fix problems with html output

[light-and-matter.git] / lm / vw / c.rbtex

<%
  require "../eruby_util.rb"
%>

<%
  chapter(
    '19',
    %q{Free Waves},
    'ch:free-waves',
    %q{\hspace{1.5mm}``The Great Wave Off Kanagawa,'' by Katsushika Hokusai (1760-1849).},
    {'opener'=>'hokusai'}
  )
%>


Your vocal cords or a saxophone reed can vibrate, but being
able to vibrate wouldn't be of much use unless the
vibrations could be transmitted to the listener's ear by
sound waves. What are waves and why do they exist? Put your
fingertip in the middle of a cup of water and then remove it
suddenly. You will have noticed two results that are
surprising to most people. First, the flat surface of the
water does not simply sink uniformly to fill in the volume
vacated by your finger. Instead, ripples spread out, and the
process of flattening out occurs over a long period of time,
during which the water at the center vibrates above and
below the normal water level. This type of wave motion is
the topic of the present chapter. Second, you have found
that the ripples bounce off of the walls of the cup, in much
the same way that a ball would bounce off of a wall. In the
next chapter we discuss what happens to waves that have a
boundary around them. Until then, we confine ourselves to
wave phenomena that can be analyzed as if the medium (e.g.,
the water) was infinite and the same everywhere.

<% marg(20) %>
<%
  fig(
    'finger-in-cup',
    %q{%
      Dipping a finger in some water, 1,
      causes a disturbance that spreads outward, 2.
    }
  )
%>
<% end_marg %>
It isn't hard to understand why removing your fingertip
creates ripples rather than simply allowing the water to
sink back down uniformly. The initial crater, (a), left
behind by your finger has sloping sides, and the water next
to the crater flows downhill to fill in the hole. The water
far away, on the other hand, initially has no way of knowing
what has happened, because there is no slope for it to flow
down. As the hole fills up, the rising water at the center
gains upward momentum, and overshoots, creating a little
hill where there had been a hole originally. The area just
outside of this region has been robbed of some of its water
in order to build the hill, so a depressed ``moat'' is
formed, (b). This effect cascades outward, producing ripples.

\pagebreak[4]

<%
  fig(
    'toes-in-pool',
    %q{%
      The two circular patterns of ripples pass
      through each other. Unlike material objects, wave patterns can overlap in space, and when
      this happens they combine by addition.
    },
    {
      'width'=>'wide',
      'sidecaption'=>true
    }
  )
%>

<% begin_sec("Wave Motion",0) %>

There are three main ways in which wave motion differs from
the motion of objects made of matter.

<% begin_sec("1. Superposition") %>

The most profound difference
is that waves do not display anything
analogous to the normal forces
between objects that come in contact. Two wave patterns can
therefore overlap in the same region of space, as shown in
figure \figref{toes-in-pool}. Where the two waves
coincide, they add together. For instance, suppose that at a
certain location in at a certain moment in time, each wave
would have had a crest 3 cm above the normal water level.
The waves combine at this point to make a 6-cm crest. We use
negative numbers to represent depressions in the water. If
both waves would have had a trough measuring -3 cm, then
they combine to make an extra-deep -6 cm trough. A +3 cm
crest and a -3 cm trough result in a height of zero, i.e.,
the waves momentarily cancel each other out at that point.
This additive rule is referred to as the 
\index{principle of superposition}principle of superposition, ``superposition''
being merely a fancy word for ``adding.''

Superposition can occur not just with sinusoidal waves like
the ones in the figure above but with waves of any shape.
The figures on the following page show superposition of wave
\index{pulse!defined}\emph{pulses}. A pulse is simply a wave of
very short duration. These pulses consist only of a single
hump or trough. If you hit a clothesline sharply, you will
observe pulses heading off in both directions. This is
analogous to the way ripples spread out in all directions
when you make a disturbance at one point on water. The same
occurs when the hammer on a piano comes up and hits a string.

Experiments to date have not shown any deviation from the
principle of superposition in the case of light waves. For
other types of waves, it is typically a very good approximation
for low-energy waves.

\startdq

\begin{dq}
In figure \subfigref{coil-spring-superposition}{3}, the fifth frame shows the spring
just about perfectly flat. If the two pulses have essentially
canceled each other out perfectly, then why does the motion
pick up again? Why doesn't the spring just stay flat?
\end{dq}

<%
  fig(
    'coil-spring-superposition',
    %q{%
      These pictures show the motion of wave pulses along a spring.
      To make a pulse, one end of the spring was shaken by hand. Movies were filmed, and a series of frame chosen
      to show the motion.\\
      1. A pulse travels to the left.\\
      2. Superposition of two colliding positive pulses.\\
      3. Superposition of two colliding pulses, one positive and one negative.
    },
    {
      'width'=>'fullpage'
    }
  )
%>

<%
  fig(
    'pete',
    %q{%
      As the wave pattern passes the rubber duck, the duck stays put. The water isn't
      moving forward with the wave.
    },
    {
      'width'=>'wide'
    }
  )
%>

<% end_sec() %>
<% begin_sec("2. The medium is not transported with the wave.") %>

Figure \figref{pete} shows a series of water
waves before it has reached a rubber duck (left), having
just passed the duck (middle) and having progressed about a
meter beyond the duck (right). The duck bobs around its
initial position, but is not carried along with the wave.
This shows that the water itself does not flow outward with
the wave. If it did, we could empty one end of a swimming
pool simply by kicking up waves! We must distinguish between
the motion of the medium (water in this case) and the motion
of the wave pattern through the medium. The medium vibrates;
the wave progresses through space.

<% marg(60) %>
<%
  fig(
    'ribbon-on-spring',
    %q{%
      As the wave pulse goes
      by, the ribbon tied to the spring is not carried along. The motion
      of the wave pattern is to the right, but the medium (spring) is
      moving up and down, not to the right.
    }
  )
%>
<% end_marg %>

<% self_check('ribbon-on-spring',<<-'SELF_CHECK'
In figure \\figref{ribbon-on-spring}, you can detect the side-to-side
motion of the spring because the spring appears blurry. At a
certain instant, represented by a single photo, how would
you describe the motion of the different parts of the
spring? Other than the flat parts, do any parts of the
spring have zero velocity?
  SELF_CHECK
  ) %>

\enlargethispage{-2\baselineskip}

__incl(eg/worm)

\enlargethispage{-2\baselineskip}

\begin{eg}{Surfing}\label{eg:surfing}
The incorrect belief that the medium moves with the wave is
often reinforced by garbled secondhand knowledge of surfing.
Anyone who has actually surfed knows that the front of the
board pushes the water to the sides, creating a wake --- the
surfer can even drag his hand through the water, as in figure
\figref{surfing-hand-drag}. If the
water was moving along with the wave and the surfer, this
wouldn't happen. The surfer is carried forward because
forward is downhill, not because of any forward flow of the
water. If the water was flowing forward, then a person
floating in the water up to her neck would be carried along
just as quickly as someone on a surfboard. In fact, it is
even possible to surf down the back side of a wave, although
the ride wouldn't last very long because the surfer and the
wave would quickly part company.
\end{eg}

