waves

4.  17 Jan 1995, a terrible earthquake struck the
Hanshin (Japan) killing over 6400 people and
injuring about 40,000 others.
 200,000 homes and buildings were damaged.
 The heaviest damage occurred in the city of
Kobe, including the buckling and collapse of an
elevated highway.
 However, geologist found that the point of origin
of the earthquake was 15-20 km below the
northern tip of Awaji Island, about 20 km south
west of Kobe.

5.  How did the energy released by the
earthquake travel that far with enough
energy to cause such great devastation?

6. A wave is a disturbance that travels;
 If it travels through a medium, the particles
in the medium oscillate around their
equilibrium positions but do not travel.

7.  Energy transport: Waves transfer energy without
transferring matter;
 This can be easily seen with water waves or
waves on ropes or strings.
 Other examples include seismic waves, which
transport energy through the Earth, sound
waves, and electromagnetic waves.

8.  Two different ways to transfer energy.
 (a) The baseball carries energy from the pitcher
to the catcher.
 (b) A wave pulse also carries energy from the
pitcher to the catcher, but the piece of rope
held by the pitcher’s hand does not move to the
catcher’s hand.

9.  Intensity: Intensity is the average power per
unit area carried by the wave;
 The intensity usually decreases with distance
from the source of the wave, due both to
dissipation and to the wave spreading over a
larger and larger area.

10.  Therefore, if energy absorption by the medium
can be neglected, the intensity of the sound is
inversely proportional to the square of the
instance from the source.
 This “inverse square law” is the result of a
conserved quantity (here, energy) radiating
uniformly from a point source in three-
dimensional space.

11.  (a) A point source of sound radiating energy
uniformly in all directions.
 (b) The intensity at a distance r2 is smaller
than the intensity at a distance r1 since the
same power is spread out over a greater
area.

12.  In a transverse wave, the motion of the
particles is perpendicular to the direction of
propagation of the wave; in a longitudinal
wave, the motion is parallel to the direction
of propagation. Both kinds of waves can be
demonstrated with a long spring such as a
Slinky.
 (a) Transverse, and (b) Longitudinal waves on
a Slinky

13.  Transverse waves cannot travel through the
bulk of a fluid, as the fluid will not exhibit any
restoring force in response to a shear stress (it
just flows);
 They can, however, travel on the surface of a
fluid, such as water waves.
 Sound waves are longitudinal, and can travel
through any material; they consist of a series of
regions of higher and lower density, called
compressions and rarefactions.

14.  Seismic waves (which travel through the solid earth)
can be either longitudinal or transverse; as the core
of the earth is liquid, transverse seismic waves
cannot travel through it, and they are reflected.
 The speeds of longitudinal and transverse waves in
the same medium are generally different;
longitudinal seismic waves travel faster than
transverse ones.

15.  Waves that combine transverse and
longitudinal motion:
 It is possible for a wave to be neither pure
transverse or pure longitudinal;
 In this case, rather than the particles moving
perpendicular or parallel to the direction of
propagation of the wave, they exhibit a more
circular motion.
 This is seen in surface seismic waves as well
as in water waves.

18.  The motion of ground particle in
 (a) P waves
 (b) S waves
 (c) one kind of surface wave
 (d) Motion of a swimmer as a water wave
passes by

19.  The speed of propagation of a wave is
determined by the mechanical properties of
the medium;
 For a string, it depends on the tension in the
string and on the mass per unit length.
 Strings under more tension and with less
mass per unit length will have higher wave
speeds.

20.  This is due to the higher restoring force when
the tension is higher, and to the lower inertia
when the mass per unit length is lower.
 The speed of the particles in the medium is
different (and variable), and will also depend
on the amplitude of the wave (the larger the
amplitude, the faster the particles will
move).
 The speed of a transverse wave on a string
is: (F = force; L = Length and m = mass)

21.  Eq. 11-2 can rewrite in another form.
 Length and mass are not independent; for a
given string composition and diameter (say, a
yellow brass string of 0.030 in diameter), the
mass of the string is proportional to its
length.
 By defining the linear mass density (mass per
unit length) of the string to be
 Thespeed of a transverse wave on a string
can be written

23. A periodic wave is one which has a repeating
pattern over time.
 Such waves are characterized by a period
(the time a complete wave takes to pass a
given point) and a frequency (the number of
waves passing a given point per unit time);
the period is the inverse of the frequency
(just as in SHM).
 At any given point, the wave repeats itself
after a time T called the period. The inverse
of the period is the frequency f.

