Wave on a Canal A wave on the surface of a long canal is described by the function
- (a) Find the function that models the position of the point x = 0 at any time t.
- (b) Sketch the shape of the wave when t = 0, 0.4, 0.8, 1.2, and 1.6. Is this a traveling wave?
- (c) Find the velocity of the wave.
(a)
To find: The function that outputs the position of the point
Answer to Problem 1P
The function
Explanation of Solution
A wave on the surface of a long canal is described by the function
The position of the wave at any point x is given by
To find the function that gives the position of the point
Therefore, the position of the point
(b)
To sketch: The shape of the wave for times
Explanation of Solution
Given:
A wave on the surface of a long canal is described by the function
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 is a sinusoidal wave.
Substitute
Use online graphing calculator and draw the graph of
From Figure 5 it can be observed that the wave is a sinusoidal wave.
From above all graphs it can be observed that the point is at different position at every instant of time.
Therefore, it can be concluded that the wave is travelling.
(c)
To find: The velocity of the wave.
Answer to Problem 1P
The velocity of the wave is
Explanation of Solution
Given:
A wave on the surface of a long canal is described by the function
Formula used:
If a wave is expressed in its standard form
Calculation:
The position of any point x on the wave is given by
Express the above expression in standard for as
Therefore, using the above formula the velocity of the wave is
Chapter 7 Solutions
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