D’ALEMBERT

AND THE DERIVATION

OF THE EQUATION FOR THE

VIBRATING STRING

SM472.5001

CHRISTOPHER M. URBAN

Midshipman First-Class

27 APRIL 2007

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Derivation of the Equation for the Vibrating String

For centuries, the motion of a wave fascinated

mathematicians, scientists and engineers. Its motion is as

commonplace as the wind or water; from the echo of music in a

church, sunlight entering through a window, the plucking a

mandolin or violin string, or the simple rolling of the ocean

- the wave and its motion can be found every day.

Mathematically, the equation of a wave was first derived

using a fixed, perfectly elastic vibrating string for a

model. Although different parameters for a vibrating string

will affect the string’s representative equation, the most

basic equations for both a wave and vibrating string are

identical. Consequently, the equation of the vibrating string

may be referred to as the wave equation, and vice versa.

Knowledge of the equation of the vibrating string has

led to the understanding of many phenomena. Water, light and

sound all propagate through waves, and, consequently, the

equations of water, light and sound waves have many

similarities to the equation of a vibrating string. The

understanding and application of the wave equation influences

many topics; optics, seismology, vibration, and sound are all

subjects linked closely to the basic understanding of the

vibrating string.

Urban.3

Derivation of the Equation for the Vibrating StringIt is obvious that an equation representing one of the

most common occurrences in nature can be important. Despite

the equation’s importance, the mathematics needed to derive

such an equation simply did not exist until the 17th century.

Its derivation involved the solution of partial differential

equations, a field not pioneered until the establishment of

calculus in the late 17th century by Leibnitz and Newton.

Partial Differential Equations

A partial differential equation is a relation involving

an unknown function and its partial derivatives. The function

depends on several independent variables (i.e. function f,

depending on x and y); its partial derivative, consequently,

is the derivative of that function with respect to one of

those variables (i.e. derivative of f(x,y) with respect to

x). Partial differential equations are used to solve

processes with a number of independent variables, and can

describe any process that is distributed in space and/or

time. The propagation of sound or heat, fluid flow and

electrostatics are all examples of processes whose solution

involves partial differential equations.

Urban.4

Derivation of the Equation for the Vibrating StringJean le Rond D’Alembert

As mathematics caught up to the dilemma of describing

the vibrating string (and more importantly, the motion of a

wave), mathematicians immediately set to work on developing a

solution. In particular, a French scholar of the

Enlightenment took up the task. As an educated thinker who

studied philosophy, physics, engineering and mathematics,

Jean le Rond d’Alembert was an influential and prominent

intellectual of the 17th century. It was d’Alembert who first

published the equation of the vibrating string.

D’Alembert contributed much to mathematics. He proposed

that the derivative of a function is the limit of a quotient

of increments, and his work with limits led him eventually to

develop a test for convergence. Known as D’Alembert’s Ratio

Test, it tested series for convergence. Theorem 1: For a

Jean le Rond d’Alembert

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Derivation of the Equation for the Vibrating String

series uk, where k is an integer:

converges if ρ<1. Conversely, the series diverges if

ρ>1.

D’Alembert’s greatest contributions to science are found

elsewhere, however. As chief editor of the Encyclopédie, ou

dictionnaire raisonné des sciences, des arts et des métiers,

or Systematic Dictionary of the Sciences, Arts and Crafts, he

is well known for the encyclopedia’s famous introduction. The

Preliminary Discourse described the purpose of the

encyclopedia: namely, “to change the way people think.”

Carrying immense political importance for its influence in

the French Revolution, it attempted to make knowledge of

philosophy and science available to the common man and to

guide the opinion of its readers. As one of the most famous

contributors and editors, d’Alembert achieved widespread

notoriety for his contributions to the encyclopedia.