<% end_sec() %>
<% begin_sec("3. A wave's velocity depends on the medium.") %>

<% marg(70) %>
<%
  fig(
    'surfing-hand-drag',
    %q{%
      Example \ref{eg:surfing}. The
      surfer is dragging his hand in the water.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'breaking-wave',
    %q{Example \ref{eg:breaking-wave}: a breaking wave.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'hull-speed',
    %q{%
      Example \ref{eg:hull-speed}. The boat has run up against
      a limit on its speed because it can't climb over its own wave. Dolphins get around the
      problem by leaping out of the water.
    }
  )
%>
<% end_marg %>

A material object can move with any velocity, and can be
sped up or slowed down by a force that increases or
decreases its kinetic energy. Not so with waves. The
magnitude of a wave's velocity depends on the properties of
the medium (and perhaps also on the shape of the wave, for
certain types of waves). Sound waves travel at about 340 m/s
in air, 1000 m/s in helium.\index{sound!speed of} If you kick up water waves in a
pool, you will find that kicking harder makes waves that are
taller (and therefore carry more energy), not faster. The
sound waves from an exploding stick of dynamite carry a lot
of energy, but are no faster than any other waves. 
Thus although both waves and physical objects carry energy as they move through space,
the energy of the wave relates to its amplitude, not to its speed.

In the
following section we will give an example of the physical
relationship between the wave speed and the properties of the medium.

\begin{eg}{Breaking waves}\label{eg:breaking-wave}
The velocity of water waves
increases with depth. The crest of a wave
travels faster than the trough, and this can cause
the wave to break.
\end{eg}

Once a wave is created, the only reason its speed will
change is if it enters a different medium or if the
properties of the medium change. It is not so surprising
that a change in medium can slow down a wave, but the
reverse can also happen. A sound wave traveling through a
helium balloon will slow down when it emerges into the air,
but if it enters another balloon it will speed back up
again! Similarly, water waves travel more quickly over
deeper water, so a wave will slow down as it passes over an
underwater ridge, but speed up again as it emerges into deeper water.

\begin{eg}{Hull speed}\label{eg:hull-speed}
 The speeds of most boats, and of some surface-swimming
animals, are limited by the fact that they make a wave due
to their motion through the water. The boat in figure \figref{hull-speed}
is going at the same speed as its own waves, and can't go any faster.
No matter how hard the boat pushes
against the water, it can't make the wave move ahead faster and get out of the
way. The wave's speed depends only on the medium. Adding energy to the wave
doesn't speed it up, it just increases its amplitude.

A water wave, unlike many other types of wave, has a speed
that depends on its shape: a broader wave moves faster. The
shape of the wave made by a boat tends to mold itself to
the shape of the boat's hull, so a boat with a longer hull
makes a broader wave that moves faster. The maximum speed of
a boat whose speed is limited by this effect is therefore
closely related to the length of its hull, and the maximum
speed is called the hull speed. Sailboats designed for racing are
not just long and skinny to make
them more streamlined --- they are also long so that their
hull speeds will be high.
\end{eg}

<% end_sec() %>
<% begin_sec("Wave patterns") %>

<% marg(35) %>
<%
  fig(
    'wave-patterns',
    %q{Circular and linear wave patterns.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'wave-patterns-three-d',
    %q{Plane and spherical wave patterns.}
  )
%>

<% end_marg %>
If the magnitude of a wave's velocity vector is preordained,
what about its direction? Waves spread out in all directions
from every point on the disturbance that created them. If
the disturbance is small, we may consider it as a single
point, and in the case of water waves the resulting wave
pattern is the familiar circular ripple, \figref{wave-patterns}/1. If, on the other
hand, we lay a pole on the surface of the water and wiggle
it up and down, we create a linear wave pattern, \figref{wave-patterns}/2. For a
three-dimensional wave such as a sound wave, the analogous
patterns would be spherical waves and plane waves,
\figref{wave-patterns-three-d}.

Infinitely many patterns are possible, but linear or plane
waves are often the simplest to analyze, because the
velocity vector is in the same direction no matter what part
of the wave we look at. Since all the velocity vectors are
parallel to one another, the problem is effectively
one-dimensional. Throughout this chapter and the next, we
will restrict ourselves mainly to wave motion in one
dimension, while not hesitating to broaden our horizons when
it can be done without too much complication.

\pagebreak

\startdqs

\begin{dq}
[see above]
\end{dq}

\begin{dq}
Sketch two positive wave pulses on a string that are
overlapping but not right on top of each other, and draw
their superposition. Do the same for a positive pulse
running into a negative pulse.
\end{dq}

\begin{dq}
A traveling wave pulse is moving to the right on a
string. Sketch the velocity vectors of the various parts of
the string. Now do the same for a pulse moving to the left.
\end{dq}

\begin{dq}
In a spherical sound wave spreading out from a point, how
would the energy of the wave fall off with distance?
\end{dq}

<% end_sec() %>
<% end_sec() %>
<% begin_sec("Waves on a String",3,'waves-on-a-string') %>

So far you have learned some counterintuitive things about
the behavior of waves, but intuition can be trained. The
first half of this section aims to build your intuition by
investigating a simple, one-dimensional type of wave: a wave
on a string. If you have ever stretched a string between the
bottoms of two open-mouthed cans to talk to a friend, you
were putting this type of wave to work. Stringed instruments
are another good example. Although we usually think of a
piano wire simply as vibrating, the hammer actually strikes
it quickly and makes a dent in it, which then ripples out in
both directions. Since this chapter is about free waves, not
bounded ones, we pretend that our string is infinitely long.

<% marg(60) %>
<%
  fig(
    'piano-hammer',
    %q{%
      Hitting a key on a piano causes a hammer
      to come up from underneath and hit a string (actually a set of
      three strings). The result is a pair of pulses moving away from
      the point of impact.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'pulse-on-string',
    %q{%
      A string is struck with a hammer, 1,
      and two pulses fly off, 2.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'ball-and-spring-model',
    %q{%
      A continuous string can be modeled as
      a series of discrete masses connected by springs.
    }
  )
%>

<% end_marg %>
After the qualitative discussion, we will use simple
approximations to investigate the speed of a wave pulse on a
string. This quick and dirty treatment is then followed by a
rigorous attack using the methods of calculus, which may be
skipped by the student who has not studied calculus. How far
you penetrate in this section is up to you, and depends on
your mathematical self-confidence. If you skip some of the math,
you should nevertheless absorb the significance of the result,
discussed on p.~\pageref{wave-on-string-significance}.

