=\]constant
(1) Consider a string of length l, stretched under tension T between two fixed points.
(2) If the string is plucked and then released, a transverse harmonic wave propagate along it's length and is reflected at the end.
(3) The incident and reflected waves will superimpose to produce transverse stationary waves in a string.
(4) Nodes (N) are formed at rigid end and antinodes (A) are formed in between them.
(5) Number of antinodes = Number of nodes \[-1\]
(6) 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 wave \[=\frac{v}{\lambda }=\frac{1}{\lambda }\sqrt{\frac{T}{m}}\]
(8) For obtaining p loops (p-segments) in string, it has to be plucked at a distance \[\frac{l}{2p}\] from one fixed end.
(9) Fundamental mode of vibration
(i) Number of loops \[p=1\]
(ii) Plucking at \[\frac{l}{2}\] (from one fixed end)
(iii) \[l=\frac{{{\lambda }_{1}}}{2}\]\[\Rightarrow \] \[{{\lambda }_{1}}=2l\]
(iv) Fundamental frequency or first harmonic \[{{n}_{1}}=\frac{1}{{{\lambda }_{1}}}\sqrt{\frac{T}{m}}=\frac{1}{2l}\sqrt{\frac{T}{m}}\]
(10) Second mode of vibration (First over tone or second harmonic)
(i) Number of loops \[p=2\]
(ii) Plucking at \[\frac{l}{2\times 2}=\frac{l}{4}\] (from one fixed end)
(iii) \[l={{\lambda }_{2}}\]
(iv) Second harmonic or first over tone \[{{n}_{2}}=\frac{1}{{{\lambda }_{2}}}\sqrt{\frac{T}{m}}=\frac{1}{l}\sqrt{\frac{T}{m}}=2{{n}_{1}}\]
(11) Third normal mode of vibration (Second over tone or third harmonic)
(i) Number of loops \[p=3\]
(ii) Plucking at \[\frac{l}{2\times 3}=\frac{l}{6}\] (from one fixed end)
(iii) \[l=\frac{3{{\lambda }_{3}}}{2}\]\[\Rightarrow \]\[{{\lambda }_{3}}=\frac{2l}{3}\]
(iv) Third harmonic or second over tone
\[{{n}_{3}}=\frac{1}{{{\lambda }_{3}}}\sqrt{\frac{T}{m}}=\frac{3}{2l}\sqrt{\frac{T}{m}}\]=3n1
(12) More about string vibration
(i) In general, if the string is plucked at length \[\frac{l}{2p},\] then it vibrates in p segments (loops) and we have the pth harmonic is give \[{{f}_{p}}=\frac{p}{2l}\sqrt{\frac{T}{m}}\]
(ii) All even and odd harmonics are present. Ratio of harmonic = 1 : 2 : 3 .....
(iii) Ratio of over tones = 2 : 3 : 4 ....
(iv) General formula for wavelength \[\lambda =\,\frac{2l}{N}\]; where N = 1,2,3, ? correspond to 1st , 2nd, 3rd modes of vibration of the string.
(v) General formula for frequency \[n=N\times \frac{v}{2l}\]
(vi) Position of nodes : \[x=0,\frac{l}{N},\frac{2l}{N},\frac{3l}{N}\,.....l\]
(vii) Position of antinodes : \[x\,=\,\frac{l}{2N},\frac{3l}{2N},\frac{5l}{2N}\,....\]\[\frac{\left( 2N-1 \right)\,l}{2N}\]
(1) Note : Any musical sound produced by the simple harmonic oscillations of the source is called note.
(2) Tone : Every musical sound consists of a number of components of different frequencies every component is known as a Tone.
(3) Fundamental note and fundamental frequency : The note of lowest frequency produced by an instrument is called fundamental note. The frequency of this note is called fundamental frequency.
(4) Harmonics : The frequency which are the integral multiple of the fundamental frequency are known as harmonics e.g. if n be the fundamental frequency, then the frequencies n, 2n, 3n .... are termed as first, second, third .... harmonics.
(5) Overtone : The harmonics other than the first (fundamental note) which are actually produced by the instrument are called overtones. e.g. the tone with frequency immediately higher than the fundamental is defined as first overtone.
(6) Octave : The tone whose frequency is doubled the fundamental frequency is defined as Octave.
