Hakim SAIBI November 4th , 2015
Waves Lecture No. 5 Hakim SAIBI November 4th , 2015 Outline
Periodic waves. Waves in three dimensions Harmonic waves.
Energy transfer via waves on a string. Harmonic sound waves. Energy
of sound waves. Electromagnetic waves. Waves in three dimensions
Wave intensity Intensity level Fig.1. Harmonic wave at some instant
in time.
Periodic waves Harmonic waves As the wave propagates along the
string move direction of propagation-in simple harmonic motion with
the frequency f of the tuning fork. During one period T of this
motion the wave moves a distance of one wavelength, so its speed is
given by: The sine function that describes the displacements in
Fig.1 is: (1) Fig.1. Harmonic wave at some instant in time. (2)
Periodic waves Where A is the amplitude, is the wavelength, and is
a phase constant that depends on the choice of the origin (where
x=0). This equation is expressed more simply as: Where k, called
the wave number, is given by: Note that k has units of m-1.
(Because the angle must be in radians, we sometimes write the units
of k as rad/m) When dealing with a single harmonic wave we usually
choose the location of the origin so that =0. For a wave traveling
in the direction of increasing x with speed v, replace x in Eq.3
with x-vt. With equal to zero, this gives: or Where Is the angular
frequency, and the argument of the sine function, (kx-vt), is
called the phase. (3) (4) (5) Harmonic Wave Function (6) Periodic
waves The angular frequency is related to the frequency f and
period T by: Substituting =2f into Eq.6 and using k=2/, we obtain:
Or v=f, which is Eq.1. If a harmonic wave traveling along a string
is described by y(x,t)=Asin(kx-t), the velocity of a point on the
string at a fixed value of x is: The acceleration of this point is
given by 2y/ t2. (7) (8) Transverse Velocity Periodic waves Energy
transfer via waves on a string
Consider again a string attached to a tuning fork. As the fork
vibrates, it transfers energy to the segment of the string attached
to it. For example, as the fork moves upward from its equilibrium
position it stretches the adjacent string segment
slightly-increasing its elastic potential energy. In addition, the
fork slows as it moves upward from it equilibrium, so it slows the
string segment closest to it. This decreases the kinetic energy of
the segment. As a wave moves along the string, energy is
transferred from one segment to the next in a similar manner. Power
is the rate of energy transfer. We can calculate the power by
considering work done by the force that one segment of the string
exerts on a neighboring segment. The rate of work done by this
force is the power. Fig.2 shows a harmonic wave moving to the right
along a string segment. That is, we assume a wave function of the
form: (9) Periodic waves Energy transfer via waves on a
string
Fig.2. The tension force FT has a component in the direction of the
transverse velocity vtr, so at this instant the force is doing work
on the end of the string that has a positive value. 2008 by W.H.
Freeman and Company Periodic waves Energy transfer via waves on a
string
The tension forceFt on the left end of the segment is directed
tangent to the string, as shown. To calculate the power transferred
by this force, we use the formula , where FT is the tension andvtr
, the transverse velocity, is the velocity of the end of the
segment. To obtain an expression for the power, we first express
the vectors in component form. That is, and Taking the scalar
product gives We obtain vy by differentiating Eq.8 From the figure,
we see that , where we have used the small angle approximation
sintan. Because tan is the slope of line tangent to the string, we
have Thus Applying Eq.10 to a harmonic wave (by taking derivatives
of Eq.9) gives: (10) Periodic waves Energy transfer via waves on a
string
Using and v=/k Eq.6, we substitute for FT and the leading k to
obtain: Where v is the wave speed. The average power at any
location x is then: Because the average value of cos2(kx-t) is 1/2.
This average is taken over an entire period T of the motion with x
held constant. The energy travels along a taut string at an average
speed equal to the wave speed v, so the average energy (E)av
flowing past point P1 during time t (Fig.3) is: The energy is
distributed over a length x=vt, so the average energy in length x
is: Note that like the average power, the average energy per unit
length is proportional to the square of the amplitude of the wave.
(11) (12) (13) Periodic waves Energy transfer via waves on a
string
Fig.3. The wave has reached point P at time t1. During time t, the
wave advanced past point P at a distance vt. 2008 by W.H. Freeman
and Company Harmonic Sound Waves Harmonic sound waves can be
generated by a tuning fork or loudspeaker that is vibrating with
simple harmonic motion. The vibrating source causes the air
molecules next to it to oscillate with simple harmonic motion about
their equilibrium positions. These molecules collide with
neighboring molecules, causing them to oscillate, which in turn
collide with their neighboring molecules, causing them to
oscillate, and so forth, thereby propagating the sound wave. Eq.5
describes a harmonic sound wave if the wave function y(x,t) is
replaced by s(x,t), which represents the displacements of the
molecules from their equilibrium positions. Thus, These
displacements are along the direction of propagation of the wave,
and lead to variations in the density and pressure of the air. Fig.