<% begin_sec("Intuitive ideas") %>

Consider a string that has been struck, \figref{pulse-on-string}/1, resulting in
the creation of two wave pulses, 2, one traveling to the
left and one to the right. This is analogous to the way
ripples spread out in all directions from a splash in water,
but on a one-dimensional string, ``all directions'' becomes
``both directions.''

We can gain insight by modeling the string as a series of
masses connected by springs. (In the actual string the mass
and the springiness are both contributed by the molecules
themselves.) If we look at various microscopic portions of
the string, there will be some areas that are flat, \figref{ball-and-spring-model}/1,
some that are sloping but not curved, 2, and some that are
curved, 3 and 4. In example 1 it is clear that both
the forces on the central mass cancel out, so it will not
accelerate. The same is true of 2, however. Only in curved
regions such as 3 and 4 is an acceleration produced. In
these examples, the vector sum of the two forces acting on
the central mass is not zero. The important concept is that
curvature makes force: the curved areas of a wave tend to
experience forces resulting in an acceleration toward the
mouth of the curve. Note, however, that an uncurved portion
of the string need not remain motionless. It may move at
constant velocity to either side.

<% end_sec() %>
<% begin_sec("Approximate treatment") %>

<% marg(0) %>
<%
  fig(
    'triangular-pulse',
    %q{A triangular pulse spreads out.}
  )
%>
<% end_marg %>
We now carry out an approximate treatment of the speed at
which two pulses will spread out from an initial indentation
on a string. For simplicity, we imagine a hammer blow that
creates a triangular dent, \figref{triangular-pulse}/1. We will estimate the amount
of time, $t$, required until each of the pulses has traveled
a distance equal to the width of the pulse itself. The
velocity of the pulses is then $\pm w/t$.

As always, the velocity of a wave depends on the properties
of the medium, in this case the string. The properties of
the string can be summarized by two variables: the tension,
$T$, and the mass per unit length, $\mu $ (Greek letter mu).

If we consider the part of the string encompassed by the
initial dent as a single object, then this object has a mass
of approximately $\mu w$ ($\text{mass}/\text{length}\times\text{length}=\text{mass})$.
(Here, and throughout the derivation, we assume that $h$ is
much less than $w$, so that we can ignore the fact that this
segment of the string has a length slightly greater than
$w$.) Although the downward acceleration of this segment of
the string will be neither constant over time nor uniform
across the string, we will pretend that it is constant for
the sake of our simple estimate. Roughly speaking, the time
interval between \figref{triangular-pulse}/1 and 2 is the amount of time required
for the initial dent to accelerate from rest and reach its
normal, flattened position. Of course the tip of the
triangle has a longer distance to travel than the edges, but
again we ignore the complications and simply assume that the
segment as a whole must travel a distance $h$. Indeed, it
might seem surprising that the triangle would so neatly
spring back to a perfectly flat shape. It is an experimental
fact that it does, but our analysis is too crude to
address such details.

The string is kinked, i.e., tightly curved, at the edges of
the triangle, so it is here that there will be large forces
that do not cancel out to zero. There are two forces acting
on the triangular hump, one of magnitude $T$ acting down and
to the right, and one of the same magnitude acting down and
to the left. If the angle of the sloping sides is $\theta $,
then the total force on the segment equals 
$2T \sin\theta $. Dividing the triangle into two right triangles,
we see that $\sin \theta $ equals $h$ divided by the length
of one of the sloping sides. Since $h$ is much less than
$w$, the length of the sloping side is essentially the same
as $w/2$, so we have $\sin \theta =2h/w$, and $F=4Th/w$.
The acceleration of the segment (actually the acceleration
of its center of mass) is
\begin{align*}
 a &= F/m \\
 &= 4Th/\mu w^2 \qquad .
\end{align*}
The time required to move a distance $h$ under constant
acceleration $a$ is found by solving $h=\frac{1}{2}at^2$ to yield
\begin{align*}
 t &= \sqrt{\frac{2h}{a}}\\
   &= w\sqrt{\frac{\mu}{2T}} \qquad .
\end{align*}
Our final result for the velocity of the pulses is
\begin{align*}
 |v| &= \frac{w}{t} \\
     &= \sqrt{\frac{2T}{\mu}}\qquad .
\end{align*}
The remarkable feature of this result is that the velocity
of the pulses does not depend at all on $w$ or $h$, i.e., any
triangular pulse has the same speed. It is an experimental
fact (and we will also prove rigorously in the following
subsection) that any pulse of any kind, triangular or
otherwise, travels along the string at the same speed. Of
course, after so many approximations we cannot expect to
have gotten all the numerical factors right. The correct
result for the velocity of the pulses is
\begin{equation*}
                v         =  \sqrt{\frac{T}{\mu}}      \qquad   .
\end{equation*}

The importance of the above derivation lies in the insight
it brings ---that all pulses move with the same speed ---
rather than in the details of the numerical result. The
reason for our too-high value for the velocity is not hard
to guess. It comes from the assumption that the acceleration
was constant, when actually the total force on the segment
would diminish as it flattened out.

<% end_sec() %>
<% begin_sec("Rigorous derivation using calculus (optional)") %>

After expending considerable effort for an approximate
solution, we now display the power of calculus with a
rigorous and completely general treatment that is nevertheless
much shorter and easier. Let the flat position of the string
define the $x$ axis, so that $y$ measures how far a point on
the string is from equilibrium. The motion of the string is
characterized by $y(x,t)$, a function of two variables.
Knowing that the force on any small segment of string
depends on the curvature of the string in that area, and
that the second derivative is a measure of curvature, it is
not surprising to find that the infinitesimal force
$\der F$ acting on an infinitesimal segment $\der x$ is given by
\begin{equation*}
                \der F         = T \frac{\der^2 y}{\der x^2} \der x      \qquad   .
\end{equation*}
(This can be proved by vector addition of the two infinitesimal
forces acting on either side.) The acceleration is then
$a=\der F/\der m$, or, substituting $\der m=\mu dx$,
\begin{equation*}
        \frac{\der^2 y}{\der t^2} = \frac{T}{\mu}\frac{\der^2 y}{\der x^2}   \qquad   .
\end{equation*}
The second derivative with respect to time is related to the
second derivative with respect to position. This is no more
than a fancy mathematical statement of the intuitive fact
developed above, that the string accelerates so as to
flatten out its curves.

Before even bothering to look for solutions to this
equation, we note that it already proves the principle of
superposition, because the derivative of a sum is the sum of
the derivatives. Therefore the sum of any two solutions
will also be a solution.

