H. SAIBI
October 28th , 2015
Fig. Deep-Ocean Assessment and Reporting of Tsunamis (NOAA) in North Pacific
©2008 by W.H. Freeman and Company
Waves are everywhere in nature
Sound waves,
telephone chord
visible light waves,
waves,
stadium waves,
earthquake waves,
waves on a string,
slinky waves
radio waves,
microwaves,
water waves,
sine waves,
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What is a wave?
 a wave is a disturbance that travels through a medium
from one location to another.
 a wave is the motion of a disturbance
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Simple Wave Motion
Transverse Waves
Transverse wave: The motion of the medium (the string) is perpendicular to the
direction of the propagation of the disturbance.
Fig.1. Transverse wave pulse
on a spring. The motion of the
propagating medium is
perpendicular to the direction
of motion disturbance.
Fig. 2.Three successive drawings of a
transverse wave on a string traveling
to the right.
©2008 by W.H. Freeman and Company
Longitudinal Waves
 Longitudinal waves: The motion of the medium is parallel to the direction of
propagation of the disturbance.
 Example: Sound waves
Fig.3. Longitudinal wave
pulse on a spring.
©2008 by W.H. Freeman and Company
Differences between Longitudinal and Transverse waves
The differences between the two can be seen
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Anatomy of a Wave
 Now we can begin to describe the anatomy of our waves.
 We will use a transverse wave to describe this since it is
easier to see the pieces.
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Anatomy of a Wave
 In our wave here the dashed line represents the
equilibrium position.
 Once the medium is disturbed, it moves away from this
position and then returns to it
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Anatomy of a Wave
crest
 The points A and F are called the CRESTS of the wave.
 This is the point where the wave exhibits the maximum
amount of positive or upwards displacement
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Anatomy of a Wave
trough
 The points D and I are called the TROUGHS of the wave.
 These are the points where the wave exhibits its maximum
negative or downward displacement.
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Anatomy of a Wave
Amplitude
 The distance between the dashed line and point A
is called the Amplitude of the wave.
 This is the maximum displacement that the wave
moves away from its equilibrium.
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Anatomy of a Wave
wavelength
 The distance between two consecutive similar points (in
this case two crests) is called the wavelength.
 This is the length of the wave pulse.
 Between what other points is can a wavelength be
measured?
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Anatomy of a Wave
 What else can we determine?
 We know that things that repeat have a frequency and a
period. How could we find a frequency and a period
of a wave?
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Wave frequency
 We know that frequency measure how often something
happens over a certain amount of time.
 We can measure how many times a pulse passes a fixed
point over a given amount of time, and this will give us
the frequency.
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Wave frequency
 Suppose I wiggle a slinky back and forth, and count that 6
waves pass a point in 2 seconds. What would the
frequency be?
 3 cycles / second
 3 Hz
 we use the term Hertz (Hz) to stand for cycles per second.
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Wave Period
 The period describes the same thing as it did with a




pendulum.
It is the time it takes for one cycle to complete.
It also is the reciprocal of the frequency.
T=1/f
f=1/T
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Wave Speed
 We can use what we know to determine how fast a wave is
moving.
 What is the formula for velocity?
 velocity = distance / time
 What distance do we know about a wave
 wavelength
 and what time do we know
 period
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Wave Speed
 so if we plug these in we get
 velocity =
length of pulse /
time for pulse to move pass a fixed point
 v=/T
 we will use the symbol  to represent wavelength
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Wave Speed
v=/T
 but what does T equal
 T=1/f
 so we can also write
 v=f
 velocity = frequency * wavelength
 This is known as the wave equation.
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Wave Behavior
 Now we know all about waves.
 How to describe them, measure them and analyze them.
 But how do they interact?
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Wave Behavior
 We know that waves travel through mediums.
 But what happens when that medium runs out?
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Boundary Behavior
 The behavior of a wave when it reaches the end of its
medium is called the wave’s BOUNDARY BEHAVIOR.
 When one medium ends and another begins, that is called
a boundary.
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Fixed End
 One type of boundary that a wave may encounter is that it
may be attached to a fixed end.
 In this case, the end of the medium will not be able to
move.
 What is going to happen if a wave pulse goes down this
string and encounters the fixed end?
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Fixed End
 Here the incident pulse is an upward pulse.
 The reflected pulse is upside-down. It is inverted.
 The reflected pulse has the same speed, wavelength, and
amplitude as the incident pulse.
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Fixed End Animation
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Free End
 Another boundary type is when a wave’s medium is
attached to a stationary object as a free end.
 In this situation, the end of the medium is allowed to slide
up and down.
 What would happen in this case?
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Free End
 Here the reflected pulse is not inverted.
 It is identical to the incident pulse, except it is moving in
the opposite direction.
 The speed, wavelength, and amplitude are the same as the
incident pulse.
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Free End Animation
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Change in Medium
 Our third boundary condition is when the medium of a
wave changes.
