2. A general expression for the wave function of transverse periodic waves
moving to the right is,
y(x, t) = A cos(kx − ωt + φ),
where φ is a phase angle satisfying 0 ≤ φ ≤ 2π.
a) For the case φ = π/2, use the axes given to sketch one full wavelength for the wave function at t = T /2 (T is the period).
b) Calculate the transverse particle velocity. With φ = π/2, what is
this velocity at x = 0, t = T /2?
c) Suppose you don’t know φ. If you are told that a particle at x = λ
is moving with transverse velocity Aω at t = 0, what is the value
of φ?
y
Solution:
A
a)
λ
x
-A
b)
vy =
∂y
= A(−ω)(−) sin(kx − ωt + φ) = Aω sin(kx − ωt + φ).
∂t
vy (0, T /2) = Aω sin[0 − (2π/T )T /2 + π/2] = Aω sin(−π/2) = −Aω.
c)
vy (λ, 0) = Aω sin[(2π/λ)λ − 0 + φ] = Aω
⇒ sin(2π+φ) = 1 ⇒ 2π+φ = (2m+1/2)π and −(2m+3/2)π, m = 0, 1, 2, . . .
The only allowed value that satisfies 0 ≤ φ ≤ 2π is φ = π/2.