Based on experiment, we expect that this equation will be
satisfied by any function $y(x,t)$ that describes a pulse or
wave pattern moving to the left or right at the correct
speed $v$. In general, such a function will be of the form
$y=f(x-vt)$ or $y=f(x+vt)$, where $f$ is any
function of one variable. Because of the chain rule, each
derivative with respect to time brings out a factor of $\pm v$.
Evaluating the second derivatives on both sides of the equation gives
\begin{equation*}
        \left(\pm v\right)^2\: f''                    =       \frac{T}{\mu} f''\qquad   .
\end{equation*}
Squaring gets rid of the sign, and we find that we have a
valid solution for any function $f$, provided that $v$ is given by
\begin{equation*}
                        v  =  \sqrt{\frac{T}{\mu}}   \qquad   .
\end{equation*}

<% end_sec() %>
<% begin_sec("Significance of the result",4) %>\label{wave-on-string-significance}
This specific result for the speed of waves on a string, $v=\sqrt{T/\mu}$, is utterly unimportant.
Don't memorize it. Don't take notes on it. Try to erase it from your memory.

What \emph{is} important about this result is that it is an example of two 
things that are usually true, at least approximately, for mechanical waves in general:
\begin{enumerate}
\item  The speed at which a wave moves does not depend on the size or shape of the wave.
\item  The speed of a mechanical wave depends on a combination of two properties of the medium: some measure of its \emph{inertia}
       and some measure of its \emph{tightness}, i.e., the strength of the force trying to bring the medium back toward equilibrium.
\end{enumerate}

<% self_check('interpret-waves-on-string',<<-'SELF_CHECK'
(a) What is it about the equation $v=\\sqrt{T/\\mu}$ that relates to fact 1 above?
(b) In the equation $v=\\sqrt{T/\\mu}$, which variable is a measure of inertia, and which is a measure of tightness?
(c) Now suppose that we produce compressional wave pulses in a metal rod by tapping the end of the rod with a hammer.
What physical properties of the rod would play the roles of inertia and tightness? How would you expect the speed of compressional waves
in lead to compare with their speed in aluminum?
  SELF_CHECK
  ) %>
<% end_sec() %>
<% end_sec() %>
<% begin_sec("Sound and Light Waves",3) %>\index{sound}\index{light}

<% begin_sec("Sound waves") %>

The phenomenon of sound is easily found to have all the
characteristics we expect from a wave phenomenon:

\begin{itemize}
\item Sound waves obey superposition. Sounds do not knock other
sounds out of the way when they collide, and we can hear
more than one sound at once if they both reach our ear simultaneously.

\item The medium does not move with the sound. Even standing in
front of a titanic speaker playing earsplitting music, we do
not feel the slightest breeze.

\item The velocity of sound depends on the medium. Sound travels
faster in helium than in air, and faster in water than in
helium. Putting more energy into the wave makes it more
intense, not faster. For example, you can easily detect an
echo when you clap your hands a short distance from a large,
flat wall, and the delay of the echo is no shorter for a louder clap.
\end{itemize}

Although not all waves have a speed that is independent of
the shape of the wave, and this property therefore is
irrelevant to our collection of evidence that sound is a
wave phenomenon, sound does nevertheless have this property.
For instance, the music in a large concert hall or stadium
may take on the order of a second to reach someone seated in
the nosebleed section, but we do not notice or care, because
the delay is the same for every sound. Bass, drums, and
vocals all head outward from the stage at 340 m/s,
regardless of their differing wave shapes.

If sound has all the properties we expect from a wave, then
what type of wave is it? It must be a vibration of a
physical medium such as air, since the speed of sound is
different in different media, such as helium or water.
Further evidence is that we don't receive sound signals that
have come to our planet through outer space. The roars and
whooshes of Hollywood's space ships are fun, but scientifically wrong.\footnote{Outer
space is not a perfect vacuum, so it is possible for sounds waves to travel through
it. However, if we want to create a sound wave, we typically do it by creating
vibrations of a physical object, such as the sounding board of a guitar,
the reed of a saxophone, or a speaker cone. The lower the density of the surrounding
medium, the less efficiently the energy can be converted into sound and
carried away. An isolated tuning fork, left to vibrate in interstellar space,
would dissipate the energy of its vibration into internal heat at a rate
many orders of magnitude greater than the rate of sound emission into the
nearly perfect vacuum around it.}

We can also tell that sound waves consist of compressions
and expansions, rather than sideways vibrations like the
shimmying of a snake. Only compressional vibrations would be
able to cause your eardrums to vibrate in and out. Even for
a very loud sound, the compression is extremely weak; the
increase or decrease compared to normal atmospheric pressure
is no more than a part per million. Our ears are apparently
very sensitive receivers! 
Unlike a wave on a string, which vibrates in the direction perpendicular
to the direction in which the wave pattern moves, a sound wave is
a longitudinal wave,\index{longitudinal wave}\index{wave!longitudinal} i.e., one in which the vibration is forward and
backward along the direction of motion.

<% end_sec() %>
<% begin_sec("Light waves") %>

Entirely similar observations lead us to believe that light
is a wave, although the concept of light as a wave had a
long and tortuous history. It is interesting to note that
Isaac Newton very influentially advocated a contrary idea
about light. The belief that matter was made of atoms was
stylish at the time among radical thinkers (although there
was no experimental evidence for their existence), and it
seemed logical to Newton that light as well should be made
of tiny particles, which he called corpuscles (Latin for
``small objects''). Newton's triumphs in the science of
mechanics, i.e., the study of matter, brought him such great
prestige that nobody bothered to question his incorrect
theory of light for 150 years. One persuasive proof that
light is a wave is that according to Newton's theory, two
intersecting beams of light should experience at least some
disruption because of collisions between their corpuscles.
Even if the corpuscles were extremely small, and collisions
therefore very infrequent, at least some dimming should have
been measurable. In fact, very delicate experiments have
shown that there is no dimming.

The wave theory of light was entirely successful up until
the 20th century, when it was discovered that not all the
phenomena of light could be explained with a pure wave
theory. It is now believed that both light and matter are
made out of tiny chunks which have \emph{both} wave and
particle properties. For now, we will content ourselves with
the wave theory of light, which is capable of explaining a
great many things, from cameras to rainbows.

If light is a wave, what is waving? What is the medium that
wiggles when a light wave goes by? It isn't air. A vacuum is
impenetrable to sound, but light from the stars travels
happily through zillions of miles of empty space. Light
bulbs have no air inside them, but that doesn't prevent the
light waves from leaving the filament. For a long time,
physicists assumed that there must be a mysterious medium
for light waves, and they called it the aether (not to be
confused with the chemical). Supposedly the aether existed
everywhere in space, and was immune to vacuum pumps. The
details of the story are more fittingly reserved for later
in this course, but the end result was that a long series of
experiments failed to detect any evidence for the aether,
and it is no longer believed to exist. Instead, light can be
explained as a wave pattern made up of electrical and magnetic fields.