(i) If \[{{n}_{2}}=2{{n}_{1}}\] it means \[{{n}_{2}}\] is an octave higher than \[{{n}_{1}}\] or \[{{n}_{1}}\] is an octave lower than \[{{n}_{2}}\].
(ii) If \[{{n}_{2}}={{2}^{3}}{{n}_{1}},\] it means \[{{n}_{2}}\] is 3-octave higher or \[{{n}_{1}}\] is 3-octave lower.
(iii) Similarly if \[{{n}_{2}}={{2}^{n}}{{n}_{1}}\] it means \[{{n}_{2}}\] is n-octave higher \[{{n}_{1}}\] is n octave lower.
(7) Unison : If the interval is one i.e. two frequencies are equal then vibrating bodies are said to be in unison.
(8) Resonance : The phenomenon of making a body vibrate with its natural frequency under the influence of another vibrating body with the same frequency is called resonance.
(1) Standing waves can be transverse or longitudinal.
(2) The disturbance confined to a particular region between the starting point and reflecting point of the wave.
(3) There is no forward motion of the disturbance from one particle to the adjoining particle and so on, beyond this particular region.
(4) The total energy associated with a stationary wave is twice the energy of each of incident and reflected wave. As in stationary waves nodes are permanently at rest. So no energy can be transmitted across then i.e. energy of one region (segment) 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 a number of segments. Each segment is vibrating up and down as a whole.
(6) All the particles in one particular segment vibrate in the same phase. Particles in two consecutive segments differ in phase by \[{{180}^{o}}\].
(7) All the particles except those at nodes, execute simple harmonic motion about their mean position with the same time period.
(8) The amplitude of vibration of particles varies from zero at nodes to maximum at antinodes \[(2a)\].
(9) All points (except nodes) pass their mean position twice in one time period.
(10) Velocity of particles while crossing mean position varies from maximum \[(\omega {{A}_{SW}}=\omega .2a)\] at antinodes to zero at nodes.
(11) In standing waves, if amplitude of component waves are not equal. Resultant amplitude at nodes will be minimum (but not zero). Therefore, some energy will pass across nodes and waves will be partially standing.
(12) Application of stationary waves
(i) Vibration in stretched string
(ii) Vibration in organ pipes (closed and open)
(iii) Kundt's tube Progressive v/s stationary wave
Progressive wave
Stationary wave
These waves transfers energy
These wave does not transfers 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 difference phase
When two sets of progressive wave trains of same type (both longitudinal or both transverse) having the same amplitude and same time period/frequency/wavelength travelling with same speed along the same straight line in opposite directions superimpose, a new set of waves are formed.
These are called stationary waves or standing waves.
In practice, a stationary wave is formed when a wave train is reflected at a boundary. The incident and reflected waves then interfere to produce a stationary wave.
(1) Suppose that the two super imposing waves are incident wave \[{{y}_{1}}=a\sin (\omega \,t-kx)\] and reflected wave \[{{y}_{2}}=a\sin (\omega \,t+kx)\]
(As \[{{y}_{2}}\] is the displacement due to a reflected wave from a free boundary)
Then by principle of superposition
\[y={{y}_{1}}+{{y}_{2}}=a\,[\sin \,(\omega \,t-kx)+\sin (\omega \,t+kx)]\]
(By using sin \[\sin C+\sin D=2\sin \frac{C+D}{2}\cos \frac{C-D}{2}\])
\[\Rightarrow \] \[y=2a\cos kx\,\sin \omega \,t\]
(If reflection takes place from rigid end, then equation of stationary wave will be \[y=2a\sin kx\,\cos \omega \,t\])
(2) As this equation satisfies the wave equation
\[\frac{{{\partial }^{2}}y}{\partial {{t}^{2}}}={{v}^{2}}\frac{{{\partial }^{2}}y}{\partial {{x}^{2}}}\]. It represents a wave.
(3) As it is not of the form \[f(ax\pm bt),\] the wave is not progressive.