 Think of a thin rope attached to a thin rope. The point
where the two ropes are attached is the boundary.
 At this point, a wave pulse will transfer from one medium
to another.
 What will happen here?
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Change in Medium
 In this situation part of the wave is reflected, and part of
the wave is transmitted.
 Part of the wave energy is transferred to the more dense
medium, and part is reflected.
 The transmitted pulse is upright, while the reflected
pulse is inverted.
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Change in Medium
 The speed and wavelength of the reflected wave remain the
same, but the amplitude decreases.
 The speed, wavelength, and amplitude of the transmitted
pulse are all smaller than in the incident pulse.
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Change in Medium Animation
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Wave Interaction
 All we have left to discover is how waves interact with
each other.
 When two waves meet while traveling along the same
medium it is called INTERFERENCE.
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Constructive Interference
 Let’s consider two waves moving towards each other, both
having a positive upward amplitude.
 What will happen when they meet?
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Constructive Interference
 They will ADD together to produce a greater
amplitude.
 This is known as CONSTRUCTIVE
INTERFERENCE.
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Destructive Interference
 Now let’s consider the opposite, two waves moving
towards each other, one having a positive (upward) and
one a negative (downward) amplitude.
 What will happen when they meet?
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Destructive Interference
 This time when they add together they will produce a
smaller amplitude.
 This is know as DESTRUCTIVE INTERFERENCE.
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Check Your Understanding
 Which points will produce constructive interference and which
will produce destructive interference?
Constructive
G, J, M, N
Destructive
H, I, K, L, O
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WAVE PULSES
Fig.4 shows a pulse on a string at time t=0. The shape of the string can be represented
by some function y=f(x).
At some later time, the pulse is farther down the string. In a new coordinate system with
origin O’ that moves to the right with the same speed as the pulse. The string is
described in this frame by f(x’) for all times.
 The x coordinates of two reference frames are
related by:
 So
. Thus, the shape of
the string in the original frame is:
…(1) (Wave Function)
wave moving in the +x direction.
The same line of reasoning for a pulse moving to
…(2)
the left leads to:
wave moving in the -x direction.
 v is the speed of propagation of the wave.
©2008 by W.H. Freeman and Company
Fig.4.
WAVE PULSES
 For waves on a string, the wave function represents the
transverse displacement of the string.
 For sound waves in air, the wave function can be the
longitudinal displacement of the air molecules, or the pressure
of the air.
 The wave function are solutions of a differential equation
called the wave equation, which can be derived using
Newton’s laws.
SPEED OF WAVES
 A general property of waves is that their speed relative to the medium depends on
the properties of the medium, but is independent of the medium of the source of
the wave.
 For example, the speed of a sound from a car horn depends only on the properties
of air and not on the motion of the car.
 For wave pulses on a rope, we can demonstrate that the greater the tension, the
faster the propagation of the waves. Furthermore, waves propagate faster in a
light rope than in a heavy rope under the same tension.
 If FT is the tension and  in the linear mass density (mass per unit length), the
wave speed is:
…(3)
Speed of waves
on a string
 For sound waves in a fluid such as air or water, the speed v is given by:
…(4) ( is the equilibrium density of the medium and B is the bulk modulus)
The bulk modulus is the negative of the ratio of the change in pressure to the fractional change in
volume:
SPEED OF WAVES
 Comparing Eq.3 and 4, we can see that, in general, the speed of waves depends on an
elastic property of the medium (the tension for string waves and the bulk modulus for
sound waves) and on an inertial property of the medium (the linear mass density or
the volume mass density).
 For sound waves in a gas such as air, the bulk modulus is proportional to the pressure,
which in turn is proportional to the density  and to the absolute temperature T of the
gas. The ratio B/ is thus independent of density and is merely proportional to the
absolute temperature T. In this case, Eq.4 is equivalent to:
…(5)
Speed of sound in a gas
(6)
T is the absolute temperature (K), which is related to Celsius tC by:
 The dimensionless constant  depends on the kind of gas.
…(7)
 The constant R is the universal gas constant:
and M is the molar mass of the gas (that is, the mass of one mole of the gas), which for
air is:
Exercise
 Surface ocean waves are possible because of gravity
and are called gravity waves. Gravity waves are called
shallow waves if the water depth is less than half a
wavelength. The wave speed for gravity waves depends
on the depth and is given by v=sqrt (gh), where h is the
depth. A gravity wave in the open ocean, where the
depth is 5 km, has a wavelength of 100 km.
 1- What is the wave speed of the wave?
 2- Is the wave a shallow wave?
Information: Tsunamis are known to travel at speeds of 800 km/h in the open
ocean.
