Derivation Of The One Dimensional Wave Equation

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{4}A_{44} Base Sequence = 12735 String Sequence = 12735 - 4 - 44 **

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Derivation Of One Dimensional Wave Equation

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There are three basic types of Linear Partial Differential Equations (PDEs). *Parobolic PDEs, Hyperbolic PDEs and Elliptic PDES*. The one dimensional wave equation is a hyperbolic PDE and is of the form:
**u _{tt} = α^{2}u_{xx}** ---------------(1)

where u(x,t) is the displacement of a point on the vibrating substance from its equilibrium position.

u

u

α is the proportinality constant.

Show that the transverse vibrations of a string of length L (figure 114.8a) fastened at each end can be described mathemathecally by equation (1).

**The strings**:
S_{7}P_{4}A_{44} (Motion - Oscillatory).
**The math**:

Pj Problem of Interest is of type *motion* (oscillatory).

Assumptions:

(1) The string is fastened at each end and stretched tightly.

(2) The string is made of homogeneous material.

(3) The string is unaffected by gravity.

(4) The vibrations take place in a plane

Consider a small subregion [x, x + Δ] of the vibrating string. Figure 114.8b is a magnification of this small subregion.

According to Newton's equation of motion:

Change in momentum of the small subregion of the string is equal to the applied forces
**The change in momentum = m∂ ^{2}u(x,t)/∂t^{2}**. Where m is mass.

Important forces acting on the string:

(1) Net force due to the tension in the string = T ( figure 114.8b)

T has a

Tension component = Tsinθ

= ≈ T[∂u(x + Δx,t)/∂x - ∂u(x,t)/∂x]

(2) External force F(x,t) = -mg (gravity).

(3) Frictional force (- β∂u(x,t)/∂t))

This is a resistance force from the medium in which the string is vibrating at a velocity of ∂u(x,t)/∂t).

(4) Restoring force (γu(x,t)).

This is the force acting opposite to the displacement of the string. The

Now, applying

Where ρ is the density of the string.

Dividing equation (2) by ρ and Δx and letting Δx ----> 0 (tends to 0), results in the following equation:

Equation (3) is known as the

The simple basic wave equation is :

Equation (4) intuitively infer that the acceleration of each point of the string is due to the tension in the string. The larger the

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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