(4) Amplitude of the wave \[{{A}_{SW}}=2a\cos kx.\] Amplitude in two different cases
Reflection at open end
Reflection at closed end
\[{{A}_{SW}}=2a\cos kx\]
\[{{A}_{SW}}=2a\sin kx\]
Amplitude is maximum when \[\cos \,kx=\pm 1\]
\[\Rightarrow \] \[kx=0,\,\,\pi ,\,\,2\pi ,\,\,...\,\,n\pi \].
\[\Rightarrow \] \[x=0,\,\frac{\lambda }{2},\,\lambda \,.......\frac{n\lambda }{2}\]
where \[k=\frac{2\pi }{\lambda }\] and n = 0, 1, 2, 3, ...
Amplitude is maximum when \[\sin \,kx=\pm 1\]
\[\Rightarrow \] \[kx=\frac{\pi }{2},\,\frac{3\pi }{2}\,.....\frac{(2n-1)\pi }{2}\]
\[\Rightarrow \]\[x=\frac{\lambda }{4},\frac{3\lambda }{4}.......\]
where \[k=\frac{2\pi }{\lambda }\]and n = 1, 2, 3, ....
Amplitude is minimum when \[\cos kx=0\]
\[\Rightarrow \]\[kx=\frac{\pi }{2},\,\frac{3\pi }{2}\,.....\frac{(2n-1)\pi }{2}\]
\[\Rightarrow \] \[x=\frac{\lambda }{4},\frac{3\lambda }{4}.......\]
Amplitude is minimum when \[\sin kx=0\]
\[\Rightarrow \] more...
This is an apparatus used to demonstrate the phenomenon of interference and also used to measure velocity of sound in air. This is made up of two U-tube A and B as shown in figure. Here the tube B can slide in and out from the tube A. There are two openings P and Q in the tube A. At opening P, a tuning fork or a sound source of known frequency n0 is placed and at the other opening a detector is placed to detect the resultant sound of interference occurred due to superposition of two sound waves coming from the tubes A and B.
Initially tube B is adjusted so that detector detects a maximum. At this instant if length of paths covered by the two waves from P and Q from the side of A and side of B are \[{{l}_{1}}\] and \[{{l}_{2}}\] respectively then for constructive interference we must have
\[{{l}_{2}}-{{l}_{1}}=N\lambda \] .... (i)
If now tube B is further pulled out by a distance \[x\] so that next maximum is obtained and the length of path from the side of B is \[{{l}_{2}}'\] then we have
\[l{{'}_{2}}={{l}_{2}}+2x\] .... (ii)
where \[x\] is the displacement of the tube. For next constructive interference of sound at point Q, we have \[l{{'}_{2}}={{l}_{1}}=(N+1)\lambda \] .... (iii)
From equation (i), (ii) and (iii), we get
\[l{{'}_{2}}-{{l}_{2}}=2\times x=\lambda \]\[\Rightarrow \]\[x=\frac{\lambda }{2}\]
Thus by experiment we get the wavelength of sound as for two successive points of constructive interference, the path difference must be \[\lambda \]. As the tube B is pulled out by \[x,\] this introduces a path difference \[2x\] in the path of sound wave through tube B. If the frequency of the source is known, \[{{n}_{0}},\] the velocity of sound in the air filled in tube can be gives as \[\nu ={{n}_{0}}.\lambda \]\[=2{{n}_{0}}x\]
(1) When two waves of same frequency, same wavelength, same velocity (nearly equal amplitude) moves in the same direction, Their superimposition results in the interference.
(2) Due to interference the resultant intensity of sound at that point is different from the sum of intensities due to each wave separately.
(3) Interference is of two type (i) Constructive interference (ii) Destructive interference.
(4) In interference energy is neither created nor destroyed but is redistributed.
(5) For observable interference, the sources (producing interfering waves) must be coherent.
