(a)
To find: The equation that outputs the combined wave and also find the nodes.
(a)
Answer to Problem 2P
The combined equation of the wave is
The nodes are at
Explanation of Solution
Given:
A wave travelling in a rope of length 24 ft is a combination of two distinct waves
Result used:
Calculation:
The combined wave equation is the sum of the two equations of the wave.
Therefore, obtain the combined equation as follows.
Therefore, the combined wave equation is
Note that the node occurs when
That is
Therefore, nodes occur at
(b)
To sketch: The shape of the wave for times
(b)
Explanation of Solution
Given:
A wave travelling in a rope of length 24 ft is a combination of two distinct waves
Calculation:
The position of any point x on the wave at time t in the canal is given by
Substitute
Use online graphing calculator and draw the graph of
From Figure 1 it can be observed that the wave is a sinusoidal wave.
Substitute
Use online graphing calculator and draw the graph of
From Figure 2 it can be observed that the wave is a sinusoidal wave.
Substitute
Use online graphing calculator and draw the graph of
From Figure 3 it can be observed that the wave is a sinusoidal wave.
Substitute
Use online graphing calculator and draw the graph of
From Figure 4 it can be observed that the wave has very small amplitude so it appears to be coinciding with the x-axis.
Substitute
Use online graphing calculator and draw the graph of
From Figure 5 it can be observed that the wave is a sinusoidal wave.
Substitute
Use online graphing calculator and draw the graph of
From Figure 6 it can be observed that the wave is a sinusoidal wave.
Substitute
Use online graphing calculator and draw the graph of
From Figure 7 it can be observed that the wave is a sinusoidal wave.
From above all graphs it can be observed that the point is at same position at every instant of time.
Therefore, it can be concluded that the wave is standing.
Chapter 7 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning