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!

ignores the last term in (9.3) for nite h one obtains the approximation (9.1) once more.

Problem c: Use expression (9.3) to show that the error made in the approximation (9.1)

depends indeed on the curvature of the function f(x).

The approximation (9.1) has a variety of applications. The rst is the numerical

solution of di erential equations. Suppose one has a di erential equation that one cannot

solve in closed form. to x out mind consider the di erential equation

df = G(f(x) x) (9.4)

dx

9.1. THREE WAYS TO ESTIMATE A DERIVATIVE 91

with initial value

f(0) = f0 : (9.5)

When this equation cannot be solved in closed form, one can solve it numerically by

evaluating the function f(x) not for every value of x, but only at a nite number of x-

values that are separated by a distance h. These points xn are given by xn = nh, and the

function f(x) at location xn is denoted by fn:

fn f(xn) : (9.6)

Problem d: Show that the derivative df=dx at location xn can be approximated by:

df (x ) = 1 (f

dx n h n+1 fn) : (9.7)

;

Problem e: Insert this result in the di erential equation (9.4) and solve the resulting

expression for fn+1 to show that:

fn+1 = fn + hG(fn xn) : (9.8)

This is all we need to solve the di erential equation (9.4) with the boundary condition

(9.5) numerically. Once fn is known, (9.8) can be used to compute fn+1 . This means

that the function can be computed at all values of the grid points xn recursively. To start

this process, one uses the boundary condition (9.5) that gives the value of the function at

location x0 = 0. This technique for estimating the derivative of a function can be extended

to higher order derivatives as well so that second order di erential equations can also be

solved numerically. In practice, one has to pay serious attention to the stability of the

numerical solution. The requirements of stability and numerical e ciency have led to many

re nements of the numerical methods for solving di erential equations. The interested

reader can consult Press et al. 47] as an introduction and many practical algorithms.

The estimate (9.1) has a second important application because it allows us to estimate

the order of magnitude of a derivative. Suppose a function f(x) varies over a characteristic

range of values F and that this variation takes place over a characteristic distance L. It

follows from (9.1) that the derivative of f(x) is of the order of the ratio of the variation

of the function f(x) divided by the length-scale over which the function varies. In other

words:

df variation of the function f(x) F : (9.9)

dx length scale of the variation L

In this expression the term F=L indicates that the derivative is of the order F=L. Note

that this is in general not an accurate estimate of the precise value of the function f(x),

it only provides us with an estimate of the order of magnitude of a derivative. However,

this is all we need to carry out scale analysis.

Problem f: Suppose f(x) is a sinusoidal wave with amplitude A and wavelength :

f(x) = A sin 2 x : (9.10)

CHAPTER 9. SCALE ANALYSIS

92

Show that (9.9) implies that the order of magnitude of the derivative of this function

is given by f=dxj O (A= ). Compare this estimate of the order of magnitude

jd

with the true value of the derivative and pay attention both to the numerical value

as well as to the spatial variation.

From the previous estimate we can learn two things. First, the estimate (9.9) is only

a rough estimate that locally can be very poor. One should always be aware that the

estimate (9.9) may break down at certain points and that this can cause errors in the

subsequent scale analysis. Second, the estimate (9.9) di ers by a factor 2 from the true

derivative. However, 2 = 6:28 which is not a small number. Therefore you must be

aware that hidden numerical factors may enter scaling arguments.

9.2 The advective terms in the equation of motion

As a rst example of scale analysis we consider the role of advective terms in the equation

of motion. As shown in expression (8.12) of section 8.3 the equation of motion for a

continuous medium is given by

@v + v = 1F : (9.11)

@t rv

Note that we have divided by the density compared to the original expression (8.12). This

equation can describe the propagation of acoustic waves when F is the pressure force, it

accounts for elastic waves when F is given by the elastic forces in the medium. We will

be interested in the situation where waves with a wavelength and a period T propagate

through the medium.

The advective terms v often pose a problem in solving this equation. The reason

rv

is that the partial time derivative @v=@t is linear in the velocity v but that the advective

terms v are nonlinear in the velocity v. Since linear equations are in general much

rv

easier to solve than nonlinear equations it is very useful to know under which conditions

the advective terms v can be ignored compared to the partial derivative @v=@t.

rv

Problem a: Let the velocity of the continuous medium have a characteristic value V .

V 2= .

V=T and that

Show that j@v=@tj jv rvj

Problem b: Show that this means that the ratio of the advective terms to the partial

time derivative is given by

V (9.12)

jv rvj

c

j@v=@tj

where c is the velocity with which the waves propagate through the medium.

This result implies that the advective terms can be ignored when the velocity of the

medium itself is much less than the velocity which the waves propagate through the

medium:

V c: (9.13)

In other words, when the amplitude of the wave motion is so small that the velocity of

the medium is much less than the wave velocity one can ignore the advective terms in the

equation of motion.

9.2. THE ADVECTIVE TERMS IN THE EQUATION OF MOTION 93

Problem c: Suppose an earthquake causes at a large distance a ground displacement of

1 mm at a frequency of 1 Hz. The wave velocity of seismic P-waves is of the order

of 5 km=s near the surface. Show that in that case V=c 10;9 .

The small value of V=c implies that for the propagation of elastic waves due to earthquakes

one can ignore advective terms in the equation of motion. Note, however, that this is not

necessarily true near the earthquake where the motion is much more violent and where

the associated velocity of the rocks is not necessarily much smaller than the wave velocity.

Figure 9.2: The shock waves generated by a T38 ying at Mach 1.1 (a speed of 1.1 times

the speed of sound) as made visible as made visibible with the schlieren method.

There are a number of physical phenomena that are intimately related to the presence

of the advective terms in the equation of motion. One important phenomenon is the

occurrence of shock waves when the motion of the medium is comparable to the wave

velocity. A prime example of shock waves is the sonic boom made by aircraft that move

at a velocity equal to the speed of sound 32]. Since the air pushed around by the aircraft

moves with the same velocity as the aircraft, shock waves are generated when the velocity

of the aircraft is equal to the speed of sound. A spectacular example can be seen in

gure 9.2 where the shock waves generated by an T38 ying at a speed of Mach 1.1 at

an altitude of 13.700 ft can be seen. These shock waves are visualised using the schlieren

method 36] which is an optical technique to convert phase di erences of light waves in

amplitude di erences.

Another example of shock waves is the formation of the hydraulic jump. You may not

known what a hydraulic jump is, but you have surely seen one! Consider water owing

down a channel such as a mountain stream as shown in gure 9.3. The ow velocity is

CHAPTER 9. SCALE ANALYSIS

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