 %%---------------------------------------------------------

<% figure_in_toc('surfing-hand-drag',{'scootx'=>-6}) %>

 %%---------------------------------------------------------

<% end_sec() %>
<% end_sec() %>
<% begin_sec("Periodic Waves",4) %>\index{wave!periodic}

<% begin_sec("Period and frequency of a periodic wave") %>

<% marg(0) %>
<%
  fig(
    'aaah',
    %q{%
      A graph of pressure versus time for a periodic sound
      wave, the vowel ``ah.''
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'shhhh',
    %q{A similar graph for a non-periodic wave, ``sh.''}
  )
%>
<% end_marg %>
You choose a radio station by selecting a certain frequency.
We have already defined period and frequency for vibrations,
but what do they signify in the case of a wave? We can
recycle our previous definition simply by stating it in
terms of the vibrations that the wave causes as it passes a
receiving instrument at a certain point in space. For a
sound wave, this receiver could be an eardrum or a
microphone. If the vibrations of the eardrum repeat
themselves over and over, i.e., are periodic, then we
describe the sound wave that caused them as periodic.
Likewise we can define the period and frequency of a wave in
terms of the period and frequency of the vibrations it
causes. As another example, a periodic water wave would be
one that caused a rubber duck to bob in a periodic manner
as they passed by it.

The period of a sound wave correlates with our sensory
impression of musical pitch. A high frequency (short period)
is a high note. The sounds that really define the musical
notes of a song are only the ones that are periodic. It is
not possible to sing a non-periodic sound like ``sh''
with a definite pitch.

The frequency of a light wave corresponds to color. Violet
is the high-frequency end of the rainbow, red the low-frequency
end. A color like brown that does not occur in a rainbow is
not a periodic light wave. Many phenomena that we do not
normally think of as light are actually just forms of light
that are invisible because they fall outside the range of
frequencies our eyes can detect. Beyond the red end of the
visible rainbow, there are infrared and radio waves. Past
the violet end, we have ultraviolet, x-rays, and gamma rays.

<% end_sec() %>
<% begin_sec("Graphs of waves as a function of position") %>

<% marg(0) %>
<%
  fig(
    'strip-chart-recorder',
    %q{A strip chart recorder.}
  )
%>
<% end_marg %>
Some waves, like sound waves, are easy to study by placing
a detector at a certain location in space and studying the
motion as a function of time. The result is a graph whose
horizontal axis is time. With a water wave, on the other
hand, it is simpler just to look at the wave directly. This
visual snapshot amounts to a graph of the height of the
water wave as a function of \emph{position}. Any wave can be
represented in either way.

An easy way to visualize this is in terms of a strip chart
recorder, an obsolescing device consisting of a pen that
wiggles back and forth as a roll of paper is fed under it.
It can be used to record a person's electrocardiogram, or
seismic waves too small to be felt as a noticeable
earthquake but detectable by a seismometer. Taking the
seismometer as an example, the chart is essentially a record
of the ground's wave motion as a function of time, but if
the paper was set to feed at the same velocity as the motion
of an earthquake wave, it would also be a full-scale
representation of the profile of the actual wave pattern
itself. Assuming, as is usually the case, that the wave
velocity is a constant number regardless of the wave's
shape, knowing the wave motion as a function of time is
equivalent to knowing it as a function of position.

<% end_sec() %>
<% begin_sec("Wavelength") %>\index{wavelength}

<% marg(0) %>
<%
  fig(
    'wavelength',
    %q{A water wave profile created by a series of repeating pulses.}
  )
%>
<% end_marg %>
Any wave that is periodic will also display a repeating
pattern when graphed as a function of position. The distance
spanned by one repetition is referred to as one \emph{wavelength}.
The usual notation for wavelength is $\lambda $, the Greek
letter lambda. Wavelength is to space as period is to time.

<%
  fig(
    'circular-and-linear-wavelengths',
    %q{Wavelengths of linear and circular water waves.},
    {
      'width'=>'wide'
    }
  )
%>

<% end_sec() %>
<% begin_sec("Wave velocity related to frequency and wavelength") %>

Suppose that we create a repetitive disturbance by kicking
the surface of a swimming pool. We are essentially making a
series of wave pulses. The wavelength is simply the distance
a pulse is able to travel before we make the next pulse. The
distance between pulses is $\lambda $, and the time between
pulses is the period, $T$, so the speed of the wave is the
distance divided by the time,
\begin{equation*}
                v    =    \lambda /T   .
\end{equation*}

This important and useful relationship is more commonly
written in terms of the frequency,

\begin{equation*}
                v    =    f \lambda    \qquad   .
\end{equation*}

\pagebreak

\begin{eg}{Wavelength of radio waves}
\egquestion The speed of light is $3.0\times10^8$ m/s. What
is the wavelength of the radio waves emitted by KMHD, a
station whose frequency is 89.1 MHz?

\eganswer Solving for wavelength, we have
\begin{align*}
 \lambda &= v/f \\
 &= (3.0\times10^8\ \munit/\sunit)/(89.1\times10^6\ \sunit^{-1}) \\
 &= 3.4\ \munit
\end{align*}
The size of a radio antenna is closely related to the
wavelength of the waves it is intended to receive. The match
need not be exact (since after all one antenna can receive
more than one wavelength!), but the ordinary ``whip''
antenna such as a car's is 1/4 of a wavelength. An antenna
optimized to receive KMHD's signal would have a length of $3.4\ \munit/4= 0.85\ \munit$.
\end{eg}

<%
  fig(
    'ultrasound',
    %q{%
      Ultrasound, i.e., sound with frequencies higher than the range of human
      hearing, was used to make this image of a fetus. The resolution of the image is related to
      the wavelength, since details smaller than about one wavelength cannot be resolved. High
      resolution therefore requires a short wavelength, corresponding to a high frequency.
    },
    {
      'width'=>'wide',
      'sidecaption'=>true
    }
  )
%>

The equation $v=f \lambda $ defines a fixed relationship
between any two of the variables if the other is held fixed.
The speed of radio waves in air is almost exactly the same
for all wavelengths and frequencies (it is exactly the same
if they are in a vacuum), so there is a fixed relationship
between their frequency and wavelength. Thus we can say
either ``Are we on the same wavelength?'' or ``Are we on
the same frequency?''

<% marg(32) %>
<%
  fig(
    'wavelength-change',
    %q{%
      A water wave traveling into a region with a different
      depth changes its wavelength.
    }
  )
%>
<% end_marg %>
A different example is the behavior of a wave that travels
from a region where the medium has one set of properties to
an area where the medium behaves differently. The frequency
is now fixed, because otherwise the two portions of the wave
would otherwise get out of step, causing a kink or
discontinuity at the boundary, which would be unphysical. (A
more careful argument is that a kink or discontinuity would
have infinite curvature, and waves tend to flatten out their
curvature. An infinite curvature would flatten out
infinitely fast, i.e., it could never occur in the first
place.) Since the frequency must stay the same, any change
in the velocity that results from the new medium must cause
a change in wavelength.

