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Speed of a Transverse Wave on a Stretched String

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Speed of a Transverse Wave on a Stretched String - Lesson Summary

A propagating wave is generated in a stretched string. The initial conditions are;
At t = 0, x = 0, y = 0 & f = 0
Let us mark a point “A” on the string.
After a time t, the wave has travelled a distance x and the particle A situated at a distance “x” from the origin and is now at the crest of the wave.
 We take a snap shot (1) with a high speed camera showing the wave form.
  After an additional time Dt, we take another snap shot (2) and mark point B on the crest of the progressive wave. The crest, which was at particle A has now moved to particle B.
The particle B is situated at (x + Dx) from the origin O.
During the time interval Dt the wave form has moved through a distance Dx.
Speed of the wave =
We have discussed about the particles A and B which were in the same phase i.e. the crest of the progressive wave.
Now if you take another pairs of elements like C & D which have the same phase, we find that their displacement in the y-direction is also the same.
y(x, t) = a sin(kx – wt + f) = a sin (kx – wt)  --------------        (As f = 0)
For the particle C & D, y is same.
\ kx – wt = constant         _______ (1)
Between particle C & D, x and t are changing
So, to keep (kx – wt) constant as x increases, t should also increase.
To find the wave speed v, we differentiate equation (1) with respect to time t and equate it to zero.

 Where, v = velocity, l = wavelength and  n = Frequency
To travel waves through a medium, the particles of the medium have to oscillate or vibrate.
The speed of the transverse wave on a string depends upon:
(1) Mass per unit length of the string - m
(2) The tension in the string – T
A relation between speed (v), mass per unit length of the string (m) and tension (T) in the string can be derived by dimensional analysis method.
Linear mass density m of the string (µ)

\ Dimension of m are [ML -1]
As Tension T = Force = N = M × a
\ Dimensions of Tension are [MLT -2]
Velocity = m/s
\ Dimensions of velocity are [LT -1]
Speed of a wave along a stretched string depends only on the tension and the linear mass density of the string and does not depend on the frequency of the wave.
The frequency of the wave is the frequency of the source which generates the waves.
The wavelength l is the determined by  


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