PES 2130 Fall 2014, Spendier
Lecture 8/Page 1
Lecture today: Chapter 16 Waves-1
1) Types of Mechanical Waves
2) Wavelength, frequency, amplitude, period, phase
3) Speed of traveling waves
Announcements:
EXAM 1 this Wednesday (chapters 18,19, and 20)
What is a wave?
It is a disturbance (or oscillation) that propagates in space and in time.
Waves are everywhere:
- Ocean
- Earth
- Matter (Probability waves, Predict behavior of e-, particles)
- Sound -> What we hear
- Sight -> What we see
- Electromagnetic waves (Power to our homes, Radio & TV , Electronics)
Waves are of three main types:
1. Mechanical waves. Common examples include water waves, sound waves, and
seismic waves. All these waves have two central features: They are governed by
Newton’s laws, and they can exist only within a material medium, such as water, air, and
rock. A mechanical wave is the propagation of energy through a medium (a material).
2. Electromagnetic waves. Common examples include visible and ultraviolet light, radio
and television waves, microwaves, x rays, and radar waves. These waves require no
material medium to exist. Light waves from stars, for example, travel through the vacuum
of space to reach us. All electromagnetic waves travel through a vacuum at the same
speed c = 299 792 458 m/s.
3. Matter waves. Although these waves are commonly used in modern technology, they
are probably very unfamiliar to you. These waves are associated with electrons, protons,
and other fundamental particles, and even atoms and molecules. Because we commonly
think of these particles as constituting matter, such waves are called matter waves.
Much of what we discuss in chapter 16 applies to waves of all kinds. However, for
specific examples we shall refer to mechanical waves.
DEMO: long spring, slinky, transverse wave vs longitudional wave (Slinky??)
PES 2130 Fall 2014, Spendier
Lecture 8/Page 2
Soliton wave: a solitary, traveling wave pulse like a Tsunami wave or audience wave in a
packed stadium.
Oscillatory waves: traveling wave pulses that repeat themselves
In physics 1 and 2 you studied examples of such oscillatory waves
- mass on a spring, pendulum, LC circuit
There are three types of waves:
1) Transverse Waves – Displacement of energy propagates perpendicular to medium’s
oscillation (wave on a string see fig 16-1 above)
2) Longitudinal Waves - Displacement of energy propagates parallel to medium’s
oscillation (sound wave)
3) Rolling Waves - A combination of transverse and longitudinal (water waves)
How to mathematically describe a wave:
To completely describe a wave on a string (and the motion of any element along its
length), we need a function that gives the shape of the wave. This means that we need a
relation in the form
y = h(x, t),
in which y is the transverse displacement of any string element as a function h of the time
t and the position x of the element along the string.
For a sinusoidal wave like that of Fig. 16-1b traveling in the positive direction of an x
axis, as the wave sweeps through succeeding elements (that is, very short sections) of the
string, the elements oscillate parallel to the y axis. At time t, the displacement y of the
element located at position x is given by the wave equation
PES 2130 Fall 2014, Spendier
Lecture 8/Page 3
You should already know what each terms in this equation mean (physics 1). Here is a
quick review:
Amplitude: ym
Maximum distance from zero
Largest value of sine and cosine is 1 and smallest -1
To modify the sine/cosine function so it goes between the maximum magnitudes of the
displacement specific for the problem we multiply sine/cosine by the amplitude ym.
Wavelength: λ [m]
The wavelength of a wave is the distance (parallel to the direction of the wave’s travel)
between repetitions of the shape of the wave (or wave shape). The wavelength is the
distance between points in the medium that are in phase.
Angular wave number: k
At time t = 0, the description of the wave shape is:
y(x,0) = ym sin(kx)
By definition, the displacement y is the same at both ends of this wavelength, that is, at x
= x1 and x = x1 + λ. Thus,
ym sin(kx1) = ymsin(k x1 +k λ)
A sine function begins to repeat itself when its angle (or argument) is increased by 2π
rad, we must have
k λ = 2π
k = 2π/ λ [rad/m] or [1/m]
k is used often in Physics – k is measured rather than λ in many experiments.
PES 2130 Fall 2014, Spendier
Lecture 8/Page 4
Period: T
We define the period of oscillation T of a wave to be the time any string element takes to
move through one full oscillation
Angular frequency: ω = 2π/ T
[rad/s]
Frequency: f
frequency f is a number of oscillations per unit time
f = 1/T = ω/2π [1/s] = [Hz]
Example: Which wave (red or blue) has the higher frequency and/or longer period?
Red wave: higher frequency
Blue wave: longer period
So less wave cycles of the blue wave pass an observer in a given time than of the red
wave.
Phase constant: ϕ [rad]
We use the phase angle to shift the sine/cosine back and forward so that it starts wherever
needed.
y(x,t) = ym sin(kx-ωt+ ϕ)
There is no shift if ϕ = 0
PES 2130 Fall 2014, Spendier
Lecture 8/Page 5
Phase of the wave: kx-ωt
The wave equation can be rewritten as
y(x,t) = ym sin(kx-ωt)= ym sin(2πx/λ-2π t/T)
y(x,t) = ym sin[2π (x/λ-t/T)]
kx-ωt = 2π (x/λ-t/T) is an angle in rad to evaluate the sine for. This term is called the
phase of the wave. As position x and time t vary, the phase changes.
The speed of a traveling wave: v
The rate at which the energy propagates (or how much fast the “bump” of wave moves)
Let’s find v = dx/dt
Although the wave is moving, the phase will be constant
kx-ωt = constant
take derivative of both sides – both x and t change:
k dx-ω dt = 0
k dx = ω dt
v = dx/dt = ω/k
or v = λ/T
Last 30min:
Comments about mistakes on HW
In-class discussion about questions regarding test 1