The velocity of water waves depends on the depth of the
water, so based on $\lambda =v/f$, we see that water waves
that move into a region of different depth must change their
wavelength, as shown in figure \figref{wavelength-change}. This effect
can be observed when ocean waves come up to the shore. If
the deceleration of the wave pattern is sudden enough, the
tip of the wave can curl over, resulting in a breaking wave.

\begin{optionaltopic}{A note on dispersive waves}
The discussion of wave velocity given here is actually an oversimplification
for a wave whose velocity depends on its frequency and wavelength. Such a wave
is called a dispersive wave. Nearly all the waves we deal with in this course
are non-dispersive, but the issue becomes important in quantum physics,
as discussed in more detail in optional section \ref{sec:dispersive-waves}.
\end{optionaltopic}

<% end_sec() %>
<% begin_sec("Sinusoidal waves") %>

Sinusoidal waves are the most important special case of
periodic waves. In fact, many scientists and engineers would
be uncomfortable with defining a waveform like the ``ah''
vowel sound as having a definite frequency and wavelength,
because they consider only sine waves to be pure examples of
a certain frequency and wavelengths. Their bias is not
unreasonable, since the French mathematician Fourier showed
that any periodic wave with frequency $f$ can be constructed
as a superposition of sine waves with frequencies $f$, $2f$,
$3f$, ... In this sense, sine waves are the basic, pure
building blocks of all waves. (Fourier's result so surprised
the mathematical community of France that he was ridiculed
the first time he publicly presented his theorem.)

However, what definition to use is a matter of utility. Our
sense of hearing perceives any two sounds having the same
period as possessing the same pitch, regardless of whether
they are sine waves or not. This is undoubtedly because our
ear-brain system evolved to be able to interpret human
speech and animal noises, which are periodic but not
sinusoidal. Our eyes, on the other hand, judge a color as
pure (belonging to the rainbow set of colors) only
if it is a sine wave.

\pagebreak

\startdq

\begin{dq}
Suppose we superimpose two sine waves with equal amplitudes
but slightly different frequencies, as shown in the figure.
What will the superposition look like? What would this sound
like if they were sound waves?

<%
  fig(
    'dq-beats',
    '',
    {
      'width'=>'wide',
      'anonymous'=>true,
      'float'=>false
    }
  )
%>
\end{dq}

<% end_sec() %>
<% end_sec() %>
<% begin_sec("The Doppler Effect",0,'doppler') %>
<% marg(3) %>
<%
  fig(
    'doppler',
    %q{%
      The pattern of waves made by a point source moving to the right
      across the water. Note the shorter wavelength of the forward-emitted waves and the longer
      wavelength of the backward-going ones.
    }
  )
%>
<% end_marg %>
Figure \figref{doppler} shows the wave pattern made by the tip of a
vibrating rod which is moving across the water. If the rod
had been vibrating in one place, we would have seen the
familiar pattern of concentric circles, all centered on the
same point. But since the source of the waves is moving, the
wavelength is shortened on one side and lengthened on the
other. This is known as the \index{Doppler effect}Doppler effect.

Note that the velocity of the waves is a fixed property of
the medium, so for example the forward-going waves do not
get an extra boost in speed as would a material object like
a bullet being shot forward from an airplane.

We can also infer a change in frequency. Since the velocity
is constant, the equation $v=f\lambda $ tells us that the
change in wavelength must be matched by an opposite change
in frequency: higher frequency for the waves emitted
forward, and lower for the ones emitted backward. The
frequency Doppler effect is the reason for the familiar
dropping-pitch sound of a race car going by. As the car
approaches us, we hear a higher pitch, but after it passes
us we hear a frequency that is lower than normal.

The Doppler effect will also occur if the observer is moving
but the source is stationary. For instance, an observer
moving toward a stationary source will perceive one crest of
the wave, and will then be surrounded by the next crest
sooner than she otherwise would have, because she has moved
toward it and hastened her encounter with it. Roughly
speaking, the Doppler effect depends only the relative
motion of the source and the observer, not on their absolute
state of motion (which is not a well-defined notion in
physics) or on their velocity relative to the medium.

Restricting ourselves to the case of a moving source, and to
waves emitted either directly along or directly against the
direction of motion, we can easily calculate the wavelength,
or equivalently the frequency, of the Doppler-shifted waves.
Let $v$ be the velocity of the waves, and $v_s$ the velocity
of the source. The wavelength of the forward-emitted waves
is shortened by an amount $v_s T$ equal to the distance
traveled by the source over the course of one period. Using
the definition $f=1/T$ and the equation $v=f\lambda $, we
find for the wavelength of the Doppler-shifted wave the equation
\begin{equation*}
        \lambda'    = \left(1-\frac{v_s}{v}\right)\lambda      \qquad   .
\end{equation*}
A similar equation can be used for the backward-emitted
waves, but with a plus sign rather than a minus sign.

\begin{eg}{Doppler-shifted sound from a race car}
\egquestion If a race car moves at a velocity of 50 m/s, and
the velocity of sound is 340 m/s, by what percentage are the
wavelength and frequency of its sound waves shifted for an
observer lying along its line of motion?

\eganswer For an observer whom the car is approaching, we find
\begin{equation*}
        1-\frac{v_s}{v}                 =    0.85   \qquad   ,
\end{equation*}
so the shift in wavelength is 15\%. Since the frequency is
inversely proportional to the wavelength for a fixed value
of the speed of sound, the frequency is shifted upward by
\begin{equation*}
                1/0.85         =    1.18  ,
\end{equation*}
i.e., a change of 18\%. (For velocities that are small
compared to the wave velocities, the Doppler shifts of the
wavelength and frequency are about the same.)
\end{eg}

\begin{eg}{Doppler shift of the light emitted by a race car}
\egquestion What is the percent shift in the wavelength of the
light waves emitted by a race car's headlights?

\eganswer Looking up the speed of light, $v=3.0\times10^8$ m/s, we find
\begin{equation*}
        1-\frac{v_s}{v} =    0.99999983   \qquad   ,
\end{equation*}
i.e., the percentage shift is only 0.000017\%.
\end{eg}

The second example shows that under ordinary earthbound
circumstances, Doppler shifts of light are negligible
because ordinary things go so much slower than the speed of
light. It's a different story, however, when it comes to
stars and galaxies, and this leads us to a story that has
profound implications for our understanding of the
origin of the universe.