Perhaps d’Alembert’s most important contribution to

mathematics was his work on the equation of the vibrating

string. D’Alembert was the first to publish the equation in

print; as with most significant innovations, however,

d’Alembert’s publication generated much controversy. Although

he clearly took the lead in deriving an equation for the

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Derivation of the Equation for the Vibrating Stringvibrating string, his findings were not conclusive. In the

Traité de l'équilibre et du mouvement des fluides, published

in 1747, d’Alembert uses mathematically simplistic boundary

conditions to arrive at his solution. D’Alembert’s over-

simplification of the problem introduces the inherent

question of how to handle actual boundary conditions. On top

of this problem, his conclusions could not mathematically

agree with the physical reality of a string’s motion. Both of

these shortcomings naturally brought doubt and discredit to

his solution.

d’Alembert and Euler

About the same time, another influential figure began

work on the derivation of the equation of the vibrating

string. Leonhard Euler, already a well known mathematician

throughout Europe and Russia, gained interest in the results

of d’Alembert’s work on the equation of the vibrating string.

However, upon publishing his work, Euler attributed no credit

to d’Alembert for his groundbreaking work on the subject.

Despite d’Alembert’s objections, Euler refused to cite him as

a contributor. Euler claimed that he was forced to start all

his work from scratch since he could not read d’Alembert’s

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Derivation of the Equation for the Vibrating Stringwriting. Thus, he did not owe d’Alembert any credit. This

argument expanded into a great feud between the two

mathematicians, ending only with d’Alembert’s death in 1783,

more than 35 years later.

Wave Equation Derivation

Despite the problems which confronted the d’Alembert

solution, it was the first time the wave equation appeared in

print. The method of d’Alembert to derive the equation is

ingenious, and is worth reproducing. The following proof is

the method d’Alembert derived the wave equation in 1747.

After several attempts, d’Alembert’s application of

differential calculus to the problem of the vibrating string

led him to a partial differentiated solution.

Theorem 2: The partial differential equation:

d2u/dt2 = c2*(d2u/dx2) (1)

is satisfied by the equation u = u(x,t) = m(x+t) + n(x-t),

where m and n are arbitrary functions.

Proof: To simplify things, d’Alembert let p = du/dx and let q

= du/dt.

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Derivation of the Equation for the Vibrating StringSince u is a function of x and t,

du = p*dx + q*dt . (2)

Both partial derivatives of u, one of the x components and

another of the t components, add together to complete the

partial derivative of u. This is half the key to the proof.

The other half of the proof is not obvious, and is the

reason for the proof’s ingenuity. If we rewrite equation (1)

in terms of p and q now, the result is

(3)

The next step of d’Alembert’s proof involves a

differential known as an exact or total differential. A

differential is exact if, for a function f, ∫df is path-

independent. For example, if f = P(x,y)+ Q(x,y), then f is an

exact differential if dp/dx = dq/dy, without loss of

generality. Regardless of the variable the function is

differentiated by, the integral of the function is always the

same.

Using equation (3), we see the possibility of another

equation, one created in a fashion similar to that of

equation (2). Realizing that p*dt + q*dx is an exact

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Derivation of the Equation for the Vibrating Stringdifferential, we can establish a second equation, denoted dv.

This is the most critical piece of the derivation.

dv = p*dt + q*dx (4)

Reviewing the derivation so far, we have described two

new equations, namely:

du = p*dx + q*dt

dv = p*dt + q*dx

Adding the two equations, one is left with:

du + dv = p*dx + q*dt + p*dt + q*dx = (p+q)*(dx+dt) (5)

Similarly, after subtracting the two equations:

du – dv = p*dx + q*dt – p*dt – q*dx = (p – q)*(dx – dt))(6)

This result is clearly a partial differential equation,

depending on the independent variables x and t. For u + v,

there exists some function m which depends on x + t.

Likewise, when u – v is the case, there exists some function

n which depends on x – t.

