16.1
Propagation of a Disturbance
In physics, a wave
is a disturbance that travels through space and time, accompanied by the transfer of
energy.
Waves travel and the wave motion transfers energy from one point to another, often with no permanent
displacement of the particles of the medium—that is, with little or no
associated mass transport. They consist, instead, of oscillations or vibrations around almost fixed
locations. For example, a cork on rippling water will bob up and down, staying in about the
same place while the wave itself moves onwards.
One type of wave is a mechanical wave,
which propagates through a medium in which the substance of this medium is
deformed. The deformation reverses itself owing to restoring forces resulting from its deformation. For
example, sound waves propagate via air molecules bumping into their neighbors.
This transfers some energy to these neighbors, which will cause a cascade of
collisions between neighbouring molecules. When air molecules collide with
their neighbors, they also bounce away from them (restoring force). This keeps
the molecules from continuing to travel in the direction of the wave.
Another type of wave can travel through a vacuum, e.g. electromagnetic
radiation
(including visible light, ultraviolet radiation, infrared radiation, gamma
rays, Xrays, and radio waves). This type of wave consists of periodic
oscillations in electrical and magnetic fields.
A main distinction can be made between
transverse and longitudinal waves. Transverse waves occur when a disturbance
sends waves perpendicular (at right angles) to the original wave. Longitudinal
waves occur when a disturbance sends waves in the same direction as the
original wave.
Waves are described by a wave equation
which sets out how the disturbance proceeds over time. The mathematical form of
this equation varies depending on the type of wave.
Two
types of waves :
•
Mechanical waves  some physical medium is
being disturbed
•
Electromagnetic waves  do not require a
medium to propagate
* the essence
of wave motion—the transfer of energy through space without the accompanying
transfer of matter
All
mechanical waves require :
•
(1) some source of disturbance,
•
(2) a medium that can be disturbed, and
•
(3) some physical mechanism through which elements
of the medium can influence each other.
One
way to demonstrate wave motion is to flick one end of a long rope that is under
tension and has its opposite end fixed
*a single bump (called a pulse) is formed and travels along the
rope with a definite speed.
If
we were to move the end of the rope up and down repeatedly, we would create a
traveling wave, which has characteristics that a pulse does not have.
A
traveling wave or pulse that causes the elements of the medium to move parallel
to the direction of propagation is called a longitudinal wave.
A
traveling wave or pulse that causes the elements of the disturbed medium to
moveperpendicular to the direction of propagation is called a transverse
wave.
The
left end of the spring is pushed briefly to the right and then pulled briefly
to the left. This movement creates a sudden compression of a region of the
coils.
Consequently, an
element of the string at x at this time has the same y
position as an element located at x vthad at
time t =0:
In general, then, we
can represent the transverse position y
for all positions and times,measured
in a stationary frame with the origin at O, as
Similarly, if the
pulse travels to the left, the transverse positions of elements of the string
are described by
The
function y,
sometimes called the wave
function, depends on the two variablesx and t. For
this reason, it is often written y(x, t), which is read “y
as a function of xand t.”
It
is important to understand the meaning of y. Consider an element of the stringat point P,
identified by a particular value of its x
coordinate. As the pulse
passesthrough P, the y coordinate of this element increases, reaches a maximum,
and thendecreases to zero. The
wave function y(x,
t)
represents the y coordinate—thetransverse position—of any element
located at position x at any time t. Furthermore,ift is fixed
(as, for example, in the case of taking a snapshot of the pulse), thenthe wave
function y(x),
sometimes called the waveform,
defines a curve representingthe
actual geometric shape of the pulse at that time.
16.2
Sinusoidal Waves
On a rope, a
sinusoidal wave could be established by shaking the end of the rope up and down
in simple harmonic motion.The same
information is more often given by the inverse of the period, which iscalled
the frequency f. In
general, the frequency of a periodic
wave is the numberf crests (or troughs, or any other point on the wave) that
pass a given point in aunit time interval. The
frequency of a sinusoidal wave is related to the period by theexpression
where the constant A
represents the wave amplitude and
the constant is the wavelength.We see that the vertical position of an
element of the medium is the samewhenever x
is increased by an integral multiple
of l. If the wave moves to the right witha speed v, then
the wave function at some later time t
is
That is, the traveling sinusoidal wave moves to the right
a distance vtin
the time t,
Note that the wave function has the form f
(x
vt) .Ifthe wave were traveling to the left, the quantity x vtwould be
replaced by x +vt,
By definition, the wave travels a distance of one
wavelength in one period T. Therefore,the wave speed, wavelength, and period are
related by the expression
Substituting this expression for v
Angular
frequency
wave
function for sinusoidal wave
16.3 The
Speed of Waves on Strings
According
to Newton’s second law, the acceleration of the element increases with
increasing tension. If the element returns to equilibrium more rapidly due to
this increased acceleration, we would intuitively argue that the wave speed is
greater. Thus, we expect the wave speed to increase with increasing tension.
Likewise,
the wave speed should decrease as the mass per unit length of the string
increases. This is because it is more difficult to accelerate a massive element
of the string than a light element. If the tension in the string is T and
its mass per unit length is μ (Greek mu), then as we shall show, the
wave speed is

First,
let us verify that this expression is dimensionally correct. The dimensions of T
are ML/T^{2} and the dimension of μ are M/L.Therefore, the
dimensions of T/μ are L^{2}/T^{2}. hence, the dimensions
of are L/T, the dimension of speed.