<% marg(20) %>
<%
  fig(
    'doppler-radar',
    %q{%
      Example \ref{eg:doppler-radar}. A Doppler
      radar image of Hurricane Katrina, in 2005.
    }
  )
%>
<% end_marg %>
\begin{eg}{Doppler radar}\label{eg:doppler-radar}
The first use of radar was by Britain during World War II: antennas on
the ground sent radio waves up into the sky, and detected the echoes when
the waves were reflected from German planes. Later, air forces wanted to
mount radar antennas on airplanes, but then there was a problem, because
if an airplane wanted to detect another airplane at a lower altitude, it
would have to aim its radio waves downward, and then it would get echoes
from the ground. The solution was the invention of Doppler radar, in which
echoes from the ground were differentiated from echoes from other aircraft
according to their Doppler shifts. A similar technology is used by
meteorologists to map out rainclouds without being swamped by reflections
from the ground, trees, and buildings.
\end{eg}

\begin{optionaltopic}{Optional topic: Doppler shifts of light}\label{qualitative-doppler-light}
If Doppler shifts depend only on the relative motion of the source and receiver, then
 there is no way for a person moving with the source and another person moving with the receiver
 to determine who is moving and who isn't. Either can blame the Doppler shift entirely
 on the other's motion and claim to be at rest herself. This is entirely in agreement
 with the principle stated originally by Galileo that all motion is relative.

On the other hand, a careful analysis of the Doppler shifts of water or sound
 waves shows that it is only approximately true, at low speeds, that the shifts  just depend
 on the relative motion of the source and observer. For instance, it is possible for
 a jet plane to keep up with its own sound waves, so that the sound waves appear to
 stand still to the pilot of the plane. The pilot then knows she is moving at exactly the
 speed of sound. The reason this doesn't disprove the relativity of motion is that the pilot
 is not really determining her absolute motion but rather her motion relative to the air, which is the medium of the sound waves.

Einstein realized that this solved the problem for sound or water waves, but would not
 salvage the principle of relative motion in the case of light waves, since light
 is not a vibration of any physical medium such as water or air. Beginning by
 imagining what a beam of light would look like to a person riding a
 motorcycle alongside it, Einstein eventually came up with a radical new way of describing
 the universe, in which space and time are distorted as measured by observers in different states
 of motion. As a consequence of this theory of relativity, he showed that light waves would have
 Doppler shifts that would exactly, not just approximately, depend only on the relative motion of the source and receiver.
The resolution of the motorcycle paradox is given in example \ref{eg:einstein-motorcycle} on p.~\pageref{eg:einstein-motorcycle},
and a quantitative discussion of Doppler shifts of light is given on p.~\pageref{subsec:doppler-light}.
\end{optionaltopic}

<% begin_sec("The Big Bang",nil,'hubble') %>\label{bigbang}
<% marg(0) %>
<%
  fig(
    'm51',
    %q{%
      The galaxy M51. Under high magnification, the milky clouds reveal
      themselves to be composed of trillions of stars.
    }
  )
%>
<% end_marg %>

As soon as astronomers began looking at the sky through
telescopes, they began noticing certain objects that looked
like clouds in deep space. The fact that they looked the
same night after night meant that they were beyond the
earth's atmosphere. Not knowing what they really were, but
wanting to sound official, they called them ``nebulae,'' a
Latin word meaning ``clouds'' but sounding more impressive.
In the early 20th century, astronomers realized that
although some really were clouds of gas (e.g., the middle
``star'' of Orion's sword, which is visibly fuzzy even to
the naked eye when conditions are good), others were what we
now call galaxies: virtual island universes consisting of
trillions of stars (for example the Andromeda Galaxy, which
is visible as a fuzzy patch through binoculars). Three
hundred years after Galileo had resolved the Milky Way into
individual stars through his telescope, astronomers realized
that the universe is made of galaxies of stars, and the
Milky Way is simply the visible part of the flat disk of our
own galaxy, seen from inside.

This opened up the scientific study of cosmology, the
structure and history of the universe as a whole, a field
that had not been seriously attacked since the days of
Newton. Newton had realized that if gravity was always
attractive, never repulsive, the universe would have a
tendency to collapse. His solution to the problem was to
posit a universe that was infinite and uniformly populated
with matter, so that it would have no geometrical center.
The gravitational forces in such a universe would always
tend to cancel out by symmetry, so there would be no
collapse. By the 20th century, the belief in an unchanging
and infinite universe had become conventional wisdom in
science, partly as a reaction against the time that had been
wasted trying to find explanations of ancient geological
phenomena based on catastrophes suggested by biblical
events like Noah's flood.

In the 1920's astronomer Edwin Hubble began studying the
Doppler shifts of the light emitted by galaxies. A former
college football player with a serious nicotine addiction,
Hubble did not set out to change our image of the beginning
of the universe. His autobiography seldom even mentions the
cosmological discovery for which he is now remembered. When
astronomers began to study the Doppler shifts of galaxies,
they expected that each galaxy's direction and velocity of
motion would be essentially random. Some would be approaching
us, and their light would therefore be Doppler-shifted to
the blue end of the spectrum, while an equal number would be
expected to have red shifts. What Hubble discovered instead
was that except for a few very nearby ones, all the galaxies
had red shifts, indicating that they were receding from us
at a hefty fraction of the speed of light. Not only that,
but the ones farther away were receding more quickly. The
speeds were directly proportional to their distance from us.
<% marg(220) %>
<%
  fig(
    'sirius',
    %q{How do astronomers know what mixture of wavelengths a star emitted originally, so that they can tell how much the Doppler shift was? This image (obtained by the author with equipment costing about \$5, and no telescope) shows the mixture of colors emitted by the star Sirius. (If you have the book in black and white, blue is on the left and red on the right.) The star appears white or bluish-white to the eye, but any light looks white if it contains roughly an equal mixture of the rainbow colors, i.e., of all the pure sinusoidal waves with wavelengths lying in the visible range. Note the black ``gap teeth.'' These are the fingerprint of hydrogen in the outer atmosphere of Sirius. These wavelengths are selectively absorbed by hydrogen. Sirius is in our own galaxy, but similar stars in other galaxies would have the whole pattern shifted toward the red end, indicating they are moving away from us.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'mount-wilson',
    %q{The telescope at Mount Wilson used by Hubble.}
  )
%>
<% end_marg %>

Did this mean that the earth (or at least our galaxy) was
the center of the universe? No, because Doppler shifts of
light only depend on the relative motion of the source and
the observer. If we see a distant galaxy moving away from us
at 10\% of the speed of light, we can be assured that the
astronomers who live in that galaxy will see ours receding
from them at the same speed in the opposite direction. The
whole universe can be envisioned as a rising loaf of raisin
 bread. As the bread expands, there is more and more space
between the raisins. The farther apart two raisins are, the
greater the speed with which they move apart.

Extrapolating backward in time using the known laws of
physics, the universe must have been denser and denser at
earlier and earlier times. At some point, it must have been
extremely dense and hot, and we can even detect the
radiation from this early fireball, in the form of microwave
radiation that permeates space. The phrase Big Bang was
originally coined by the doubters of the theory to make it
sound ridiculous, but it stuck, and today essentially all
astronomers accept the Big Bang theory based on the very
direct evidence of the red shifts and the cosmic microwave
background radiation.

