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







Related Discussions:- Transverse wave stretched string, Assignment Help, Ask Question on Transverse wave stretched string, Get Answer, Expert's Help, Transverse wave stretched string Discussions

Write discussion on Transverse wave stretched string
Your posts are moderated
Related Questions
Assume the production function for a rm is given by the equation Q = L √K. Graph the isoquants corresponding to Q = 10, Q = 20, and Q = 50. Do these isoquants exhibit diminishing


Tension in the strings: Sol.: When weight is attached to string then it will be in tension. The diagrams are shown below to describe this particular concept.

State Hook's law: Sol.: It states that when material is loaded within the elastic limit, the stress is proportional to strain produced by the stress. This means that the rat

simply supported beam subjected to point loads results

In considering a particular fermentation there are a number of choices regarding process operation. Typical  initial considerations are listed below. Again, the choices made will d

Technical Limitations of Extension of screw conveyor Technical Limitations: Breakdown of hinged joint at intermediate condition of link. ?Not proper extension of coaxi

define elastomer.explain in details?

Determine the Properties of foundation Soil profile together with natural type and properties of each strata will have a bearing on the choice of foundation. The fact that loa

Compoun d shaft: compoun d shaft? How is compound shaft classified?                                                               Sol.: Shaft made up of 2 or more mat