(6) Let at a given point two waves arrives with phase difference \[\phi \] and the equation of these waves is given by
\[{{y}_{1}}={{a}_{1}}\sin \omega t,\,\,{{y}_{2}}={{a}_{2}}\sin \,\,(\omega t+\phi )\,\] then by the principle of superposition \[\vec{y}\,=\,{{\vec{y}}_{1}}\,+\,{{\vec{y}}_{2}}\]
\[\Rightarrow \] \[y={{a}_{1}}\sin \omega t+{{a}_{2}}\sin \,(\omega t+\phi )=A\,\,\sin \,(\omega t+\theta )\]
where \[A\,=\,\sqrt{{{a}_{1}}^{2}\,+\,{{a}_{2}}^{2}\,+\,2{{a}_{1}}{{a}_{2}}\,\cos \,\varphi }\]
and tan \[\theta =\frac{{{a}_{2}}\sin \,\varphi }{{{a}_{1}}\,+\,a{}_{2}\cos \,\,\varphi }\]
since Intensity \[(l)\propto \] (Amplitude A)\[^{2}\]\[\Rightarrow \]\[\frac{{{I}_{1}}}{{{I}_{2}}}={{\left( \frac{{{a}_{1}}}{{{a}_{2}}} \right)}^{2}}\]
Therefore, the resultant intensity is given by
\[I=\,{{I}_{1}}\,+\,{{I}_{2}}\,+\,2\,\sqrt{{{I}_{1}}{{I}_{2}}}\,\cos \,\varphi \]
Constructive and destructive interference
Constructive interference
Destructive interference
When the waves meets a point with same phase, constructive interference is obtained at that point (i.e. maximum sound).
When the wave meets a point with opposite phase, destructive interference is obtained at that point (i.e. minimum sound)
Phase difference between the waves at the point of observation \[\phi ={{0}^{o}}\] or \[2n\pi \]
Phase difference \[\phi ={{180}^{o}}\] or \[(2n-1)\,\pi \,;\,\] n = 1, 2, ...
Path difference between the waves at the point of observation \[\Delta =n\lambda \](i.e. even multiple of \[\lambda /2\])
(1) The displacement at any time due to any number of waves meeting simultaneously at a point in a medium is the vector sum of the individual displacements due each one of the waves at that point at the same time.
(2) If \[\overrightarrow{{{y}_{1}}}\,,\overrightarrow{{{y}_{2}}},\overrightarrow{{{y}_{3}}}\].... are the displacements at a particular time at a particular position, due to individual waves, then the resultant displacement.
\[\overrightarrow{y\,}\,=\overrightarrow{{{y}_{1}}}+\overrightarrow{{{y}_{2}}}+\overrightarrow{{{y}_{3}}}\,+\,.....\]
(3) Important applications of superposition principle
(i) Interference of waves : Adding waves that differs in phase
(ii) Formation of stationary waves : Adding wave that differs in direction.
(iii) Formation of beats : Adding waves that differs in frequency.
(iv) Formation of Lissaju's figure : Adding two perpendicular simple harmonic motions. (See S.H.M. for more detail)
An echo is simply the repetition of speaker's own voice caused by reflection at a distance surface e.g. a cliff. a row of building or any other extended surface. If there is a sound reflector at a distance d from source, then the time interval between original source and it's echo at the site of source will be
\[t=\frac{d}{v}+\frac{d}{v}=\frac{2d}{v}\]
As the persistence of hearing for human ear is 0.1 sec, therefore in order that an echo of short sound (e.g. clap or gun fire) will be heard if \[t>0.1\Rightarrow \frac{2d}{\upsilon }>0.1\Rightarrow d>\frac{\upsilon }{20}\]
If \[\upsilon =\] Speed of sound \[=340\,\,m/s\] then \[d>17\,m\].
(3) The box serves the purpose of increasing the loudness of the sound produced by the vibrating wire.
(4) 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)
(5) Resonance : When a vibrating tuning fork is placed on the box, and if the length between the bridges is properly adjusted then if \[{{(n)}_{Fork}}={{(n)}_{String}}\to \] rider is thrown off the wire.
