derivation of wave equation

Let a string of homogeneous matter be tightened between the points x = 0 and x = p of the x-axis and let the string be made vibrate in the xy-plane. Let the line density of mass of the string be the constant σ. We suppose that the amplitude of the vibration is so small that the Tension T of the string can be regarded to be constant.

The position of the string may be represented as a function

y = y(x, t)

where t is the time. We consider an element dm of the string situated on a tiny interval [x, x+dx]; thus its mass is σ dx. If the angles the vector T at the ends x and x+dx of the element forms with the direction of the x-axis are α and β, then the scalar components of the resultant force F of all forces on dm (the gravitation omitted) are

Fx = − T cosα + T cos β, Fy = − T sin α + T sin β.

Since the angles α and β are very small, the ratio

F cosβ − cos α − 2sin β−-αsin β+α --x = ------------- = -----β−-2α----β+2α-, Fy sin β − sinα 2sin--2-cos -2--

having the expression − tan β+2α- , also is very small. Therefore we can omit the horizontal component F_x and think that the vibration of all elements is strictly vertical. Because of the smallness of the angles α and β, their sines in the expression of F_y may be replaced with their tangents, and accordingly

′ ′ ′′ Fy = T ⋅ (tan β − tan α) = T [yx(x+dx, t) − yx(x, t)] = T yxx(x, t)dx,

the last form due to the mean-value theorem.

On the other hand, by Newton the force equals the mass times the acceleration:

Fy = σ dxy ′′tt(x, t)

Equating both expressions, dividing by T dx and denoting ∘ ---
T-
σ = c, we obtain the partial differential equation

y′′_xx =

y′′_tt

(1)

for the equation of the transversely vibrating string.

But the equation (1) don’t suffice to entirely determine the vibration. Since the end of the string are immovable,the function y(x, t) has in addition to satisfy the boundary conditions

y(0, t) = y(p, t) = 0

(2)

The vibration becomes completely determined when we know still e.g. at the beginning t = 0 the position f(x) of the string and the initial velocity g(x) of the points of the string; so there should be the initial conditions

y(x, 0) = f(x), y′_t(x, 0) = g(x).

(3)

The equation (1) is a special case of the general wave equation

∇²u =

u′′_tt

(4)

where u = u(x, y, z, t). The equation (4) rules the spatial waves in ℝ. The number c can be shown to be the velocity of propagation of the wave motion.

References

[1] K. Väisälä: Matematiikka IV. Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).

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