4 shows the displacement of air molecules and the density
variations caused by a sound wave at some fixed time. The pressure
is maximum where the density is maximum. We see from this figure
that the density wave, and thus the pressure wave, is 90o out of
phase with displacement wave. (in the arguments of sine or cosine
functions, we will always express phase angles in radians. However,
in verbal descriptions, we usually say that: two waves are 90o out
of phase, rather than two waves are out of phase by /2 rad. (14)
Harmonic Sound Waves Where the displacement s is zero, the density,
and thus the pressure, is either maximum or minimum, and where the
displacement is a maximum or a minimum, the density, and thus the
pressure, is equal to its equilibrium value. A displacement wave
given by Eq.14 thus implies a pressure wave given by: Where p
stands for the pressure minus the local equilibrium pressure, and
p0, the maximum value of p, is called the pressure amplitude. It
can be shown that the pressure amplitude p0 is related to the
displacement amplitude s0 by: Where v is the speed of propagation
and is the equilibrium density of the gas. Thus, as a harmonic
sound wave travels through air, the displacement of air molecules,
the pressure, and the density all vary sinusoidally with the
frequency of the vibrating source. (15) (16) Harmonic Sound Waves
Figure 4: a) displacement from equilibrium of air molecules in a
harmonic sound wave versus position at some instant. Points x1 and
x2 are points of zero displacement. b) some representative
molecules equally spaced at their equilibrium position cycle
earlier. The arrows indicate the directions of their velocities at
that instant. c) molecules near points x1, x2 and x3 after the
sound wave arrives. d) density of the air at this instant. e)
Pressure change, which is proportional to the density change,
versus position. The pressure change and displacement are 90
degrees out of phase. 2008 by W.H. Freeman and Company Energy of
sound waves The average energy of a harmonic sound wave in a volume
element V is given by Eq.13 with A replaced by s0 and x, replaced
by V, where is the equilibrium density of the medium. The energy
per unit volume is the average energy density av: (17) (18)
Electromagnetic Waves
Waves of energy emitted from any accelerating charges Any object
that is above absolute zero emits electromagnetic waves The entire
range of possibilities is called the Electromagnetic Spectrum Waves
in Three-Dimensions (3D)
Fig.5 shows 2D circular waves on the surface of water in a ripple
tank. These waves are generated by drops of water striking the
surface. The wave crests from concentric circles called wavefronts.
For a point source of sound, the waves move out in 3D, and the
wavefronts are concentric spherical surfaces. Fig.5. Circular
wavefronts diverging from a point source in a ripple tank. 2008 by
W.H. Freeman and Company 2008 by W.H. Freeman and Company Waves in
Three-Dimensions
The motion of any set of wavefronts can be indicated by rays, which
are directed lines perpendicular to the wavefronts (Fig.6). For
circular or spherical waves, the rays are radial lines. Fig.6. The
motion of wavefronts can be represented by rays drawn perpendicular
to the wavefronts. For a point source, the rays are radial lines
diverging from the source. 2008 by W.H. Freeman and Company Waves
in Three-Dimensions
In a homogeneous medium, such as air at constant density, the
wavefronts travels in straight lines in the direction of the rays,
much like a beam of particles. At a great distance from a point
source, a sufficiently small section of the wavefront can be
approximated by a flat source (a plane), and the rays are
approximately parallel lines; such a wave is called a plane wave
(Fig.7). The two-dimensional analog of a plane wave is a line wave,
which is a small part of a circular wavefront at a great distance
from the source. Line waves can also be produced in a ripple tank
by a line source, as in Fig.8. Waves in Three-Dimensions
Fig.7. Plane waves. At great distances from a point source, the
wavefronts are approximately parallel lines perpendicular to the
wavefronts. 2008 by W.H. Freeman and Company Waves in
Three-Dimensions
Fig.8. A two-dimensional analog of a plane wave can be generated in
a ripple tank by a flat board that oscillates up and down in the
water to produce the wavefronts, which are straight lines. 2008 by
W.H. Freeman and Company Wave Intensity If a point source emits
waves uniformly in all directions, then the energy at a distance r
from the source is distributed uniformly on a spherical surface of
radius r and area A=4r2. If Pav is the average power emitted by the
source is Pav/(4r2). The average power per unit area that is
incident perpendicular to the direction of propagation is called
the intensity. The SI units of intensity are watts per square meter
(W/m2). At a distance r from a point source, the intensity is: The
intensity of a three-dimensional wave varies inversely with the
square of the distance from a point source. (19) Intensity Defined
(20) Intensity due to a point source Wave Intensity There is a
simple relation between the intensity of a wave and the energy
density in the medium through which it propagates. Fig.9 shows a
spherical wave that has just reached the radius r1. The volume
inside the radius r1 contains energy because the particles in that
region are oscillating. The region outside r1 contains no energy
because the wave has not yet reached it. After a short time t, the
wave moves out a short distance r=v t past r1. The average energy
in the spherical shell of surface area A, thickness v t, and volume
V=A r=Av t is: The rate of transfer of energy is the power passing
into the shell. The average incident power is And the intensity of
the wave is: (21) Wave Intensity Figure 9 2008 by W.H. Freeman and
Company Wave Intensity Thus, the intensity equals the product of
the wave speed v and the average energy density av. Substituting
av=1/22s02 from Eq.18 for the energy density in a harmonic sound
wave, we obtain: where we have used s0=p0/(v) from Eq.16 This
result-that the intensity of a sound wave is proportional to the
square of the amplitude-is a general property of harmonic waves.
(22) Wave Intensity The human ear car accommodate a large range of
sound-wave intensities, from about 10-12W/m2 (which is usually
taken to be the threshold of hearing) to about 1 W/m2 (an intensity
great enough to estimate pain in most people). The pressure
amplitudes that correspond to these extreme intensities are about
3x10-5 Pa for the hearing threshold and 30 Pa for the pain
threshold. (Recall that a Pascal is a Newton per square meter).
These very small pressure variations add to or subtract from the
normal atmosphere pressure of about kPa. Wave Intensity Fig.10.
Sound waves from a telephone handset spreading out in the air. The
waves have been made visible by sweeping out the space in front of
the handset with a light source whose brightness is controlled by a
microphone. 2008 by W.H. Freeman and Company 2008 by W.H. Freeman
and Company
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