chapter 16 Wave and their Fundamentals of Waves

A wave is a disturbance in a medium that carries energy without a net movement of particles. It may take the form of elastic deformation, a variation of pressure, electric or magnetic intensity, electric potential, or temperature.

2. WHAT IS PHYSICS? ——
Il || One of the primary subjects of physics is waves. To see how important
waves are in the modern world, just consider the music industry. Every piece of
music you hear, from some retro-punk band playing in a campus dive to the most
eloquent concerto playing on the Web, depends on performers producing waves
and your detecting those waves. In between production and detection, the info
mation carried by the waves might need to be transmitted (as in a live perfo
mance on the Web) or recorded and then reproduced (as with CDs, DVDs, or the
other devices currently being developed in engineering labs worldwide). The
financial importance of controlling music waves is staggering, and the rewards to

engineers who develop new control techniques can be rich

This chapter focuses on waves traveling along a stretched string, such as on
a guitar. The next chapter focuses on sound waves, such as those produced by
a guitar string being played. Before we do all this, though, our first job is to clas-
sify the countless waves of the everyday world into basic types.

3. 16-2 Types of Waves

Waves are of three main types:

Mechanical waves. These waves are most familiar because we encounter
them almost constantly; common examples include water waves, sound waves,
and seismic waves. All these waves have two central features: They are gov
erned by Newton’s laws, and they can exist only within a material medium,

such as water, air, and rock.

Electromagnetic waves. These waves are less familiar, but you use them
constantly; common examples include visible and ultraviolet light, radio and
television waves, microwaves, x rays, and radar waves. These waves require no
material medium to exist. Light waves from stars, for example, travel through
the vacuum of space to reach us. All electromagnetic waves travel through a
vacuum at the peed c = 299 792 458 m/s.

3. Matter waves. Although these waves are commonly used in modern tech-
nology, they are probably very unfamiliar to you. These waves are associated
with electrons, protons, and other fundamental particles, and even atoms and
molecules. Because we commonly think of these particles as constituting mat-
ter,such waves are called matter waves.

4. Transverse and Longitudinal Waves

5. Fig. 16-2 A sound wave is set up in an
air-filled pipe by moving a piston back and
forth. Because the oscillations of an ele
ment of the air (represented by the dot) are
parallel to the direction in which the wave
travels, the wave is a longitudinal wave.

6. 6-4 Wavelength and Frequency

To completely describe a wave on a string (and the motion of any element along
its length), we need a function that gives the shape of the wave. This means that

we need a relation in the form
y = h(x, à, (16-1)

sin(ky — mí

Angular
wave number
| | ‘
Position — Angular
frequency

Fig. 16-3 The names of the quantities in
Eq. 16-2, for a transverse sinusoidal wave.

7. Watch this spot in this
series of snapshots.

(e)
Fig. 16-4 Five “snapshots” of a string
wave traveling in the positive direction of
an x axis. The amplitude y,, is indicated. A
typical wavelength A, measured from an ar-
bitrary position x. is also indicated.

8. The wavelength A of a wave is the distance (parallel to the direction of the wave’s
travel) between repetitions of the shape of the wave (or wave shape). A typical
wavelength is marked in Fig. 16-4a, which is a snapshot of the wave at time £ = 0.
At that time, Eq. 16-2 gives, for the description of the wave shape,

y(x,0) = y,, sin kx. (16-3)

This is a graph,
not a snapshot.

Ym SIN kx, = y, Sin k(x; + A)

Fig. 16-5 A graph of the displacement
of the string element at x = 0 as a function
of time, as the sinusoidal wave of Fig. 16-4
passes through the element. The amplitude
s indicated. A typical period T,mea-
red from an arbitrary * oo
indicated. We call k the angular wave number of the wave; its SI unit is the radian per meter,

9. Period, Angular Frequency, and Frequency

= y„sin(-ot)
—Ym sin wt (x = 0).

sin(—a)

We define the period of oscillation T of a wave to be the time any string
element takes to move through one full oscillation. A typical period is marked on
the graph of Fig. 16-5. Applying Eq. 16-6 to both ends of this time interval and
equating the results yield

—y,, Sin ot = —y,, Sin w(t, + T)

10. Y sin wf, = —y,, Sin w(t, + T)
=—y,,sin(wt, + wT).

This can be true only if 7 = 27, or
(angular frequency). (16-8)

We call w the angular frequency of the wave; its SI unit is the radian per second.
Look back at the five snapshots of a traveling wave in Fig. 16-4. The time
between snapshots is iT. Thus, by the fifth snapshot, every string element has
made one full oscillation.
The frequency f of a wave is defined as 1/T and is related to the angular
frequency w by

(frequency). (16-9)

o
27

Like the frequency of simple harmonic motion in Chapter 15, this frequency fis a
number of oscillations per unit time—here, the number made by a string element
as the wave moves through it. As in Chapter 15, fis usually measured in hertz or
its multiples, such as kilohertz.

11. 16-5 The Speed of a Traveling Wave

kx — wt = aconstant.

Wave at {= At
0

12. Eq. 16-2 with —1. This corresponds to the condition

kx + wt = aconstant,

m Sin(kx + ot).

y(x, t) = h(kx + wf),

The minus sign (compare Eq. 16-12) verifies that the wave is indeed moving in
the negative direction of x and justifies our switching the sign of the time variable.

Consider now a wave of arbitrary shape, given by

13. 16-9 The Principle of Superposition for Waves

Fig. 1
show two pul

This is another example of the principle of superposition, which says that when sev-
eral effects occur simultane eir net effect is the sum of the individual effects.

14. 1 Interference of Waves

e traveling along a stretched string be g

sin(kx — wt)

sin(kx — wt + $).
From the principle of superposition (Eq. 16-46), the resultant wave is the
algebraic sum of the two interfering waves and has displacement

= y,(x, 0) + px 0)

Ym COS 56] sin(kx— wt +

sina ,

The resultant wave differs from the interfering waves in two respects:

15. (1) its phase constant is i,

(2) its amplitude y,

yi, = 12y,, cosióbl (amplitude)

0 rad (or 0°), the two interfering waves are exactly in phase,
n Eq. 16-51 reduces to

y (x, 1) = 2y,, sin(kx — ot) (¢= 0).

16. Being exactly out of This is an intermediate
phase, they produce situation, with an
a flat string. intermediat

vila 0
and
ya(x, 0)

rep

(a)

17. If $ = rr rad (or 180°), the interfering waves 2 exactly out of phase as in

Fig. 16-13b. Then cos 54 becomes cos 7/2 ), and the amplitude of the resultant
wave as given by Eq. 16-52 is zero. We then have, for all valu and,

Y'(x 9 =0 (d= rad).

This type of interference is called fully destructive interference

use a sinusoidal wave repeats its shape every 27rrad,

18. Phase Difference and Resulting Interference Types*

Phase Difference, in mplitude |
of Resultant Type of
Radians Wavelength: Wav Interference

Intermediate

Fully destructive

Intermediate
Fully constructive

Intermediat

“The phase difference is between two otherwise identical waves, with amplitude y,,. moving in the
same direction.