So, we now have: u +v = 2*m(x + t) (7)

and, u – v = 2*n(x – t). (8)

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Derivation of the Equation for the Vibrating StringAdding these two identities gives us the result:

2*u = 2*m(x +t) + 2*n(x – t) (9)

which is simplified to

u = m(x + t) + n(x – t). Q.E.D. (10)

Jean le Rond d’Alembert‘s solution explains an important

part of the behavior of a vibrating string or wave. The

function m describes a left-traveling wave, while the

function n describes a right-traveling wave. The addition of

these two waves produces the resulting visible wave.

Partial Differential Equation Derivation

Theorem 3: The work remaining to be done on the wave

equation is the derivation of equation (1):

d2u/dt2 = c2*(d2u/dx2) (11)

Although this equation was derived by d’Alembert through

trial and observation, we will derive this partial

differential equation using mathematics and physics.

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Derivation of the Equation for the Vibrating StringProof (by physics):

Figure 1

Figure 1 shows a segment of a perfectly elastic string,

secured at both ends, in motion. The function u(x,t) is the

Δu

Δx

x

T(x,t)

T(x+Δx, t)

u(x,t)θ(x,t)

θ(x+Δx,t )

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Derivation of the Equation for the Vibrating Stringvertical displacement of the string from the x-axis at

position x and time t. The function T(x,t) is the tension in

the string at position x and time t. θ(x,t)is the angle

between the string and a horizontal line at position x and

time t. We will define ρ(x) as the mass density of the string

at position x.

There are three forces acting on the string segment. The

first force is the tension pulling to the right, which has

magnitude T(x+Δx,t) and acts at angle θ(x+Δx,t). The second

force is the tension pulling to the left, which has magnitude

T(x,t) and acts at angle θ(x,t). Finally, all other external

forces, like gravity will be grouped together as a single

force. We shall assume that all of the external forces act

vertically and we shall denote by F(x,t)*Δx as the net

magnitude of the external forces acting on the element of

string.

We begin this proof using Newton’s Second Law. Briefly

stated, Newton’s Second Law states that the force acting on

an object is the object’s mass times its acceleration. By

summing the three forces acting on the string and determining

its mass and acceleration, we can describe the force acting

on the string.

First we will determine the mass of the string. Since

the string element is a tiny segment of the string, there is

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Derivation of the Equation for the Vibrating Stringlittle curvature in the string. Using this assumption, we can

treat the string element as a right triangle, where the

changes in x and u act as the sides of the triangle. Using

the Pythagorean Theorem, the length of the string is thus

√(Δx2 +Δu2). Since density is mass divided by length, the mass

of the element of string is:

ρ(x)*√(Δx2 +Δu2). (12)

Acceleration is the other undetermined quantity in our

equation. We know that velocity is the distance covered over

time of an object. So, velocity is the time derivative of the

function of an object’s position. Likewise, acceleration is

change in velocity over time. Thus, the second time

derivative of the position function is an object’s

acceleration.

According to Newton’s Second Law, the net force on an

object is its mass times acceleration. So, adding the

vertical components of tension and external forces:

ρ(x)*√(Δx2 +Δu2) * d2u/dt2 =

T(x+x,t)*sinθ(x+x,t)-T(x,t)* sinθ(x,t)+ F(x,t)*Δx (13)

If we divide equation (13) by Δx and take the limit of x

as it approaches 0, we get :

ρ(x)*√(12+(du/dx)2)*d2u/dt2(x,t) = (14)

=d/dx[T(x,t)*sinθ(x,t)] + F(x,t).

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Derivation of the Equation for the Vibrating String=dT/dx(x,t)*sinθ(x,t) + T(x,t)*cosθ(x,t)*dθ/dx(x,t)+F(x,t)

Figure 2

Using Figure 2, we can observe that

tanθ(x,t) = limx->0Δu/Δx = du/dx(x,t). (15)

At this point, our equation is too complicated to solve.

