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Derivation Of The One Dimensional Wave Equation
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Derivation Of The One Dimensional Wave Equation

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 = α²u_xx ---------------(1)
where u(x,t) is the displacement of a point on the vibrating substance from its equilibrium position.
u_tt is the second partial derivative of u(x,t) with respect ot t
u_xx (concavity) is the second partial derivative of u(x,t) with respect to x
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: S₇P₄A₄₄ (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∂²u(x,t)/∂t². 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 which has a net transverse force on the small subregion of the string as follows:
Tension component = Tsinθ₂ - Tsinθ₁
= ≈ 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 restoring force is negative and downward if the displacement is positive and above the x-axis.

Now, applying Newton's equation of motion to the small subregion of the string, yields the following equation:
Δxρ∂²u(x,t)/∂t² = T[∂u(x + Δx,t)/∂x - ∂u(x,t)/∂x] + ΔxF(x,t) -Δxβ∂u(x,t)/∂t - Δxγu(x,t)----(2)
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:
∂²u(x,t)/∂t² = α²∂²u(x,t)/∂x² [-β∂u(x,t)/∂t - γu(x,t) + F(x,t)]/ρ----(3).
Equation (3) is known as the telephone equation and is sometimes written without the density ρ (it is implied even if it is not written).

The simple basic wave equation is :
∂²u(x,t)/∂t² = α²∂²u(x,t)/∂x²----(4)

Equation (4) intuitively infer that the acceleration of each point of the string is due to the tension in the string. The larger the concativity (u_xx), the stronger the force (where α² = T/ρ is the proportionality constant).

Math

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