A transverse wave travels through a medium at 500 m/s. A particle in the medium is at its equilibrium position (zero displacement) as the wave passes. What can you conclude about the particle's speed at this moment?
AThe particle is also moving at approximately 500 m/s, matching the wave speed
BThe particle's speed is at its maximum value of Aω, since it is at the equilibrium position
CThe particle's speed is zero, since the wave has just reached it
DThe particle's speed equals the wave speed divided by the amplitude
Particle velocity is v = -Aω cos(kx - ωt). At zero displacement, sin(kx - ωt) = 0, which means cos(kx - ωt) = ±1, so particle speed is at its maximum |Aω|. Wave speed (ω/k) is an entirely separate quantity describing how fast the pattern moves through the medium — determined by medium properties, not by the particle's oscillation phase. The two speeds have different causes and different formulas.
Question 2 Multiple Choice
A wave passes through a medium and a particle is observed to be at its maximum displacement from equilibrium (a crest). What is the particle's velocity at this instant?
AMaximum, since the wave is pushing it hardest at the crest
BEqual to the wave speed — the particle rides the crest
CZero, since the particle is momentarily at its turning point
DAω, the maximum particle speed
At maximum displacement (crest or trough), the particle is at its turning point — like a ball thrown upward at its peak, momentarily at rest before reversing. From v = -Aω cos(kx - ωt): when displacement is maximum, sin(kx - ωt) = ±1, which forces cos(kx - ωt) = 0, giving zero velocity. The 90° phase difference between displacement and velocity is the key relationship: maximum displacement coincides with zero speed, and zero displacement coincides with maximum speed.
Question 3 True / False
A particle in a medium that carries a wave should be moving whenever the wave is moving.
TTrue
FFalse
Answer: False
At maximum displacement (a crest or trough), a particle is momentarily at rest even as the wave pattern continues traveling. The particle's velocity is governed by its phase in the oscillation cycle, not by whether the wave is propagating. This is the fundamental distinction: wave propagation does not require particles to be in constant motion — at any given instant, particles at the crests and troughs are stationary while those at equilibrium positions are moving fastest.
Question 4 True / False
A wave can have rapidly oscillating particles even if the wave itself propagates slowly through the medium.
TTrue
FFalse
Answer: True
Maximum particle velocity = Aω, which depends on amplitude and frequency — not on wave speed. Wave speed v_wave = ω/k depends on the medium's physical properties (tension, density, etc.). These are independent quantities. A slow-propagating wave with large amplitude or high frequency will have fast-moving particles. Conversely, seismic waves can travel thousands of meters per second while particles oscillate with tiny velocities. The cork bobs up and down; the pattern travels to shore.
Question 5 Short Answer
Explain why particle velocity and wave speed are fundamentally different quantities, and what determines each.
Think about your answer, then reveal below.
Model answer: Particle velocity (v_particle = ∂u/∂t) describes how fast an individual piece of the medium oscillates about its equilibrium position; its maximum is Aω (amplitude × angular frequency). Wave speed (v_wave = ω/k) describes how fast the wave pattern — the crests and troughs — moves through the medium; it is determined by the medium's physical properties (e.g., tension and linear density for a string). A cork on water illustrates the difference: the cork bobs up and down (particle velocity) while the wave pattern travels toward shore (wave speed). The cork does not travel to shore; only the pattern does.
The distinction matters for energy calculations: wave intensity is proportional to (particle velocity)² × (density × wave speed), so both quantities appear together. Getting them confused leads to errors in predicting how much energy a wave carries. It also resolves the apparent paradox of how energy can flow through a medium without any net transport of matter — the pattern moves, but the matter stays near its equilibrium position.