Stewart Precalc 6e Section 7.FM: Focus on Modeling: Traveling and Standing Waves

1 Problem - Wave on a Canal · Level 3

Wave on a Canal. A wave on the surface of a long canal is described by the function \(y(x, t) = 5 \sin\left(2 x - \dfrac{\pi}{2} t\right)\), \(x \geq 0\)

(a) Find the function that models the position of the point \(x = 0\) at any time \(t\).

Answer for 1(a)

(b) Sketch the shape of the wave when \(t = 0, 0.4, 0.8, 1.2\), and \(1.6\). Is this a traveling wave?

Answer for 1(b)

(c) Find the velocity of the wave.

Answer for 1(c)

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2 Problem - Wave in a Rope · Level 3

Wave in a Rope. Traveling waves are generated at each end of a tightly stretched rope 24 ft long, with equations \(y = 0.2 \sin(1.047 x - 0.524 t)\) and \(y = 0.2 \sin(1.047 x + 0.524 t)\)

(a) Find the equation of the combined wave, and find the nodes.

Answer for 2(a)

(b) Sketch the graph for \(t = 0, 1, 2, 3, 4, 5\), and \(6\). Is this a standing wave?

Answer for 2(b)

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3 Problem - Traveling Wave from Graph · Level 3

Traveling Wave. A traveling wave is graphed at the instant \(t = 0\). If it is moving to the right with velocity \(6\), find an equation of the form \(y(x, t) = A \sin(k x - k v t)\) for this wave.

4 Problem - Traveling Wave from Parameters · Level 3

Traveling Wave. A traveling wave has period \(\dfrac{2 \pi}{3}\), amplitude \(5\), and velocity \(0.5\).

(a) Find the equation of the wave.

Answer for 4(a)

(b) Sketch the graph for \(t = 0, 0.5, 1, 1.5\), and \(2\).

Answer for 4(b)

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5 Problem - Standing Wave with Frequency · Level 3

Standing Wave. A standing wave with amplitude \(0.6\) is graphed at several times \(t\) as shown in the figure. If the vibration has a frequency of \(20\) Hz, find an equation of the form \(y(x, t) = A \sin(\alpha x) \cos(\beta t)\) that models this wave.

6 Problem - Standing Wave with Specified Nodes · Level 3

Standing Wave. A standing wave has maximum amplitude \(7\) and nodes at \(0, \dfrac{\pi}{2}, \pi, \dfrac{3 \pi}{2}, 2 \pi\), as shown in the figure. Each point that is not a node moves up and down with period \(4 \pi\). Find a function of the form \(y(x, t) = A \sin(\alpha x) \cos(\beta t)\) that models this wave.

7 Problem - Vibrating String · Level 4

Vibrating String. When a violin string vibrates, the sound produced results from a combination of standing waves that have evenly placed nodes. The figure illustrates some of the possible standing waves. Let's assume that the string has length \(\pi\).

(a) For fixed \(t\), the string has the shape of a sine curve \(y = A \sin(\alpha x)\). Find the appropriate value of \(\alpha\) for each of the illustrated standing waves.

Answer for 7(a)

(b) Do you notice a pattern in the values of \(\alpha\) that you found in part (a)? What would the next two values of \(\alpha\) be? Sketch rough graphs of the standing waves associated with these new values of \(\alpha\).

Answer for 7(b)

(c) Suppose that for fixed \(t\), each point on the string that is not a node vibrates with frequency \(440\) Hz. Find the value of \(\beta\) for which an equation of the form \(y = A \cos(\beta t)\) would model this motion.

Answer for 7(c)

(d) Combine your answers for parts (a) and (c) to find functions of the form \(y(x, t) = A \sin(\alpha x) \cos(\beta t)\) that model each of the standing waves in the figure. (Assume that \(A = 1\).)

Answer for 7(d)

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8 Problem - Waves in a Tube · Level 4

Waves in a Tube. Standing waves in a violin string must have nodes at the ends of the string because the string is fixed at its endpoints. But this need not be the case with sound waves in a tube (such as a flute or an organ pipe). The figure shows some possible standing waves in a tube. Suppose that a standing wave in a tube \(37.7\) ft long is modeled by the function \(y(x, t) = 0.3 \cos\left(\dfrac{x}{2}\right) \cos(50 \pi t)\) Here \(y(x, t)\) represents the variation from normal air pressure at the point \(x\) feet from the end of the tube, at time \(t\) seconds.

(a) At what points \(x\) are the nodes located? Are the endpoints of the tube nodes?

Answer for 8(a)

(b) At what frequency does the air vibrate at points that are not nodes?

Answer for 8(b)

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9 Example - A Traveling Wave · Level 2

A traveling wave is described by the function \(y(x, t) = 3 \sin\left(2 x - \dfrac{\pi}{2} t\right)\), \(x \geq 0\)

(a) Find the function that models the position of the point \(x = \dfrac{\pi}{6}\) at any time \(t\). Observe that the point moves in simple harmonic motion.

Answer for 9(a)

(b) Sketch the shape of the wave when \(t = 0, 0.5, 1.0, 1.5\), and \(2.0\). Does the wave appear to be traveling to the right?

Answer for 9(b)

(c) Find the velocity of the wave.

Answer for 9(c)

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10 Example - A Standing Wave · Level 2

Traveling waves are generated at each end of a wave tank 30 ft long, with equations \(y = 1.5 \sin\left(\dfrac{\pi}{5} x - 3 t\right)\) and \(y = 1.5 \sin\left(\dfrac{\pi}{5} x + 3 t\right)\)

(a) Find the equation of the combined wave, and find the nodes.

Answer for 10(a)

(b) Sketch the graph for \(t = 0, 0.17, 0.34, 0.51, 0.68, 0.85\), and \(1.02\). Is this a standing wave?

Answer for 10(b)

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