Consider
a pulse moving on a taut string to the right with a uniform speed v measured
relative to a stationary frame of reference. Instead of staying in this
reference frame, it is more convenient to choose as our reference frame one
that moves along with the pulse with the same speed as the pulse, so that the
pulse is at rest within the frame. This change of reference frame is permitted
because Newton’s laws are valid in either a stationary frame or one that moves
with constant velocity. In our new reference frame, all elements of the string
move to the left—a given element of the string initially to the right of the
pulse moves to the left, rises up and follows the shape of the pulse, and then
continues to move to the left. Figure 16.11a shows such an element at the
instant it is located at the top of the pulse.
The small
element of the string of length ∆s shown in Figure 16.11a, and magnified
in Figure 16.11b, forms an approximate arc of a circle of radius R. In
our moving frame of reference (which is moving to the right at a speed v along
with the pulse), the shaded element is moving to the left with a speed v.
This element has a centripetal acceleration equal to v^{2}/R,
which is supplied by components of the force T whose magnitude is the tension
in the string. The force T acts on both sides of the element and is tangent to
the arc, as shown in Figure 16.11b. The horizontal components of T cancel, and
each vertical component T sin θ acts radially toward the center of the
arc.
Hence, the
total radial force on the element is 2T sin θ. Because the element is
small, θ is small, and we can use the smallangle approximation sin θ ≈ θ.
Therefore, the total radial force is
The element has
a mass m =μ.s. Because the element forms part of a circle and
subtends an angle 2θ at the center, ∆s = R(2θ), we find that
If we apply
Newton’s second law to this element in the radial direction, we have.
This expression
for v is Equation 16.18. Notice that this derivation is based on the
assumption that the pulse height is small relative to the length of the string.
Using this assumption, we were able to use the approximation sin θ ≈ θ.
Furthermore, the model assumes that the tension T is not affected by the
presence of the pulse; thus, T is the same at all points on the string.
Finally, this proof does not assume
any particular shape for the pulse. Therefore, we conclude that a pulse of any
shape travels along the string with speed without any change in pulse
shape
16.4 Reflection and Tranmsission
We have discussed waves traveling
through a uniform medium. We now consider how a traveling wave is
affected when it encounters a change in the medium. For example, consider a pulse
traveling on a string that is rigidly attached to a support at one end as in Figure 16.14. When
the pulse reaches the support, a severe change in the medium occurs—the string ends.
The result of this change is that the pulse undergoes reflection— that is, the pulse
moves back along the string in the opposite direction.
Now
consider another case :
this time, the pulse arrives at the end of a string that is free to move
vertically, as in Figure 16.15. The tension at the free end is maintained
because the
string is tied to a ring of negligible mass that is free to slide vertically on
a smooth
post without friction. Again, the pulse is reflected, but this time it is not
inverted. When
it reaches the post, the pulse exerts a force on the free end of the string, causing the ring to
accelerate upward. The ring rises as high as the incoming pulse,
Finally,
we may have a situation in which the boundary is intermediate between these two extremes. In
this case, part of the energy in the incident pulse is reflected and part undergoes transmission—that
is, some of the energy passes through the boundary. For instance,
suppose a light string is attached to a heavier string, as in Figure 16.16. When a pulse
traveling on the light string reaches the boundary between the two, part of the
pulse is reflected and inverted and part is transmitted to the heavier string. The reflected
pulse is inverted for the same reasons described earlier in the case of the string
rigidly attached to a support.
According
to Equation 16.18, the speed of a wave on a string increases as the mass per
unit length of the string decreases. In other words, a wave travels more slowly
on a heavy string than on a light string if both are under the same tension.
The following general rules apply to reflected waves : when a wave or pulse
travels from medium A to medium B and V_{A} > V_{B} (that
is, when B is denser than A), it is inverted upon reflection. When a wave or
pulse travels from medium A to medium B and V_{A} < V_{B}
(that is, when A is denser than B), it is not inverted upon reflection.
16.5 Rate
of Energy Transfer By Sinusoidal Waves
on Strings
Waves
transport energy when they propagate through a medium. We can easily
demonstrate this
by hanging an object on a stretched string and then sending a pulse downthe
string, as in Figure 16.18a. When the pulse meets the suspended object, the
objectis momentarily displaced upward, as in Figure 16.18b. In the process,
energy is transferredto the object and appears as an increase in the
gravitational potential energy ofthe object–Earth system. This section examines
the rate at which energy is transportedalong a string. We shall assume a
onedimensional sinusoidal wave in the calculation ofthe energy transferred
This
expression shows that the rate of energy transfer by a sinusoidal wave on a
string is proportional to (a) the square of the frequency, (b) the square of
the amplitude and (c)the wave speed.
The rate of energy transfer in any sinusoidal wave is proportional to the square of the angular frequency and to the square of the amplitude
The rate of energy transfer in any sinusoidal wave is proportional to the square of the angular frequency and to the square of the amplitude
Answer: (d). Doubling the amplitude of the wave causes the power to be larger by a
factor of 4. In (a), halving the linear mass density of the string causes the
power to change by a factor of 0.71—the rate decreases. In (b), doubling the
wavelength of the wave halves the frequency and causes the power to change by a
factor of 0.25—the rate decreases. In (c), doubling the tension in the string
changes the wave speed and causes the power to change by a factor of 1.4—not as
large as in part (d).
16.6 The
Linier Wave Equation
All wave
functions y(x, t) represent solutions of an equation
calledthe linear wave equation. This equation gives a complete
description of the wave motion,and from it one can derive an expression for the
wave speed.Furthermore, the linearwave equation is basic to many forms of wave
motion. In this section, we derive this equation as applied to waves on
strings.
Suppose a
traveling wave is propagating along a string that is under a tension T.
Let us consider one small string element of length ∆x (Fig. 16.20). The
ends of the element
make small
angles θ_{A} and θ_{B} with the x axis.
Wave
functions are solutions to a differential equation called the linier wave
equation
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