‫األول الباب‬
‫‪Chapter 1‬‬
‫‪Wave Motion‬‬
‫الحركة الموجية‬
Introduction

A wave is a disturbance from an equilibrium
state that moves or propagates with time
from one region to of space to another.
Examples


Dropping a stone into the water produces a
disturbance which spreads out horizontally in
all directions along the surface.
A source of sound produces a fluctuation in
pressure in the surrounding atmosphere, and
this disturbance is propagated to distant
points.


Light, radio waves, x-rays, and γ rays are all
examples of electromagnetic waves.
A characteristic of all waves is the ability to
transport energy from one region of space to
another.
Propagation of a Disturbance



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.
A pulse traveling down a stretched rope.

A traveling wave or pulse that causes the
elements of the disturbed medium to move
perpendicular to the direction of propagation
is called a
transverse wave.
Stretched spring.

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.
The motion of water elements on the surface of deep
water in which a wave is propagating is a combination of
transverse and longitudinal displacements.
Elements at the surface move in nearly circular paths.
Each element is displaced both horizontally and vertically
from its equilibrium position.
Various forms of waves
Consider a pulse traveling to the right on a
long string,

Consequently, an element of the string at x at
this time has the same y position as an
element located at (x – vt) had at time t = 0:

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,
and depends on the two variables x and t.
it is written:
y(x, t)


The wave function y(x, t) represents the y
coordinate, the transverse position of any
element located at position x at any time t.
the wave function y(x), sometimes called the
waveform,
Example

A pulse moving to the right along the x axis is
represented by the wave function

where x and y are measured in centimeters
and t is measured in seconds. Plot the wave
function at t = 0, t = 1.0 s, and t = 2.0 s.
Solution




this function is of the form:
y = f (x - vt).
The wave speed is: v = 3.0 cm/s.
The maximum value of y is given by :
A = 2.0 cm.
Representing y by letting (x - 3.0 t = 0.)
The wave function expressions are:

We now use these expressions to plot the
wave function versus x at these times. For
example, let us evaluate y(x, 0) at x = 0.50
cm:
Sinusoidal Waves


The point at which the displacement of the
element from its normal position is highest is
called the crest of the wave.
The distance from one crest to the next is
called:
the wavelength

The wavelength is the minimum distance
between any two identical points (such as
the crests) on adjacent waves.

The period T is the time interval required for
two identical points (such as the crests) of
adjacent waves to pass by a point.
The period of the wave is the same as the
period of the simple harmonic oscillation of
one element of the medium.


The same information is more often given by
the inverse of the period, which is:
the frequency f.
The frequency of a periodic wave is the
number of crests (or troughs, or any other
point on the wave) that pass a given point in
a unit time interval.

The frequency of a sinusoidal wave is related
to the period by:

The most common unit for frequency, is
second-1, or hertz (Hz).
The corresponding unit for T is seconds.


Because the wave is sinusoidal, we expect
the wave function at this instant to be
expressed as:
y(x, 0) = A sin ax,
where A is the amplitude and a is a constant
to be determined. At x = 0, we see that
y(0, 0) = A sin a(0) = 0,

The next value of x for which y is zero is
Thus,

For this to be true, we must have

If the wave moves to the right with a speed v,
then the wave function at some later time t is

the wave speed, wavelength, and period are
related by the expression

Substituting

We can express the wave function in a
convenient form by defining two other
quantities:
the angular wave number k (usually called
simply the wave number)

angular frequency

we generally express the wave function in the
form

where is the phase constant,. This constant
can be determined from the initial conditions.
4- Sinusoidal Waves on Strings

at t = 0, the wave function can be written as

Therefore, the transverse speed vy and the
transverse acceleration ay of elements of the
string are:

The maximum values of the transverse speed
and transverse acceleration are simply the
absolute values of the coefficients of the
cosine and sine functions:

The transverse speed reaches its maximum,
whereas the magnitude of the transverse
acceleration reaches its maximum value:
(ω2A) when y = ± A.
5- The Speed of Waves on Strings

If the tension in the string is T and its mass
per unit length is μ, then the wave speed is
6- Reflection and Transmission

A pulse traveling
on a string that is
rigidly attached to
a support at one
end
the pulse arrives at the end of a string
that is free to move vertically
A light string is attached to a heavier string:

According to the Equation of the speed of a
wave on a string:
the speed increases as the mass per unit
length of the string decreases.

