Two waves are described by u₁ = A sin(kx − ωt) and u₂ = A sin(kx + ωt). They have identical amplitude, wavenumber, and angular frequency. What is the fundamental physical difference between them?
AThey travel in opposite directions
BThey have different wave speeds, since one has a minus sign and one a plus sign
Cu₂ does not satisfy the wave equation, because the sign must be negative
DThey oscillate with different angular frequencies at any fixed point
The sign between kx and ωt encodes the direction of propagation. To 'ride' a crest — keep the phase constant — you set kx − ωt = C. As t increases, x must increase at rate ω/k, so u₁ travels in the +x direction. For u₂, setting kx + ωt = C requires x to decrease as t increases: it travels in the −x direction. Both waves have the same speed v = ω/k and both satisfy the wave equation. The minus sign is not just convention — it is the physical statement that the wave moves right.
Question 2 Multiple Choice
For the wave u(x, t) = 5 sin(3x − 6t + π/4), what is the wave speed?
A6 m/s, because ω = 6 is the rate of oscillation
B2 m/s, because v = ω/k = 6/3
C3 m/s, because k = 3 sets the spatial scale
D5 m/s, because the amplitude determines the energy and thus the speed
Wave speed is v = ω/k. Reading off the parameters: k = 3 (coefficient of x), ω = 6 (coefficient of t), so v = 6/3 = 2 m/s. The amplitude A = 5 sets the displacement magnitude but has no effect on propagation speed — speed is determined entirely by ω and k (or equivalently by the medium's properties). Choosing ω = 6 as the speed is the most common error; ω is the angular frequency (radians per second), not meters per second.
Question 3 True / False
For a harmonic wave u = A sin(kx − ωt + φ), a crest (point of maximum displacement) moves in the +x direction at speed ω/k as time advances.
TTrue
FFalse
Answer: True
True. A crest corresponds to a fixed value of the phase: kx − ωt + φ = π/2 (or any value giving sin = 1). Differentiating with respect to t: k(dx/dt) − ω = 0, so dx/dt = ω/k. The crest moves in the +x direction at speed v = ω/k. This is exactly what 'wave propagation' means: patterns of phase (crests, troughs, zero-crossings) translate through space at this speed.
Question 4 True / False
Increasing the amplitude A of a harmonic wave causes the wave to propagate faster through the medium.
TTrue
FFalse
Answer: False
False. Wave speed v = ω/k depends only on the angular frequency and wavenumber — or equivalently, on the medium's physical properties (tension and density for a string, bulk modulus and density for sound, etc.). The amplitude controls how far the medium is displaced from equilibrium but has no effect on how fast the pattern travels. A louder sound wave and a quiet sound wave in the same air travel at the same speed.
Question 5 Short Answer
What does it mean for the phase of a harmonic wave to be constant, and how does this connect to the concept of wave speed?
Think about your answer, then reveal below.
Model answer: The phase of u(x, t) = A sin(kx − ωt + φ) is the argument kx − ωt + φ. Holding the phase constant means setting kx − ωt + φ = C and asking how x must change as t increases to keep C fixed. Differentiating: k(dx/dt) = ω, so dx/dt = ω/k. A surface of constant phase — a crest, trough, or zero-crossing — moves through space at this speed v = ω/k. Wave speed is precisely the speed at which phase patterns propagate.
This framing unifies the mathematical and physical descriptions of a wave. The wave equation's solutions are sinusoids because they are the functions whose spatial and temporal oscillations are locked together in a fixed ratio ω/k = v. Fourier's theorem then extends this: any periodic waveform is a superposition of sinusoids, so understanding how one sinusoid propagates gives you the tools for arbitrary periodic waves.