**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 on**AB** are

**F**_{v} = T (x + δx) sin (θ + δ θ) - T (x) sin θ

and, **F**_{H} = 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 **F**_{H} = 0

and, **F**_{v} = 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

where m is the mass of the portion AB of string. If we write as the mass per unit length of the string, we have m = **δ x**. Hence, we get

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

The wave velocity depends only on the characteristics of string, i.e. its mass per unit length and tension **T**. All kinds of disturbances travel with the same velocity **v**, for a given and T. Hence, if we move one end of the string up and down in**SHM** 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 = t**_{1} to **t = t**_{2} 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 = x**_{1} to **x = x**_{2} such that

**x**_{2} - x_{1} = v (t_{2} - t_{1})

We call such a limited disturbance as a wave train which contains **(x**_{2} - x_{1})/λ number of waves, corresponding to **(t**_{2} - t_{1})/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.