JEE Main & Advanced Physics Wave Mechanics 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 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 : \[x\,=\,\frac{l}{2N},\frac{3l}{2N},\frac{5l}{2N}\,....\]\[\frac{\left( 2N-1 \right)\,l}{2N}\]