Lesson 24 - Waves
I.
General Information
A.
Basic Definition
The movement of a disturbance with the transfer of energy and linear momentum
without the transfer of mass.
B.
Types
1.
Longitudinal waves
Longitudinal waves are waves in which the motion of the individual atoms is
parallel to the direction of propagation of the disturbance.
EXAMPLES:
2.
Transverse Waves
Transverse waves are waves in which the motion of the individual atoms or field
is perpendicular to the direction of propagation of the disturbance.
EXAMPLES:
3.
One Dimensional Traveling Wave
A one dimensional traveling wave is any function that can be described by either
f(x-vt) or f(x+vt)
or as a linear combination of the two. The first function represents a wave
traveling in the +x-direction while the second function represents a wave traveling
in the -x-direction.
EXAMPLE: Sinusoidal Wave
Consider the following transverse wave shown below:

2



y  A Sin  x  υ t 








We plot the wave a time t=0.0 s below

2

y(t  0 s)  ASin x 




y



A
x


2
-A
1.
Wavelength - The minimal distance between two peaks of a wave.
Symbol -
2.
Period - The time it takes for two identical points (like the peaks) on a wave to
pass by a point. Alternatively, it is the time it takes for a individual point to make
a single cycle of disturbance.
Symbol -
3.
Frequency - The number of waves passing a point in 1 second or the number of
oscillations at any point in 1 second.
Symbol Units -
4.
Angular Frequency - Related to frequency for waves in the same manner as it
was for rotation.
=
5.
Wave Number - Related to the wavelength of a wave in the same manner that
angular frequency is related to period.
Symbol Units -
EXAMPLE : Plane Waves
One important traveling wave is the plane wave given by the equations below:
The wave is called a plane wave because the surface of constant phase angle at any
instant of time is a plane for a given value of x as shown below:
EXAMPLE: Spherical Waves
A tree dimensional spherical wave is the output due to any point source and is given by
the equations:
The 1/r dependence is required by conservation of energy as the sphere propagates in the
radial direction since we will see later that the energy density of a wave is proportional to
the square of the wave's amplitude.
II.
Superposition and Interference of Waves
A.
Superposition When two waves move through a medium, the total wave is the algebraic sum of
the individual waves.
B.
Totally Constructive Interference
When two waves of the same frequency are in phase, the result will be a single
larger wave of the same frequency whose amplitude is the sum of the amplitudes
of the two individual waves.
YTOTAL  A1 Sin(kx t  )  A 2 Sin(kx t  )
YTOTAL 
C.
Total Destructive Interference
Two waves traveling in the same direction with the same frequency and amplitude
will completely cancel if they are 180 degrees out if phase. This is called total
destructive interference. Destructive interference is the tell-tale experimental
proof of wave phenomena.
III.
Transmission and Reflection of Waves
When a wave strikes the interface between two different media, the wave may be
reflected, transmitted, or both.
A.
From A Light to An Infinitely Dense Medium
_________________ Transmission and _______________ Reflection
Phase Results:
B.
Across An Interfaces Between Two Media of The Same Density
_______________ Transmission and ________________ Reflection
Phase Results:
C.
From A Dense Media To An Almost Zero Density Media
_________________ Transmission and ____________________ Reflection
Phase Results:
D.
From A Dense To A Light Media
________________Transmission and __________________ Reflection
Phase Results :