SPEED OF WAVES
Derivation of v for waves on a string
 Eq.3 (v=SQRT FT/) can be obtained by applying the pulse-momentum theorem to
the motion of a string. Suppose you are holding one end of a long taut string with
tension FT and uniform mass per unit length . (the other end of the string is attached
to a distant wall). Suddenly, you begin to move your hand upward at a constant speed
u. After a short time, the string appears as shown in Fig.5 with the rightmost point of
the inclined segment of the string moving to the right at the wave speed v and the
entire inclined segment moving upward at speed u. By applying the impulsemomentum theorem (
) to the string, we obtain:
…(8)
where Fy is the upward component of the force of your hand on the string, m is the
mass of the inclined segment, and t is the time that your hand has been moving
upward. The two triangles in the figure are similar, so:
or
Substituting for Fy in Eq.8 gives:
where vt has been substituted for m. Solving for v gives:
Which is the expression for the wave speed that is given in Eq.3.
SPEED OF WAVES
Derivation of v for waves on a string
Fig.5. As the end of the spring moves upward at constant speed u, the point
where the string changes from horizontal to inclined moves to the right at
the wave speed v.
©2008 by W.H. Freeman and Company
THE WAVE EQUATION
We can apply Newton’s second law to a segment of the string to derive a differential
equation known as the wave equation, which relates the spatial derivatives of y(x,t) to
its time derivatives. Fig.6 shows one segment of a string. We consider only small
angles 1 and 2. Then the length of the segment is approximately x and its mass is
m=x, where  is the string’s mass per unit length. First, we show that, for small
vertical displacements, the net horizontal force on a segment is zero and the tension
is uniform and constant. The net force in the horizontal direction is zero. That is,
Where 2 and 1 are the angles shown and FT is the tension in string. Because the
angles are assumed to be small, we may approximate cos by 1 for each angle. Then,
the net horizontal force on the segment can be written
Thus,
the segment moves vertically, and the net force in this direction is:
THE WAVE EQUATION
 Because the angles are assumed to be small, we may approximate sin by tan for
each angle. Then the net vertical force on the string segment can be written
 The tangent of the angle made by the string with the horizontal is the slope of the
line tangent to the string. The slope S is the first derivative of y(x,t) with respect to
x for constant t. A derivative of a function of two variables with respect to one of
the variables with the other held constant is called a partial derivative. The partial
derivative of y with respect to x is written y/x. Thus, we have:
Then
Where S1 and S2 are slopes of either end of the string segment and S is the
change in the slope. Setting this net force equal to the mass x times the
acceleration 2y/t2 gives:
or
…(9)
THE WAVE EQUATION
Fig.6. Segment of stretched string used for derivation of the wave
equation. The net vertical force on the segment is FT2 sin2-FT1 sin1,
where F is the tension in the string. The wave equation is derived by
applying Newton’s second law to the segment.
©2008 by W.H. Freeman and Company
THE WAVE EQUATION
 In the limit as x0, we have:
 Thus, in the limit as x0, Eq.9 becomes
…(10-a) Wave equation for a taut string.
 We now show that the wave equation is satisfied by any function x-vt.
Let =x-vt and consider any wave function:
We use y’ for the derivative of y with respect to . Then, by the chain rule for the
derivatives,
and
Because
and
and
We have
Taking the second derivatives, we obtain
and
THE WAVE EQUATION
Thus,
…(10-b)
Wave Equation
The same result (Eq.10-b) can be obtained for any
function of x+vt as well. Comparing Eq.10-a and 10-b,
we see that the speed of propagation of the wave is
…(3)
SPEED OF WAVES
Derivation of v for sound waves
(Eq.4), where B and  are the bulk
modulus and density of the medium, respectively. This equation can be obtained
by applying the impulse-momentum theorem to the motion of the air in a long
cylinder (Fig.7) with a piston at one end with the other end open to the
atmosphere. Suddenly, you begin to move the piston to the right at constant speed
u. After a short time, t from the initial position of the piston is moving to the
right with speed u. By applying the impulse-momentum theorem (
)
to the air in the cylinder we obtain:
…(11)
 Where m is the mass of the air moving with speed u and F is the net force on the
air in the cylinder. The air was initially at rest. The net force F is related to the
pressure increase P of the air near the moving piston by:
 The speed of sound is given by:
 Where A is the cross-sectional area of the cylinder. The bulk modulus of the air is
given by:
so
SPEED OF WAVES
Derivation of v for sound waves
 Where Aut is the volume swept out by the piston and Avt is the initial
volume of the air that is now moving with speed u. Substituting for F in Eq.11
gives:
or
 Where Avt have been substituted for m. Solving for v gives:
 Which is the same as the expression for v in Eq.4.
 A wave equation for sound waves can be derived using Newton’s laws. In one
dimension, this equation is:
 Where s is the displacement of the medium in the x direction and vs is the
speed of the sound in the medium.
SPEED OF WAVES
Derivation of v for sound waves
Fig.7. The air near to the piston is moving to the right at the same constant
speed u as the piston. The right edge of this pressure pulse moves to the
right with the wave speed v. The pressure in the pulse is higher than the
pressure in the rest of the cylinder by P.
©2008 by W.H. Freeman and Company