Correct Answer to Question 5

For this motion, the displacement y of the string is given by

y = (10 cm) cos[(4 m−1)x + (2π s−1)t].

The amplitude of the wave is the maximum displacement. The cosine function can at most be 1; so its coefficient (10 cm) is the amplitude. The statement "a" is not true.

The quantity y measures the transverse displacement of a particle of the string at the position x. The wave then propagates from one particle to the next along either the +x or −x direction. So statements "d" and "e" are not true.

To find the direction of propagation (+x or −x), we can use this argument: consider the particle at x = 0 at the instant t = 0. The displacement y of this particle at this moment is

y = 10 cm × cos(0) = 10 cm

This is a wave crest since the displacement is maximum. Now we wait a small time interval Δt and ask: to what position Δx has the crest moved? For that crest,

(4 m−1) Δx + (2π s−1) Δt = 0.

Hence,

Δx = −[(2π s−1)/(4 m−1)]Δt = −(π/2 m/s) Δt.

The important result is the sign: as time goes on, that crest moves in the NEGATIVE x direction. All crests and troughs do the same. So statement "b" is false and "c" is true.

We just found that the magnitude of v is π/2 m/s so the statement "f" is true. Alternatively, we could recognize that ω = (2π s−1) and k = (4 m−1) and find the wave speed from v = ω/k.

Formula Sheet
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