(6) Laws of string
(i) Law of length : If T and m are constant then \[n\propto \frac{1}{l}\]
\[\Rightarrow \] nl = constant \[\Rightarrow \] \[{{n}_{1}}{{l}_{1}}={{n}_{2}}{{l}_{2}}\]
(ii) Law of mass : If T and l are constant then \[n\propto \frac{1}{\sqrt{m}}\]
\[\Rightarrow \] \[n\sqrt{m}=\]constant \[\Rightarrow \] \[\frac{{{n}_{1}}}{{{n}_{2}}}=\sqrt{\frac{{{m}_{2}}}{{{m}_{1}}}}\]
(iii) Law of density : If T, l and r are constant then \[n\propto \frac{1}{\sqrt{d}}\]
\[\Rightarrow \] \[n\sqrt{d}=\] constant \[\Rightarrow \] \[\frac{{{n}_{1}}}{{{n}_{2}}}=\sqrt{\frac{{{d}_{2}}}{{{d}_{1}}}}\]
(iv) Law of tension : If l and m are constant then \[n\propto \sqrt{T}\] \[\Rightarrow \] \[\frac{n}{\sqrt{T}}=\]constant
\[\Rightarrow \] \[\frac{{{n}_{1}}}{{{n}_{2}}}=\sqrt{\frac{{{T}_{2}}}{{{T}_{1}}}}\] 
(i) Number of loops \[p=1\]
(ii) Plucking at \[\frac{l}{2}\] (from one fixed end)
(iii) \[l=\frac{{{\lambda }_{1}}}{2}\]\[\Rightarrow \] \[{{\lambda }_{1}}=2l\]
(iv) Fundamental frequency or first harmonic \[{{n}_{1}}=\frac{1}{{{\lambda }_{1}}}\sqrt{\frac{T}{m}}=\frac{1}{2l}\sqrt{\frac{T}{m}}\]
(10) Second mode of vibration (First over tone or second harmonic)
(i) Number of loops \[p=2\]
(ii) Plucking at \[\frac{l}{2\times 2}=\frac{l}{4}\] (from one fixed end)
(iii) \[l={{\lambda }_{2}}\]
(iv) Second harmonic or first over tone \[{{n}_{2}}=\frac{1}{{{\lambda }_{2}}}\sqrt{\frac{T}{m}}=\frac{1}{l}\sqrt{\frac{T}{m}}=2{{n}_{1}}\]
(11) Third normal mode of vibration (Second over tone or third harmonic)
(i) Number of loops \[p=3\]
(ii) Plucking at \[\frac{l}{2\times 3}=\frac{l}{6}\] (from one fixed end)
(iii) \[l=\frac{3{{\lambda }_{3}}}{2}\]\[\Rightarrow \]\[{{\lambda }_{3}}=\frac{2l}{3}\]
(iv) Third harmonic or second over tone
\[{{n}_{3}}=\frac{1}{{{\lambda }_{3}}}\sqrt{\frac{T}{m}}=\frac{3}{2l}\sqrt{\frac{T}{m}}\]=3n1
(12) More about string vibration
(i) In general, if the string is plucked at length \[\frac{l}{2p},\] then it vibrates in p segments (loops) and we have the pth harmonic is give \[{{f}_{p}}=\frac{p}{2l}\sqrt{\frac{T}{m}}\]
(ii) All even and odd harmonics are present. Ratio of harmonic = 1 : 2 : 3 .....
(iii) Ratio of over tones = 2 : 3 : 4 ....
(iv) General formula for wavelength \[\lambda =\,\frac{2l}{N}\]; where N = 1,2,3, ? correspond to 1st , 2nd, 3rd modes of vibration of the string.
(v) General formula for frequency \[n=N\times \frac{v}{2l}\]
(vi) Position of nodes : \[x=0,\frac{l}{N},\frac{2l}{N},\frac{3l}{N}\,.....l\]
(vii) Position of antinodes : \[x\,=\,\frac{l}{2N},\frac{3l}{2N},\frac{5l}{2N}\,....\]\[\frac{\left( 2N-1 \right)\,l}{2N}\] 
In practice, a stationary wave is formed when a wave train is reflected at a boundary. The incident and reflected waves then interfere to produce a stationary wave.
(1) Suppose that the two super imposing waves are incident wave \[{{y}_{1}}=a\sin (\omega \,t-kx)\] and reflected wave \[{{y}_{2}}=a\sin (\omega \,t+kx)\]
(As \[{{y}_{2}}\] is the displacement due to a reflected wave from a free boundary)
Then by principle of superposition
\[y={{y}_{1}}+{{y}_{2}}=a\,[\sin \,(\omega \,t-kx)+\sin (\omega \,t+kx)]\]
(By using sin \[\sin C+\sin D=2\sin \frac{C+D}{2}\cos \frac{C-D}{2}\])
\[\Rightarrow \] \[y=2a\cos kx\,\sin \omega \,t\]
(If reflection takes place from rigid end, then equation of stationary wave will be \[y=2a\sin kx\,\cos \omega \,t\])
(2) As this equation satisfies the wave equation
\[\frac{{{\partial }^{2}}y}{\partial {{t}^{2}}}={{v}^{2}}\frac{{{\partial }^{2}}y}{\partial {{x}^{2}}}\]. It represents a wave.
