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HW4
Due: 11:00pm on Thursday, March 4, 2010
Note:
To understand how points are awarded, read your instructor's
Grading Policy
.
Creating a Standing Wave
Description:
Show that the superposition of two traveling waves can be a standing wave; then answer qualitative multiple choice questions.
Learning Goal:
To see how two traveling waves of the same frequency create a standing wave.
Consider a traveling wave described by the formula
.
This function might represent the lateral displacement of a string, a local electric field, the position of the surface of a body of water, or any of a number of other physical manifestations of waves.
Part A
Which one of the following statements about the wave described in the problem introduction is correct?
ANSWER:
The wave is traveling in the
direction.
The wave is traveling in the
direction.
The wave is oscillating but not traveling.
The wave is traveling but not oscillating.
Part B
Which of the expressions given is a mathematical expression for a wave of the same amplitude that is traveling in the opposite direction? At time
this new wave should have the same displacement as
, the wave described in the problem introduction.
ANSWER:
The principle of
superposition
states that if two functions each separately satisfy the wave equation, then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every
term in the wave equation is linear in the amplitude of the wave.
Consider the sum of two waves
, where
is the wave described in Part A and
is the wave described in Part B. These waves have been chosen so that their sum can be
written as follows:
.
This form is significant because
, called the envelope, depends only on position, and
depends only on time. Traditionally, the time function is taken to be a trigonometric function with unit
amplitude; that is, the overall amplitude of the wave is written as part of
.
Part C
Find
and
. Keep in mind that
should be a trigonometric function of unit amplitude.
Hint C.1
A useful identity
A useful trigonometric identity for this problem is
.
Hint C.2
Applying the identity
Since you really need an identity for
, simply replace
by
in the identity from Hint C.1, keeping in mind that
.
Express your answers in terms of
, , ,
, and . Separate the two functions with a comma.
ANSWER:
,
=
Part D
Which one of the following statements about the superposition wave
is correct?
ANSWER:
This wave is traveling in the
direction.
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View Full DocumentThis wave is traveling in the
direction.
This wave is oscillating but not traveling.
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 Fall '10
 MichaelChen
 Physics

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