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Chapter 15: Problem 1

Fans at a local football stadium are so excited that their team is winning that they start "the wave" in celebration. Which of the following four statements is (are) true? I. This wave is a traveling wave. II. This wave is a transverse wave. III. This wave is a longitudinal wave. IV. This wave is a combination of a longitudinal wave and a transverse wave. a) I and II c) III only e) I and III b) II only d) I and IV

Short Answer

Expert verified
a) Transverse wave only b) Longitudinal wave only c) Both transverse and longitudinal waves d) Combination of transverse and longitudinal waves in a traveling wave Answer: d) Combination of transverse and longitudinal waves in a traveling wave

Step by step solution

01

Understand the characteristics of a traveling wave

A traveling wave is a type of wave that moves through a medium, transferring energy from one point to another. In this case, "the wave" is initiated by the fans and moves around the stadium in celebration. Since the wave is moving through the medium (i.e., the fans in the stadium), it is a traveling wave.
02

Define transverse and longitudinal waves

Transverse waves are characterized by oscillations or vibrations occurring perpendicular to the direction of energy transfer, while longitudinal waves have oscillations parallel to the direction of energy transfer. In other words, in a transverse wave, the particles of the medium move up and down, and in a longitudinal wave, the particles of the medium compress and expand along the direction of the wave.
03

Analyze the wave in the context of a football stadium

In "the wave" created by fans in a stadium, fans stand up and sit down as the wave passes, causing a noticeable motion that moves horizontally along the stadium rows. We observe up and down motion of fans (transverse motion), but we also see that fans transfer energy by influencing others to stand up and sit down in the same direction as the wave's movement (longitudinal motion). Thus, "the wave" in a football stadium context has both transverse and longitudinal characteristics.
04

Select the correct answer

Based on our analysis, we concluded that "the wave" is a traveling wave (I), has characteristics of both transverse and longitudinal waves, making it a combination of both types (IV). Therefore, the correct answer is d) I and IV.

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Most popular questions from this chapter

a) Starting from the general wave equation (equation 15.9 ), prove through direct derivation that the Gaussian wave packet described by the equation \(y(x, t)=(5.00 m) e^{-0.1(x-5 t)^{2}}\) is indeed a traveling wave (that it satisfies the differential wave equation). b) If \(x\) is specified in meters and \(t\) in seconds, determine the speed of this wave. On a single graph, plot this wave as a function of \(x\) at \(t=0, t=1.00 \mathrm{~s}, t=2.00 \mathrm{~s},\) and \(t=3.00 \mathrm{~s}\) c) More generally, prove that any function \(f(x, t)\) that depends on \(x\) and \(t\) through a combined variable \(x \pm v t\) is a solution of the wave equation, irrespective of the specific form of the function \(f\)

A string with a mass of \(30.0 \mathrm{~g}\) and a length of \(2.00 \mathrm{~m}\) is stretched under a tension of \(70.0 \mathrm{~N}\). How much power must be supplied to the string to generate a traveling wave that has a frequency of \(50.0 \mathrm{~Hz}\) and an amplitude of \(4.00 \mathrm{~cm} ?\)

A traveling wave propagating on a string is described by the following equation: $$ y(x, t)=(5.00 \mathrm{~mm}) \sin \left(\left(157.08 \mathrm{~m}^{-1}\right) x-\left(314.16 \mathrm{~s}^{-1}\right) t+0.7854\right) $$ a) Determine the minimum separation, \(\Delta x_{\min }\), between two points on the string that oscillate in perfect opposition of phases (move in opposite directions at all times). b) Determine the separation, \(\Delta x_{A B}\), between two points \(A\) and \(B\) on the string, if point \(B\) oscillates with a phase difference of 0.7854 rad compared to point \(A\). c) Find the number of crests of the wave that pass through point \(A\) in a time interval \(\Delta t=10.0 \mathrm{~s}\) and the number of troughs that pass through point \(B\) in the same interval. d) At what point along its trajectory should a linear driver connected to one end of the string at \(x=0\) start its oscillation to generate this sinusoidal traveling wave on the string?

The middle-C key (key 52 ) on a piano corresponds to a fundamental frequency of about \(262 \mathrm{~Hz},\) and the sopranoC key (key 64) corresponds to a fundamental frequency of \(1046.5 \mathrm{~Hz}\). If the strings used for both keys are identical in density and length, determine the ratio of the tensions in the two strings.

Suppose that the tension is doubled for a string on which a standing wave is propagated. How will the velocity of the standing wave change? a) It will double. c) It will be multiplied by \(\sqrt{2}\). b) It will quadruple. d) It will be multiplied by \(\frac{1}{2}\).
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