<% end_sec() %>
<% begin_sec("What the big bang is not") %>

Finally it should be noted what the Big Bang theory is not.
It is not an explanation of \emph{why} the universe exists.
Such questions belong to the realm of religion, not science.
Science can find ever simpler and ever more fundamental
explanations for a variety of phenomena, but ultimately
science takes the universe as it is according to observations.


Furthermore, there is an unfortunate tendency, even among
many scientists, to speak of the Big Bang theory as a
description of the very first event in the universe, which
caused everything after it. Although it is true that time
may have had a beginning (Einstein's theory of general
relativity admits such a possibility), the methods of
science can only work within a certain range of conditions
such as temperature and density. Beyond a temperature of
about $10^9$ degrees C, the random thermal motion of
subatomic particles becomes so rapid that its velocity is
comparable to the speed of light. Early enough in the
history of the universe, when these temperatures existed,
Newtonian physics becomes less accurate, and we must
describe nature using the more general description given by
Einstein's theory of relativity, which encompasses Newtonian
physics as a special case. At even higher temperatures,
beyond about $10^{33}$ degrees, physicists know that
Einstein's theory as well begins to fall apart, but we don't
know how to construct the even more general theory of nature
that would work at those temperatures. No matter how far
physics progresses, we will never be able to describe nature
at infinitely high temperatures, since there is a limit to
the temperatures we can explore by experiment and observation
in order to guide us to the right theory. We are confident
that we understand the basic physics involved in the
evolution of the universe starting a few minutes after the
Big Bang, and we may be able to push back to milliseconds or
microseconds after it, but we cannot use the methods of
science to deal with the beginning of time itself.

\pagebreak

\startdqs

<% marg(40) %>
<%
  fig(
    'shock-wave',
    %q{%
      Shock waves from by the X-15
      rocket plane, flying at 3.5 times the speed of sound.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'sonic-boom',
    %q{%
      As in figure \figref{shock-wave}, this plane shows a shock wave. The
      sudden decompression of the air causes water droplets to
      condense, forming a cloud.
    }
  )
%>

<% end_marg %>

\begin{dq}\label{dq:sonic-boom}
If an airplane travels at exactly the speed of sound,
what would be the wavelength of the forward-emitted part of
the sound waves it emitted? How should this be interpreted,
and what would actually happen? What happens if it's going
faster than the speed of sound? Can you use this to explain what
you see in figure \figref{shock-wave}?
\end{dq}

\begin{dq}
If bullets go slower than the speed of sound, why can a
supersonic fighter plane catch up to its own sound, but
not to its own bullets?
\end{dq}

\begin{dq}
If someone inside a plane is talking to you, should their
speech be Doppler shifted?
\end{dq}

\begin{dq}
The plane in figure \figref{sonic-boom} was photographed when it was traveling
at a speed close to the speed of sound. Comparing figures \figref{shock-wave}
and \figref{sonic-boom}, how can we tell from the angles of the cones
that the speed is much lower in figure \figref{sonic-boom}?
\end{dq}

<% end_sec() %>
<% end_sec() %>\begin{summary}

\begin{vocab}

\vocabitem{superposition}{the adding together of waves that overlap with each other}

\vocabitem{medium}{a physical substance whose vibrations constitute a wave}

\vocabitem{wavelength}{the distance in space between repetitions of a periodic wave}

\vocabitem{Doppler effect}{the change in a wave's frequency and
wavelength due to the motion of the source or the observer or both}

\end{vocab}

\begin{notation}

\notationitem{$\lambda$}{wavelength (Greek letter lambda)}

\end{notation}

\begin{summarytext}

Wave motion differs in three important ways from the motion
of material objects:

(1) Waves obey the principle of superposition. When two
waves collide, they simply add together.

(2) The medium is not transported along with the wave. The
motion of any given point in the medium is a vibration about
its equilibrium location, not a steady forward motion.

(3) The velocity of a wave depends on the medium, not on the
amount of energy in the wave. (For some types of waves,
notably water waves, the velocity may also depend on
the shape of the wave.)

Sound waves consist of increases and decreases (typically
very small ones) in the density of the air. Light is a wave,
but it is a vibration of electric and magnetic fields, not
of any physical medium. Light can travel through a vacuum.

A periodic wave is one that creates a periodic motion in a
receiver as it passes it. Such a wave has a well-defined
period and frequency, and it will also have a wavelength,
which is the distance in space between repetitions of the
wave pattern. The velocity, frequency, and wavelength of a
periodic wave are related by the equation
\begin{equation*}
                v  =  f \lambda .
\end{equation*}
A wave emitted by a moving source will be shifted in
wavelength and frequency. The shifted wavelength is
given by the equation
\begin{equation*}
\lambda'    = \left(1-\frac{v_s}{v}\right)\lambda \qquad ,
\end{equation*}
where $v$ is the velocity of the waves and $v_s$ is the
velocity of the source, taken to be positive or negative so
as to produce a Doppler-lengthened wavelength if the source
is receding and a Doppler-shortened one if it approaches. A
similar shift occurs if the observer is moving, and in
general the Doppler shift depends approximately only on the
relative motion of the source and observer if their
velocities are both small compared to the waves' velocity.
(This is not just approximately but exactly true for light
waves, and as required by Einstein's
Theory of Relativity.)

\end{summarytext}

\end{summary}

<% begin_hw_sec %>

<% begin_hw('alterations') %>__incl(hw/alterations)<% end_hw() %>

<% marg(0) %>
<%
  fig(
    'hw-water-wave-pulse',
    %q{Problem \ref{hw:waterpulse}.}
  )
%>

<% end_marg %>
<% begin_hw('waterpulse') %>__incl(hw/waterpulse)<% end_hw() %>

\pagebreak

<% marg(0) %>
<%
  fig(
    'hw-sine-single-period',
    %q{Problem \ref{hw:stringkinematics}.}
  )
%>

<% end_marg %>
<% begin_hw('stringkinematics') %>__incl(hw/stringkinematics)<% end_hw() %>

<% begin_hw('dopplerf') %>__incl(hw/dopplerf)<% end_hw() %>

<% begin_hw('superpositionexperiment') %>__incl(hw/superpositionexperiment)<% end_hw() %>

<% begin_hw('middlec') %>
 The musical note middle C has a frequency of 262 Hz.
What are its period and wavelength?\answercheck
<% end_hw() %>

<% begin_hw('dopplersing') %>__incl(hw/dopplersing)<% end_hw() %>

<% begin_hw('soundtemp') %>__incl(hw/soundtemp)<% end_hw() %>

<% end_hw_sec %>
<% end_chapter() %>