1051/1017-455-20073 Homework #2 Due 27 March 2008 (Th)
Read (for meaning, i.e., don’t “skim”) P3§5 Superposition of Waves
1. For the following two harmonic waves:
(a) Show on a phasor diagram:
f1[t] = 2 cos [!0t]
f2[t] = 7 cos h!0t
4i
(b) Find the mathematical expression for the superposition f1[t] + f2[t]in the form of a cosine.
2. Two plane waves of the same frequency that vibrate along the zdirection are:
f1[x; t] = A1cos 2x
X1
1t+;A1= 40 mm,X1= 30 mm,1= 20 Hz
f2[y; t] = A2cos 2y
Y2
2t+;A2= 20 mm,Y2= 40 mm,2= 20 Hz
Evaluate the resultant waveform f1[x; t] + f2[y; t]at [x; y] = [50 mm;20 mm]
3. Consider the superposition of two sinusoidal traveling waves:
f1[z; t] = A1cos [k1z!1t];
A1= 101 mm; 1= 100 Hz;v1= 250 m
s
f2[z; t] = A2cos [k2z!2t];
A2= 99 mm; 2= 150 Hz;v2= 500 m
s
(a) Find an expression for the resulting wave in terms of the average wave, the modulation wave, plus
any remaining amplitude.
(b) Calculate the wavelengths of the average and modulation waves.
(c) Find the velocities of the average and modulation waves.
(d) Does this system exhibit normal or anomalous dispersion?
4. The phase velocity of waves in some medium is proportional to !+1
2. Find an expression for the
modulation velocity and determine whether the waves exhibit normal or anomalous dispersion.
5. Plot and write the equation of the superposition of the following harmonic waves:
E1= sin h
18 !ti
E2= 3 cos 5
9!t
E3= 2 sin h
6!ti
where the period of each is 2 s:
6. A laser emits a monochromatic beam of wavelength 0, which is re‡ected normally from a plane mirror
that recedes from the light source at velocity v.
(a) Determine the beat frequency between the incident and re‡ected light.
(b) Determine the beat frequency between the incident and re‡ected light if the light is incident on
the plane mirror at angle :
1
