PHYS 355 - Homework # 1
(due in class on Sept 04)
(15 points)
1) Starting from our original expression for a harmonic wave:
(
)(
[]
tvxksinAt,x −
)
=
ψ
derive the equivalent expressions:
a)
()
⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛−=
τλ
π
ψ
tx
2sinAt,x
b)
()
[]
txksinAt,x
ω
ψ
−=
d)
()
⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛−= t
v
x
2sinAt,x
νπ
ψ
(15 points)
2) Consider a monochromatic green lightwave with wavelength λ = 550 nm (1 nm =
1×10-9 m). Assuming the speed of light in vacuum (which is close to the one in air) to be
c = 3×108 m/s, then:
a) Calculate the period τ and the optical frequency ν for green light? Make sure to include
the proper units.
b) Write a sinusoidal harmonic wave with unit amplitude propagating towards the
negative x-axis. Consider that for x = 0 and t = 0, the (initial) phase is π/2. Make sure to
include all the units.
c) Plot the wave profile against x for following times: t = 0, 0.1 τ, and 0.2 τ. Make sure to
properly label the x-axis and display the units used. Assume an arbitrary unit for the y-
axis.
(5 points)
3) Consider the 3D wave equation:
()()
(
)
(
)
2
2
22
2
2
2
2
2
t
t,z,y,x
v
1
z
t,z,y,x
y
t,z,y,x
x
t,z,y,x
∂
∂
=
∂
∂
+
∂
∂
+
∂
∂
ψ
ψ
ψ
ψ
.
If 1
ψ
and 2
ψ
are both solutions of the wave equation then show that a linear combination
21
ψ
ψ
ba + is also a solution (where a and b are constants independent of x,y,z, and t).
1/2
