Title: Objectives: After completion of this module, you should be able to:
1Objectives After completion of this module, you
should be able to
- Demonstrate your understanding of transverse and
longitudinal waves. - Define, relate and apply the concepts of
frequency, wavelength, and wave speed. - Solve problems involving mass, length, tension,
and wave velocity for transverse waves. - Write and apply an expression for determining the
characteristic frequencies for a vibrating string
with fixed endpoints.
2Mechanical Waves
A mechanical wave is a physical disturbance in an
elastic medium.
Consider a stone dropped into a lake.
Energy is transferred from stone to floating log,
but only the disturbance travels.
Actual motion of any individual water particle is
small.
Energy propagation via such a disturbance is
known as mechanical wave motion.
3A Transverse Wave
In a transverse wave, the vibration of the
individual particles of the medium is
perpendicular to the direction of wave
propagation.
4 Longitudinal Waves
In a longitudinal wave, the vibration of the
individual particles is parallel to the direction
of wave propagation.
5Wave speed in a string.
The wave speed v in a vibrating string is
determined by the tension F and the linear
density m, or mass per unit length.
L
m m/L
v speed of the transverse wave (m/s) F
tension on the string (N) m or m/L mass per
unit length (kg/m)
6Example 1 A 5-g section of string has a length
of 2 M from the wall to the top of a pulley. A
200-g mass hangs at the end. What is the speed of
a wave in this string?
F (0.20 kg)(9.8 m/s2) 1.96 N
v 28.0 m/s
Note Be careful to use consistent units. The
tension F must be in newtons, the mass m in
kilograms, and the length L in meters.
7Periodic Wave Motion
A vibrating metal plate produces a transverse
continuous wave as shown.
For one complete vibration, the wave moves a
distance of one wavelength l as illustrated.
8Velocity and Wave Frequency.
The period T is the time to move a distance of
one wavelength. Therefore, the wave speed is
The frequency f is in s-1 or hertz (Hz).
The velocity of any wave is the product of the
frequency and the wavelength
9Production of a Longitudinal Wave
- An oscillating pendulum produces condensations
and rarefactions that travel down the spring.
- The wave length l is the distance between
adjacent condensations or rarefactions.
10Velocity, Wavelength, Speed
Wave equation
11The Superposition Principle
- When two or more waves (blue and green) exist in
the same medium, each wave moves as though the
other were absent.
- The resultant displacement of these waves at any
point is the algebraic sum (yellow) wave of the
two displacements.
Constructive Interference
Destructive Interference
12Formation of a Standing Wave
Incident and reflected waves traveling in
opposite directions produce nodes N and antinodes
A.
The distance between alternate nodes or
anti-nodes is one wavelength.
13Possible Wavelengths for Standing Waves
Fundamental, n 1
1st overtone, n 2
2nd overtone, n 3
3rd overtone, n 4
n harmonics
14Possible Frequencies f v/l
Fundamental, n 1
1st overtone, n 2
2nd overtone, n 3
3rd overtone, n 4
n harmonics
15Characteristic Frequencies
Now, for a string under tension, we have
Characteristic frequencies
16Example 2. A 9-g steel wire is 2 m long and is
under a tension of 400 N. If the string vibrates
in three loops, what is the frequency of the wave?
For three loops n 3
Third harmonic 2nd overtone
f3 224 Hz
17Summary for Wave Motion