24.  The distance between corresponding points
on successive waves is called the
wavelength;
 the wave speed is then the frequency
multiplied by the wavelength.
 The maximum displacement of a particle
from its equilibrium position is called the
amplitude of the wave.
 If the wave shape is sinusoidal, the wave is
called harmonic, and every point in the
material undergoes SHM.

25.  During one period T, a periodic wave traveling at
speed v moves a distance vT.
 In Figure 11.7, note that, at any instant, points
separated by a distance vT along the direction of
propagation of a wave move “in sync” with each
other.
 Thus, vT is the repetition distance of the wave,
just as the period is the repetition time.
 Sinusoidal wave moving with speed v in the x-
direction. The amplitude A and the wavelength
are shown.

26.  The max. displacement of any particle from
its equilibrium position is the amplitude A of
the wave.
 For a sinusoidal wave traveling along a
stretched string in the x-direction, the
amplitude A is the maximum displacement in
the positive or negative y-direction.
 For surface water waves, the amplitude is
the height of a crest (a high point) above of
the depth of a through ( a low point) below
the undisturbed water level .
 Sinusoidal wave moving with speed v in the
x-direction. The amplitude A and the
wavelength are shown.

27.  Harmonic waves are a special kind of periodic wave
in which the disturbance is sinusoidal (sine or
cosine).
 In a harmonic transverse wave on a string, for
instance, every point on the string moves in SHM
with the same amplitude and angular frequency,
although different points reach their maximum
dispalcement at different times.
 The maximum speed and maximum acceleration of a
point on the string depend on both the angular
frequency and the amplitude of the wave

28.  Since the total energy of an object moving in
SHM is proportional to the amplitude
squared, the the total energy of a harmonic
wave is propotional to the square of its
amplitude.
 That turns out to be a general result not
limited to harmonic waves
 The intensity of a wave is proportional to the
square of its amplitude

29.  If we consider a transverse wave traveling
along a one-dimensional string which is
oriented along the x-axis and where the
particles are displaced in the y-direction, the
displacement of the particles (and therefore
the wave) is a function of both time and
position along the string;
 that is, y is a function of both x and t. If the
wave retains its shape as it travels, those
variables must appear in the combination
(t - x/v);
 such a wave is called a traveling wave.

31.  Sinusoidal wave moving with speed v in the
direction.
 The amplitude A and the wavelength are
shown

32.  Ifa wave is harmonic, its shape is sinusoidal;
we can show that the argument of the sine
or cosine function is (wt ± kx), where k =
2p/l and l is the wavelength.
 The figure below shows a wave pulse, with
the same shape, at successive times. The
motion of the point x repeats the motion of
the point x= 0 with a time delay t = x/v.

34. A wave can be graphed either at a single
point or at a single time;
 if the wave is harmonic, both graphs will be
sinusoidal.

35.  Two graphs of a harmonic wave y(x,t) = A sin
( t-kx) on a string
 (a) The vertical displacement of a particular
point on the string (x=0) as a function of
time.
 (b) The vertical displacement as a function
of horizontal position at a single instant of
time (t=0)

37.  Iftwo waves are traveling through the same
medium, and their amplitudes are not too large (so
that the medium still obeys Hooke's law),
 The net disturbance at any point is the sum of the
individual disturbances due to each wave. This can
be seen, for example, by dropping two pebbles into
a pond.
 PS = When two or more waves overlap, the net
disturbance at any point is the sum of the individual
disturbances due to each wave.

38.  (a) Two identical wave pulses traveling toward and
through each other.
 (b), (c) Details of the wave pulse summation:
dasehed lines are the separate wave pulses and solid
line is the sum

39.  Suppose two wave pulses are traveling toward
each other on a string (Fig.11.11a). If one of
the pulses (acting alone) would produce a
displacement y1 at a certain point and the
other would produce a displacement y2 at the
same point.

40.  The result when the two overlap is a displacement of
y1+y2.
 Fig. 11.11b, c show in greater detail how y1 and y2 add
together to produce the net displacement when the
pulses overlap.
 The dashes curves represent the individual pulses; the
solid line represents the superposition of the pulses.
 In Fig. 11.11b the pulses are starting to overlap and in
Fig. 11.11c they are just about to coincide.

