St. Vincent College
PH 241: Optics
Exam 1
9/11/2009
1. Given the wave shape f(x) = Ae−αx,
a) Write a wave function corresponding to a wave of this shape traveling to the right along the x-axis
with speed v.
b) Show that your wave function from (a) satisfies the one-dimensional wave equation.
2. An ultrasonic wave travels in a certain crystal. The wavelength is λ= 5 ×10−5cm with a frequency
of ν= 6 ×108Hz.
a) Work out the velocity of this wave.
b) Assuming that this is a harmonic wave, write a wave function that corresponds to it. You need not
make any assumptions about the exact phase of the wave at x= 0 and t= 0, thus you may choose to
use either sin or cos as the basis of your function. Take the amplitude of the wave to be 3×10−6cm.
3. A transverse wave on a string has the form
ψ(x, t) = 30 cos(6.28 x−20 t)
where the amplitude has units of cm.
a) What are the units of the coefficients of xand t?
b) Compute the frequency, wavelength, and period of this wave.
c) Determine the direction of motion of the wave.
4.
a) Write the complex value ˜z= 3 + 4iin exponential form, ˜z=Aeiθ .
b) Show that, if ˜z=a+bi, then |˜z|= [˜z∗˜z]1/2gives the graphically expected value |˜z|= (a2+b2)1/2.
5. Given a harmonic wave function ψ, where the function φ(x, t) = kx −ωt +εdescribes the phase of
the wave, derive the phase velocity
vp=∂x
∂t φ
=ω
k
where vpis defined as the velocity of a point of constant phase.
