Ch 16 Waves 1
2 Types of Waves. There are many types of waves, such as mechanical waves (water, sound,
seismic…), electromagnetic waves (light, radio, microwave…), matter waves…
3 Transverse and Longitudinal Waves. Traveling waves carry energy (not material) from one place to
another. Transverse waves: motion is perpendicular to the direction of propagation.
Longitudinal waves: motion is parallel to the direction of propagation. In general, waves (e.g., water
waves) have a combination of transverse and longitudinal motion.
4 Wavelength and Frequency.
General form of a wave travelling in the positive
direction:
( , )=
(
−
+ ).
The amplitude is . Depending on the context, the amplitude may have any units. If the waveform
represents the transverse motion of a string, the units are of distance:[ ] = [ ] = .
The phase is
−
+ . The phase constant is . The (angular) wave number is , closely related to
the wavelength λ, where = 2 /. The angular frequency (angular velocity) is , closely related to the
period
•
= 2 / . Units: [ ] = [ ℎ
]=
, [ ]=
,[ ] =
, [] =
,[ ] = .
Checkpoint 1
= / = / =  .
For a wave traveling in the ± direction with speed , the waveform can be written as
( , ) = sin[ ( ∓ ) + ].
5 The Speed of a Traveling Wave. The speed of propagation is
•
•
•
Checkpoint 2
Sample Problem: Transverse wave, amplitude, wavelength, period, velocity
Sample Problem: Transverse wave, transverse velocity, transverse acceleration
6 Wave Speed on a Stretched String. Derivation from Newton’s 2nd Law: It follows that, for small
displacements, the speed of transverse waves on a string does not depend on the frequency (or
wavelength), and is given by
density. Units: [ ] =
•
, [ ]=
=
, where is the string tension, and
,[ ] =
=
/ is its linear mass
.
Checkpoint 3
9 The Principle of Superposition for Waves. When two waves that obey a linear differential
equation are travelling through the same region of space, their waveforms are added to get the
resultant waveform. Thus, the propagation of a wave is not altered by the presence of another wave.
10 Interference of Waves. Constructive interference: When the phases of two waves in the same
place are (exactly or nearly) the same, or differ by a multiple of 2 (0, ±2 , ±4 , ±6 , … ), their
waveforms have the same sign, and their addition creates a larger wave.
Destructive interference: When the phases of two waves in the same place differ by (exactly or nearly)
an odd multiple of (± , ±3 , ±5 , … ), their waveforms have opposite sign, and their addition creates
a smaller wave.
•
•
•
Table 1: Phase Difference and Resulting Interference Type
Checkpoint 4
Sample Problem: Interference of two waves, same direction, same amplitude
12 Standing Waves. If two sinusoidal waves with the same amplitude and wavelength travel in
opposite directions along a stretched string, their interference produces a standing wave. At the nodes,
there is no string motion; at the anti-nodes, the string motion is maximal. If there is a node at the origin
( = 0), then the resulting waveform takes the form ( , ) = 2 sin( )cos( + ).
The nodes are located at = /2, for = 0, ±1, ±2, ±3, …
The antinodes are halfway between the nodes, at
= ( + )/2, for
= 0, ±1, ±2, ±3, …
Reflections at a boundary. In this context, a boundary is a place where the string comes to an end. If the
boundary is “hard,” i.e. the string is fixed in place, the waveform is inverted as it bounces from the
boundary. So a positive (upward) pulse becomes a negative (downward) pulse. If the boundary is “soft,”
i.e. the string is free to move without friction, the waveform is not inverted as it bounces from the
boundary. So a positive (upward) pulse remains a positive (upward) pulse.
•
Checkpoint 5
13 Standing Waves and Resonance. A finite string has boundaries at either end. If the wavelength
(or frequency) has the right value, then standing wave patterns will fit neatly on the string. These
waveforms are said to have resonant frequencies. The correct wavelengths are those that will match the
boundary conditions. If the string is held fixed at both ends, then the string length must equal a multiple
of a half-wavelength: = /2, for = 1, 2, 3 … The wavelengths are  = 2 / and the frequencies
are = / = /2 . The lowest frequency is called the fundamental frequency, or 1st harmonic:
= /2 ; any other frequency is called the nth harmonic frequency.
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•
Checkpoint 6
Sample Problem: Resonance of transverse waves, standing waves, harmonics
----------------Material below is not covered in this course----------------------------------------------------7 Energy and Power of a Wave Travelling Along a String
Kinetic energy. Elastic potential energy. Energy transport. The rate of energy transmission.
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Sample Problem: Average power of a transverse wave
8 The Wave Equation
11 Phasors
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Sample Problem: Interference of two waves, same direction phasors, any amplitude