44.  Reflection: If a wave encounters a boundary
between two media, some or all of the wave will
be reflected (that is, some or all of the energy will
travel back into the first medium).
 How much of the energy is reflected depends on
how different the properties of the two media are
(in particular, on the wave speed in the two
media); the more different, the more reflection
takes place.
 The reflected wave will be inverted if it reflects
from a medium with a lower wave speed; if the
reflecting medium has a higher wave speed, the
reflected wave will not be inverted.

45.  Snapshots of the reflection of a wave pulse
from a fixed end.
 The reflected pulse is upside down.

46.  Refraction: In general, part of the wave will be
transmitted into the second medium;
 both the reflected and transmitted waves will
have the same frequency as the incident
wave, but the wavelength of the transmitted
wave will be different (since the frequency is
the same and the speed is different).
 Unless the incident wave is perpendicular to the
boundary, the transmitted wave will also be at a
different angle than the incident wave; this is
called refraction, and the transmitted wave is
also called the refracted wave.

47.  The angles are defined with respect to the
normal to the boundary surface; the ratio of
the sines of the angles of incidence and of
refraction is the same as the ratio of the
wave speeds in the respective media.
 Since = f, and the frequencies are the
same
 this equation applies to any kind of wave
and is of particular importance in the study
of optics.

48.  The angle in Eq. (11-10) are called the angle
of incidence and the angle of refraction.
 Note that these angles are measured
between the propagation direction of the
wave and the normal

49.  (a) A broad beam of light refracts when it passes from
air into water. The reflected wave is omitted for
clarity.
 The normal is the direction perpendicular to the
boundary. The wave’s propagation direction is closer
to the normal in the slower medium (water, in this
case).

52.  Interference: If two waves being superposed
have the same frequency and have a fixed phase
relationship, they are called coherent.
 In this case, superposition will yield what is
called interference - the resultant wave will
have the same frequency as the original
waves, and will have an amplitude somewhere
between the sum of the two amplitudes and the
absolute magnitude of their difference.
 If the waves are in phase, the interference is
called constructive; if they are 180º out of
phase, the interference is called destructive.

53.  Coherent waves (a) in phase and (b) 180 out of
phase.
 (one wave is drawn with a lighter line to
distinguish it from the other?)

54.  Suppose two coherent waves start out in phase with each
other.
 In Fig.11.18, two rods vibrate up and down in step with each
other to generate circular waves on the surface of the
water.
 If two waves travel the same distance to reach a point on
the water surface, they arrive in phase and interfere
constructively.
 At points where the distances are different, we calculate
the phase difference as follows.
 One wavelength of path difference corresponds to a phase
difference of 2 radians (one full cycle).
 Then, working by proportions
 (d1-d2)/ = phase difference / 2 rad.
 If the phase difference is an even integral multiple of
rad, then constructive interference occurs at point P; if the
phase difference is an odd integral multiple of rad, then
destructive interference occurs at point P.

55.  Overhead view of coherent surface water waves.
 The two waves travel different distances d1 and
d2 to reach a point P.
 The phase difference between the waves at
point P is k (d1-d2)

56.  Intensity effects for interfering waves:
 If two waves which are superposed are
coherent, their amplitudes add; their
intensities (which are proportional to the
square of the amplitude) do not, in general.
 However, if two waves which are superposed
are incoherent, their intensities do add.

58.  Diffraction: If a wave encounters an
obstacle, it will bend around it; the amount
of bending depends on the size of the
obstacle compared to the wavelength of the
wave.
 This bending is called diffraction.

59.  Diffraction of a wave when it encounters an
obstacle.

62. A wave can be reflected at a boundary in
such a way that the wave appears to stand
still; this is typical of waves on finite strings,
such as those in musical instruments. In this
case, the string vibrates as a whole;
 every point reaches its maximum amplitude
simultaneously, and every point also reaches
its minimum amplitude (namely, zero)
simultaneously as well. Therefore, there are
points on the string which never move; these
are called nodes. Between the nodes are
points which have the maximum amplitude;
these are called antinodes.

63.  Ifboth ends of the string are fixed, both are
nodes; if one end is free to move, it is an
antinode. A string with both ends fixed can
have an integral number of half-wavelengths
along its length; the lowest frequency occurs
when the wavelength is twice the length of
the string (so that the only nodes are at the
ends); this frequency is called the
fundamental. All other natural frequencies of
the string are integer multiples of the
fundamental frequency.

64. A standing wave at various times, where T is
the period.

65. L is length of string and n =1,2,3,…the
possible wavelength for standing waves on a
string are
 The Frequencies are