Transverse wave stretched string, Mechanical Engineering

Transverse Wave Stretched String

As a typical example of wave motion in one-dimension, let us consider the transverse harmonic waves on a long, taut string. We assume that in equilibrium position, the string is horizontal (along X-axis) under sufficient tension T so that we can neglect the effect of gravity.


If the string is sufficiently long, it is possible to set up of a harmonic wave moving in one direction on it by making transverse SHM of one of its free ends, at x = 0. (The other end is too far off so that we are not immediately concerned whether it is free or fixed.)

We look at a small portion AB of length δ x which is displaced in the vertical plane by a small amount from its equilibrium position. In the displaced position of the string, the tensions on the part AB are T (x) and T (x + δ x), as shown in fig. acting tangentially to the string at points A and B. Neglecting gravity, the vertical and horizontal components of resultant force onAB are

Fv = T (x + δx) sin (θ + δ θ) - T (x) sin θ

and, FH = T (x + δx) cos (θ + δ θ) - T (x) cos θ

For small displacements, we assume that tension in the string does not vary appreciably from point to point, so that T (x + δx) ? T (x) = T. Further, small displacement implies that angles θ etc. are small so that we approximately put

sin (θ + δ θ) ? tan (θ + δ θ)     and sin θ ? tan θ

cos (θ + δ θ) ? cos (θ) ? 1

Hence, we get FH = 0

and, Fv = T(tan (θ + δ θ) - tan θ) = T δ (tan θ)

Since tan θ = ∂y/∂x , where y denotes the displacement along vertical direction, we have

2196_download.png 

where m is the mass of the portion AB of string. If we write  563_download (4).png  as the mass per unit length of the string, we have m =  563_download (4).png  δ x. Hence, we get

1858_download (2).png 

That is, the equation of motion for the small piece of string is a wave equation of the type, where the wave velocity is given by

70_download (3).png 

The wave velocity depends only on the characteristics of string, i.e. its mass per unit length  563_download (4).png  and tension T. All kinds of disturbances travel with the same velocity v, for a given  563_download (4).png  and T. Hence, if we move one end of the string up and down inSHM of amplitude A and frequency v, the disturbance moves along the string as a travelling harmonic wave given by

y (x, t) = A sin (2 π)/λ (x - vt)

where wavelength λ = v/v'.

If we vibrate the x = 0 end of the string between time t = t1 to t = t2 and then stop, then there would appear on the string a train of sine (or cosine) waves of limited extent, contained at any instant, between x = x1 to x = x2 such that

x2 - x1 = v (t2 - t1)

We call such a limited disturbance as a wave train which contains (x2 - x1)/λ number of waves, corresponding to (t2 - t1)/Toscillations performed at x = 0 end. On the other hand, if there is continuous vibration of free end, we get a continuous stream of waves. Note that in general a disturbance may be a continuous pattern, a finite wave train, or just a brief pulse.

Posted Date: 9/18/2012 1:15:06 AM | Location : United States







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