D ALEMBERT AND THE DERIVATION OF THE EQUATION FOR THE VIBRATING STRING

(1)

AND THE DERIVATION OF THE EQUATION FOR THE

VIBRATING STRING

SM472.5001

CHRISTOPHER M. URBAN Midshipman First-Class

27 APRIL 2007

(2)

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.

(3)

It 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 17

^th

century.

Its derivation involved the solution of partial differential equations, a field not pioneered until the establishment of calculus in the late 17

^th

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.

(4)

Jean 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 17

^th

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

(5)

series u

k

, 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

(6)

vibrating 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

(7)

writing. 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:

d

²

u/dt

²

= c

²

*(d

²

u/dx

²

) (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.

(8)

Since u is a function of x and t,

du = pdx + qdt . (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 pdt + qdx is an exact

(9)

differential, we can establish a second equation, denoted dv.

This is the most critical piece of the derivation.

dv = pdt + qdx (4)

Reviewing the derivation so far, we have described two new equations, namely:

du = pdx + qdt dv = pdt + qdx

Adding the two equations, one is left with:

du + dv = pdx + qdt + pdt + qdx = (p+q)(dx+dt)* (5)

Similarly, after subtracting the two equations:

du – dv = pdx + qdt – pdt – qdx = (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 = 2m(x + t)* (7)

and, u – v = 2n(x – t).* (8)

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Adding these two identities gives us the result:

2u = 2m(x +t) + 2n(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):

d

²

u/dt

²

= c

²

*(d

²

u/dx

²

) (11)

Although this equation was derived by d’Alembert through

trial and observation, we will derive this partial

differential equation using mathematics and physics.

(11)

Proof (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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vertical 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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little 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

√(Δx

²

+Δu

²

). Since density is mass divided by length, the mass of the element of string is:

ρ(x)*√(Δx

²

+Δu

²

). (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)*√(Δx

²

+Δu

²

) * d

²

u/dt

^{2 =}

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)*√(1

²

+(du/dx)

²

)*d

²

u/dt

²

(x,t) = (14)

=d/dx[T(x,t)*sinθ(x,t)] + F(x,t).

(14)

=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) = lim

x->0

Δu/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+tan

²

θ)

(15)

(15), that │du/dx(x,t)│<<1. Thanks to this conclusion, we know:

√(1

²

+(du/dx)

²

)≈1 , sinθ(x,t)≈du/dx(x,t),

cosθ(x,t)≈1 , dθ/dx(x,t) ≈ d

²

u/dx

²

(x,t) (16) Substituting into equation (14) gives the equation:

ρ(x)* d

²

u/dt

²

(x,t) =

dT/dx(x,t)* du/dx(x,t) + T(x,t)* d

²

u/dx

²

(x,t) + F(x,t). (17) Thanks to our conclusion that │du/dx(x,t)│<<1, we can

simplify this equation further:

ρ(x)* d

²

u/dt

²

(x,t) = T(x,t)* d

²

u/dx

²

(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)* d

²

u/dt

²

(x,t) = T(t)* d

²

u/dx

²

(x,t) + F(x,t). (21)

(16)

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:

d

²

u/dt

²

(x,t) = c

²

* d

²

u/dx

²

(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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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

(18)

D ALEMBERT AND THE DERIVATION OF THE EQUATION FOR THE VIBRATING STRING

AND THE DERIVATION OF THE EQUATION FOR THE

VIBRATING STRING

SM472.5001

CHRISTOPHER M. URBAN Midshipman First-Class

27 APRIL 2007

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.

It 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 17

century.

Its derivation involved the solution of partial differential equations, a field not pioneered until the establishment of calculus in the late 17

century by Leibnitz and Newton.

Partial Differential Equations

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.

Jean 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 17

century. It was d’Alembert who first published the equation of the vibrating string.

Jean le Rond d’Alembert

series u

, 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.”

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

vibrating string, his findings were not conclusive. In the Traité de l'équilibre et du mouvement des fluides, published

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

writing. 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:

d

u/dt

= c

*(d

u/dx

) (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.

Since 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)

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

differential, 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 = pdx + qdt . (2)

equation (2). Realizing that pdt + qdx is an exact

dv = pdt + qdx (4)

du = pdx + qdt dv = pdt + qdx

du + dv = pdx + qdt + pdt + qdx = (p+q)(dx+dt)* (5)

du – dv = pdx + qdt – pdt – qdx = (p – q)(dx – dt))(6)*

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

and, u – v = 2n(x – t).* (8)

2u = 2m(x +t) + 2n(x – t)* (9)

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

=dT/dx(x,t)sinθ(x,t) + T(x,t)cosθ(x,t)*dθ/dx(x,t)+F(x,t)