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 vA > vB (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 vA < vB (that is, when A is
denser than B), it is not inverted upon
reflection.
Rate of Energy Transfer by
Sinusoidal Waves on Strings

an element of the string of length Δx and
mass Δm

The kinetic energy K associated with a
moving particle is

an element of length Δx and mass Δm, we
see that the kinetic energy ΔK of this element
is:

at time t = 0, then the kinetic energy of a
given element is

the total kinetic energy Kλ in one wavelength

A similar analysis to that above for the total
potential energy Uλ in one wavelength will
give exactly the same result:

The total energy in one wavelength of the
wave is the sum of the potential and kinetic
energies:

The power, or rate of energy transfer,
associated with the wave is
8- The Linear Wave Equation
The ends of the
element make
small angles θA
and θB with the x
axis.The net force
acting on the
element in the
vertical direction is


Imagine undergoing an infinitesimal
displacement outward from the end of the
rope element along the line representing the
force T.
This displacement has infinitesimal x and y
components and can be represented by the
vector:


Because we are evaluating this tangent at a
particular instant of time, we need to express
this in partial form as
Substituting for the tangents gives:

The tangent of the angle with respect to the x
axis for this displacement is dy/dx.

Apply Newton’s second law to the element,
with mass given by m = μ Δx

Combining the two last equations, we obtain

the partial derivative of any function is
defined as



If we associate f(x + Δx) with and
f(x) with , we see that, in the limit Δx → 0,
this becomes:
If we take the sinusoidal wave function to be
of the form

Substituting



Both sides of the equation depend on x and t
through the same function .
Because this function divides out, we do
indeed have an identity, provided that:
Using the relationship v = ω/k , in this
expression, we see that:
The linear wave equation is often
written in the form:
9- The principle of superposition
The composite wave is resolved using Fourier
analysis to a fundamental wave and harmonics
Chapter 2
Sound Waves


30
≤
f
≤
17 000 Hz.
Longitudinal Waves


We call sounds waves with frequencies
above about 17,000 Hz
 ultrasonic waves,
and those with frequencies less than about
30 Hz
 infrasonic waves.


The circular arcs represent the crests of the
wave. They are also called wave fronts.
The energy given to the wave by the sound
source flows out along the straight lines,
called rays.
2- Speed of Sound Waves
In air: 340 meters/second, 760 miles/hour
 Mach 1
The speed of sound in ideal gases is given by:



p is the absolute pressure of the gas ρ is its
density. γ is related to the nature of the gas.
It is about 1.67 for monoatomic gases such
as He. For diatomic gases as N2 or O2 it is
about 1.4.

ρ = m / v, p v = n R T, and n = m / M ;
we obtain :

the speed of sound in an ideal gas depends
on temperature but not on pressure. In the
case of air this gives:
vair = [ 331.5 + 0.61 TC ] m/s
where TC is the Celsius temperature. For
example, at 20°C the speed of sound in air is:
v = 331.5 + (0.61) 20 = 243.7 m/s





The speed of sound in liquids and solids also
depends on the material, and is given by:
ρ is the density of the material and B is the
bulk modulus.
For sound waves in rods the bulk modulus is
replaced by Young's modulus Y. We have:
3- Sound intensity









The decibel sound level scale is defined by:
L = sound loudness level in decibels
= 10 log ( I / Io )
where I is the sound intensity and Io is a
reference sound intensity = 1x10-12 W/m2 .
L = 10 ( log I - log Io )
Because log (1x10-12 ) = - 12
L = 120 + 10 log I
where I must be in watts per square meter
and L must be in dB.
Sometimes the loudness is expressed in Bels
where 10 decibels = 1 Bel.
4- Interference of sound waves;
Beats

Let the two waves, of frequencies f1 and f2,
be represented at a fixed point in space by

The resultant displacement, by the principle
of superposition, is


Using the identity
5- Doppler effect
Doppler Effect
Change in frequency of a wave due to relative
motion between source and observer.
A sound wave frequency change is noticed as a
change in pitch.

If this motion occurs at a frequency f0 with
period T0, the speed of the waves on the
water surface is v0, we can find the
wavelength of the waves: λ0 = v0 T0

for approaching moving source:

for receding moving source:

for approaching moving listener:

for receding moving listener:

the general formula:

The upper sign in each case is to be used if
the velocity is one of approach.
If it is one of recession, then the lower sign
is to be used

6- Response of the ear

Shock Wave

The ratio (vS /v) is referred to as the Mach
number, and the conical wave front
produced when vS > v (supersonic speeds) is
known as a shock wave.
8- Applications of sound
Sonar
Ultrasound in medicine
Doppler flow meter