So, since vibrating strings usually have small amplitudes or

vibrations (i.e. guitar string, violin string, etc.) we will

assume that the vibrations in this system are small. This

implies that │θ(x,t)│<<1 for all x and t. Applying this

conclusion, it follows that │tanθ(x,t)│<<1, and from equation

θ

1

tanθ

√(1+tan2θ)

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Derivation of the Equation for the Vibrating String(15), that │du/dx(x,t)│<<1. Thanks to this conclusion, we

know:

√(12+(du/dx)2)≈1 , sinθ(x,t)≈du/dx(x,t),

cosθ(x,t)≈1 , dθ/dx(x,t) ≈ d2u/dx2(x,t) (16)

Substituting into equation (14) gives the equation:

ρ(x)* d2u/dt2(x,t) =

dT/dx(x,t)* du/dx(x,t) + T(x,t)* d2u/dx2(x,t) + F(x,t). (17)

Thanks to our conclusion that │du/dx(x,t)│<<1, we can

simplify this equation further:

ρ(x)* d2u/dt2(x,t) = T(x,t)* d2u/dx2(x,t) + F(x,t). (18)

This settles the vertical component of the string element for

now.

Now we follow similar steps in solving the horizontal

element of the string. Beginning with Newton’s sum of the

forces, T(x+Δx,t)*cosθ(x+Δx,t) – T(x,t)*cosθ(x,t) = 0 (19)

Similarly dividing by Δx and taking its limit as Δx

approaches 0, we have: d/dx[T(x,t)*cosθ(x,t)]=0. (20)

Next, we make the case that the amplitudes of the string in

our system will be small, or │θ(x,t)│<<1. For a small θ,

vibrations/waves will be directed only horizontally. This

observation implies that tension is a function solely of

time, further simplifying our equation to:

ρ(x)* d2u/dt2(x,t) = T(t)* d2u/dx2(x,t) + F(x,t). (21)

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Derivation of the Equation for the Vibrating String

Assuming that string density, ρ, will be constant

throughout the string, and that there are no acting external

forces, F, on the string, we are left with:

d2u/dt2(x,t) = c2* d2u/dx2(x,t) (22)

where c = √(T/ρ). Q.E.D.

This conclusion, the same made by d’Alembert, is the

partial differential equation that describes the propagation

of a wave over time. Its importance cannot be underestimated.

The equation’s applicability extends far beyond the initial

scope of the vibrating string. Since the days of d’Alembert,

we now know that waves describe the motion of many phenomena;

sound, water and light all travel in waves, and may be

described using variations of the above 1-D vibrating string

partial differential equation. D’Alembert’s success in 1747

in publishing the solution to the equation of a vibrating

string was the first step in understanding the important

topic of wave propagation.

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Derivation of the Equation for the Vibrating String

WORKS CITED

Falstad, Paul. “Loaded String Simulation”

Version 1.5, posted 7/23/05

http://www.falstad.com/loadedstring/

Sharman, R.V. Vibrations and Waves. London:

Butterworths, 1963.

Weisstein, Eric W. "d'Alembert's Solution."

From MathWorld -A Wolfram Web Resource.

http://mathworld.wolfram.com/dAlembertsSolution.html

Wilkins, D.R. “Jean le Rond D’Alembert.”

School of Mathematics: Trinity College, Dublin

http://www.maths.tcd.ie/pub/HistMath/People/DAlembert/Ro

useBall/RB.DAlembert.htmll

Urban.18

Derivation of the Equation for the Vibrating String

O'Connor, J.J. and Robertson, E.F. School

of Mathematics and Sciences: University of St. Andrews,

Scotland

http://www.history.mcs.standrews.ac.uk/Biographies/D'Ale

mbert.html

Feldman, J. “Derivation of the Wave Equation”

Mathematics Department: University of British Columbia,

Vancouver http://www.math.ubc.ca/~feldman/apps/wave.pdf

The American Mathematical Monthly > Vol. 64, No. 3

(Mar., 1957), pp. 155-157

Stable URL:http://links.jstor.org/sici?sici=0002890%28