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
We shutdown our furnace and always encountered crack on our piping at the weld joints. I investigated that upon shutdown the furnace cracking, our maintenance immediately remove th

Classification Based on Type of Welding  Fusion welding : These processes involve fusion of the base metal to complete the weld. Fusion welds generally  do  not  require  thea

Free Body Diagram: while a body or link is isolated, the forces (active and reactive) are also illustrated on it. It is called free body diagram of the body. The essential condit


(a) What is the function of a spark plug? (b) What are the types of spark plugs? Point out the differences between them. (c) Describe the construction of a spark plug with th

ELECTRIC DISCHARGE PHENOMENON The flow of electricity through metallic solids, or through liquid metals is called Conduction. The electricity  flow through ionic solutions

Evaluate the least pull, through the center of wheel: A uniform wheel having 600 mm diameter, weighing 5KN rests against rigid rectangular block of 150mm height as shown in t

Find the minimum weight to keep equilibrium: A belt supports the two weights W 1 an d W 2 over a pulley as shown in the figure given below. If W 1 = 1000N, fi

Represent moment Graphically: How you represent moment Graphically? Sol.: Consider force F which is represented in magnitude and direction, by   line AB . L

A jet of water moving at 8 m/sec. impinges on a concave-shaped vane to deflect the jet through 120 o when stationary. If the vane is moving at 5 m/sec., determine the angle of jet