(3) As it is not of the form \[f(ax\pm bt),\] the wave is not progressive.
(4) Amplitude of the wave \[{{A}_{SW}}=2a\cos kx.\] Amplitude in two different cases
Initially tube B is adjusted so that detector detects a maximum. At this instant if length of paths covered by the two waves from P and Q from the side of A and side of B are \[{{l}_{1}}\] and \[{{l}_{2}}\] respectively then for constructive interference we must have
\[{{l}_{2}}-{{l}_{1}}=N\lambda \] .... (i)
If now tube B is further pulled out by a distance \[x\] so that next maximum is obtained and the length of path from the side of B is \[{{l}_{2}}'\] then we have
\[l{{'}_{2}}={{l}_{2}}+2x\] .... (ii)
where \[x\] is the displacement of the tube. For next constructive interference of sound at point Q, we have \[l{{'}_{2}}={{l}_{1}}=(N+1)\lambda \] .... (iii)
From equation (i), (ii) and (iii), we get
\[l{{'}_{2}}-{{l}_{2}}=2\times x=\lambda \]\[\Rightarrow \]\[x=\frac{\lambda }{2}\]
Thus by experiment we get the wavelength of sound as for two successive points of constructive interference, the path difference must be \[\lambda \]. As the tube B is pulled out by \[x,\] this introduces a path difference \[2x\] in the path of sound wave through tube B. If the frequency of the source is known, \[{{n}_{0}},\] the velocity of sound in the air filled in tube can be gives as \[\nu ={{n}_{0}}.\lambda \]\[=2{{n}_{0}}x\] 
where \[A\,=\,\sqrt{{{a}_{1}}^{2}\,+\,{{a}_{2}}^{2}\,+\,2{{a}_{1}}{{a}_{2}}\,\cos \,\varphi }\]
and tan \[\theta =\frac{{{a}_{2}}\sin \,\varphi }{{{a}_{1}}\,+\,a{}_{2}\cos \,\,\varphi }\]
since Intensity \[(l)\propto \] (Amplitude A)\[^{2}\]\[\Rightarrow \]\[\frac{{{I}_{1}}}{{{I}_{2}}}={{\left( \frac{{{a}_{1}}}{{{a}_{2}}} \right)}^{2}}\]
Therefore, the resultant intensity is given by
\[I=\,{{I}_{1}}\,+\,{{I}_{2}}\,+\,2\,\sqrt{{{I}_{1}}{{I}_{2}}}\,\cos \,\varphi \]
Constructive and destructive interference
(2) If \[\overrightarrow{{{y}_{1}}}\,,\overrightarrow{{{y}_{2}}},\overrightarrow{{{y}_{3}}}\].... are the displacements at a particular time at a particular position, due to individual waves, then the resultant displacement.
\[\overrightarrow{y\,}\,=\overrightarrow{{{y}_{1}}}+\overrightarrow{{{y}_{2}}}+\overrightarrow{{{y}_{3}}}\,+\,.....\]
(3) Important applications of superposition principle
(i) Interference of waves : Adding waves that differs in phase
(ii) Formation of stationary waves : Adding wave that differs in direction.
(iii) Formation of beats : Adding waves that differs in frequency.
(iv) Formation of Lissaju's figure : Adding two perpendicular simple harmonic motions. (See S.H.M. for more detail) 
An echo is simply the repetition of speaker's own voice caused by reflection at a distance surface e.g. a cliff. a row of building or any other extended surface. If there is a sound reflector at a distance d from source, then the time interval between original source and it's echo at the site of source will be
\[t=\frac{d}{v}+\frac{d}{v}=\frac{2d}{v}\]
As the persistence of hearing for human ear is 0.1 sec, therefore in order that an echo of short sound (e.g. clap or gun fire) will be heard if \[t>0.1\Rightarrow \frac{2d}{\upsilon }>0.1\Rightarrow d>\frac{\upsilon }{20}\]
If \[\upsilon =\] Speed of sound \[=340\,\,m/s\] then \[d>17\,m\]. 
