Question

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

a) This wave is a traveling wave, and this wave is a transverse wave. b) This wave is a traveling wave, and this wave is a combination of longitudinal and transverse waves. c) This wave is a longitudinal wave, and this wave is a transverse wave. d) This wave is not a traveling wave, and this wave is a longitudinal wave. Answer: a) This wave is a traveling wave, and this wave is a transverse wave.

Step-by-step solution

Understanding different types of waves

In general, there are two types of waves: longitudinal and transverse. A longitudinal wave is a wave in which the particles of the medium vibrate parallel to the direction of the wave's motion (e.g., sound waves). A transverse wave is a wave in which the particles of the medium vibrate perpendicular to the direction of the wave's motion (e.g., light waves).

Analyzing the wave created by fans

In the case of "the wave" at a football stadium, fans stand up and raise their arms in sequence, creating a wave-like motion. Fans don't physically move to create the wave. Instead, they move up and down at their respective positions while the overall motion of the wave travels around the stadium.

Determining if the wave is a traveling wave

A traveling wave is a wave that moves from one place to another in space. Since the excitement wave created by fans moves around the stadium, it can be considered a traveling wave. Therefore, statement I is true.

Determining if the wave is a transverse wave

Recall that transverse waves have particles of the medium vibrating perpendicular to the direction of the wave's motion. In this case, the fans move up and down while the wave travels horizontally around the stadium. The motion of fans is perpendicular to the motion of the wave. Therefore, statement II is true.

Determining if the wave is a longitudinal wave

In a longitudinal wave, the particles of the medium vibrate parallel to the direction of the wave's motion. The fans' motion in the case of the stadium wave is up and down, not parallel to the motion of the wave. Therefore, statement III is false.

Determining if the wave is a combination of longitudinal and transverse waves

Since the fans' motion is not parallel to the wave's motion, we can rule out the possibility of the wave being a combination of longitudinal and transverse waves. Therefore, statement IV is false. Hence, the correct answer is a) I and II (This wave is a traveling wave, and this wave is a transverse wave).

Key concepts

Traveling Waves

Traveling waves are fascinating phenomena observed in various physical settings, from the rippling surface of a pond to the invisible signals carrying our cell phone conversations. At their core, traveling waves are disturbances or oscillations that move through a medium (or in some cases, even in a vacuum) from one location to another, carrying energy without transporting matter.

When fans at a football stadium perform 'the wave', they essentially create a textbook example of a traveling wave. Each fan represents a particle of the medium; as they stand up and sit down, they pass on the energy of the wave to their neighbors, creating a chain reaction that propagates the wave around the stadium. Notably, the individual fans don't travel with the wave; they merely transfer its energy, capturing the essence of how traveling waves operate in other mediums, such as water waves or sound waves.

Transverse Waves

Now, let's dive into the nature of transverse waves. These waves are characterized by particle motion which is perpendicular to the direction in which the wave travels. Common examples include waves on a string, light waves, and indeed the human 'wave' at sporting events.

In the stadium 'wave', fans stand up and sit down, moving vertically while the wave travels horizontally around the stadium. The fans don't move forward with the wave; instead, they oscillate up and down—just like the particles in a plucked guitar string. This up-and-down motion, at right angles to the travel direction of the wave, hallmarks the transverse wave. To put this into perspective, imagine a water wave: the water's surface goes up and down, while the wave itself moves across the pond.

Longitudinal Waves

While transverse waves are all about perpendicular motion, longitudinal waves have a different mode of oscillation. In these waves, the particles of the medium move parallel to the direction in which the wave propagates. Picture a slinky: when you compress and release one end, waves travel along the length of the slinky as coils move closer together and then apart, in the same direction the wave travels.

Sound waves are a classic example of longitudinal waves in action, where air molecules compress and rarefy to transmit sound from one place to another. However, in the context of 'the wave' at a football stadium, the motion of fans (up and down) does not align with the defining characteristics of a longitudinal wave (back and forth), and hence 'the wave' cannot be classified as a longitudinal wave.

Wave Motion

Understanding wave motion is crucial to grasp how energy is transported in various systems—from ocean surfing to the transmission of electricity. Wave motion is the transport of a wave's energy and momentum through a medium or space, which can occur via transverse waves, longitudinal waves, or even a combination of both in complex systems.

The motion of a wave encapsulates such diverse phenomena as the seismic waves traveling through Earth's crust during an earthquake, the vibrational energy moving through a guitar string, and the magnetic and electric field oscillations in electromagnetic waves like light. Each of these instances demonstrates the wave's capability to transfer energy in distinctive but fundamentally underlying patterns dictated by the nature of their medium and the forces involved. The 'wave' phenomenon at a football stadium, whereby a consistent pattern emerges as each fan participates, captivates an audience both as a display of unity and an example of synchronized wave motion.