# Problems

# Section 1.4: Traveling Waves

1. *1.1 A harmonic wave traveling along a string is generated by an oscillator that completes 180 vibrations per minute. If it is observed that a given crest, or maximum, travels 300 cm in 10 s, what is the wavelength?

2. 1.2 For the pressure wave described in Example 1-1, plot the following:

   1. p(x,t) versus x at t=0

   2. p(x,t) versus t at x=0

   Be sure to use appropriate scales for x and t so that each of your plots covers at least two cycles.

3. *1.3 A 2 kHz sound wave traveling in the x direction in air was observed to have a differential pressure p(x,t)=10 N/m 2 at x=0 and t=50 μ s. If the reference phase of p(x,t) is 36°, find a complete expression for p(x,t). The velocity of sound in air is 330 m/s.

4. 1.4 A wave traveling along a string is given by

   y(x,t)=2 sin(4 π t+10 π x)(cm),

   where x is the distance along the string in meters and y is the vertical displacement. Determine: (a) the direction of wave travel, (b) the reference phase ϕ 0, (c) the frequency, (d) the wavelength, and (e) the phase velocity.

5. 1.5 Two waves, y 1(t) and y 2(t), have identical amplitudes and oscillate at the same frequency, but y 2(t) leads y 1(t) by a phase angle of 60°. If

   y 1(t)=4 cos(2 π×10 3 t),

   write the expression appropriate for y 2(t) and plot both functions over the time span from 0 to 2 ms.

6. *1.6 The height of an ocean wave is described by the function

   y(x,t)=1.5 sin(0.5 t−0.6 x)(m).

   Determine the phase velocity and wavelength, and then sketch y(x,t) at t=2 s over the range from x=0 to x=2 λ.

7. 1.7 A wave traveling along a string in the +x direction is given by

   y 1(x,t)=A cos(ω t−β x),

   where x=0 is the end of the string, which is tied rigidly to a wall, as shown in Fig. P1.7.

   # Figure P1-7 Wave on a string tied to a wall at x=0 (Problem 1.7).

   When wave y 1(x,t) arrives at the wall, a reflected wave y 2(x,t) is generated. Hence, at any location on the string, the vertical displacement y s is the sum of the incident and reflected waves:

   y s(x,t)=y 1(x,t)+y 2(x,t).

   1. Write an expression for y 2(x,t), keeping in mind its direction of travel and the fact that the end of the string cannot move.

   2. Generate plots of y 1(x,t), y 2(x,t), and y s(x,t) versus x over the range −2 λ≤x≤0 at ω t=π/4 and at ω t=π/2.

8. 1.8 Two waves on a string are given by the functions:

   y 1(x,t)=4 cos(20 t−30 x)(cm)y 2(x,t)=−4 cos(20 t+30 x)(cm),

   where x is in centimeters. The waves are said to interfere constructively when their superposition |y s|=|y 1+y 2| is a maximum, and they interfere destructively when |y s| is a minimum.

   1. *a. What are the directions of propagation of waves y 1(x,t) and y 2(x,t)?

   2. b.At t=(π/50) s, what location x do the two waves interfere constructively, and what is the corresponding value of |y s|?

   3. c.At t=(π/50) s, what location x do the two waves interfere destructively, and what is the corresponding value of |y s|?

9. 1.9 Find expressions for y(x,t) for a sinusoidal wave traveling along a string in the negative x direction, given that y m a x=40 cm, λ=30 cm, f=10 Hz, and

   1. y(x,0)=0 at x=0

   2. y(x,0)=0 at x=3.75 cm

10. *1.10 An oscillator that generates a sinusoidal wave on a string completes 20 vibrations in 50 s. The wave peak is observed to travel a distance of 2.8 m along the string in 5 s. What is the wavelength?

11. 1.11 The vertical displacement of a string is given by the harmonic function:

    y(x,t)=2 cos(16 π t−20 π x)(m),

    where x is the horizontal distance along the string in meters. Suppose a tiny particle were attached to the string at x=5 cm. Obtain an expression for the vertical velocity of the particle as a function of time.

12. *1.12 Given two waves characterized by

    y 1(t)=3 cos ω t,y 2(t)=3 sin(ω t+60°),

    does y 2(t) lead or lag y 1(t) and by what phase angle?

13. 1.13 The voltage of an electromagnetic wave traveling on a transmission line is given by

    υ(z,t)=5 e−α z sin(4 π×10 9 t−20 π z)(V),

    where z is the distance in meters from the generator.

    1. Find the frequency, wavelength, and phase velocity of the wave.

    2. At z=2 m, the amplitude of the wave was measured to be 2 V. Find α.

14. *1.14 A certain electromagnetic wave traveling in seawater was observed to have an amplitude of 98.02 (V/m) at a depth of 10 m, and an amplitude of 81.87 (V/m) at a depth of 100 m. What is the attenuation constant of seawater?

15. 1.15 A laser beam traveling through fog was observed to have an intensity of 1 (μ W/m 2) at a distance of 2 m from the laser gun and an intensity of 0.2 (μ W/m 2) at a distance of 3 m. Given that the intensity of an electromagnetic wave is proportional to the square of its electric-field amplitude, find the attenuation constant α of fog.

16. 1.16 In Module 1.2, set f=1 Hz, λ=2 cm, and α=0.5 Np/cm. Estimate the amplitude y(x) of the blue wave at its first maximum (at x=0) and at its second maximum (at x=2 cm). Compute the ratio y(2 cm)/y(0). Does the result match your expectations given the values of f, λ, and α?

# Section 1.5: Complex Numbers

1. 1.17 Complex numbers z 1 and z 2 are given as

   z 1=3−j 2,z 2=−4+j 3.

   1. a.Express z 1 and z 2 in polar form.

   2. b.Find |z 1| by first applying Eq. (1.41) and then by applying Eq. (1.43).

   3. *c.Determine the product z 1 z 2 in polar form.

   4. d.Determine the ratio z 1/z 2 in polar form.

   5. e.Determine z 1 3 in polar form.

2. 1.18 Evaluate each of the following complex numbers and express the result in rectangular form:

   1. a.z 1=8 e j π/3

   2. *b.z 2=3 e j 3 π/4

   3. c.z 3=2 e−j π/2

   4. d.z 4=j 3

   5. e.z 5=j−4

   6. f.z 6=(1−j)3

   7. g.z 7=(1−j)1/2

3. 1.19 Complex numbers z 1 and z 2 are given by

   z 1=−3+j 2,z 2=1−j 2.

   Determine (a) z 1 z 2, (b) z 1/z 2∗, (c) z 1 2, and (d) z 1 z 1∗ all in polar form.

4. 1.20 If z=−2+j 4, determine the following quantities in polar form:

   1. a.1/z

   2. b.z 3

   3. *c.|z|2

   4. d.I m{z}

   5. e.I m{z∗}

5. 1.21 Find complex numbers t=z 1+z 2 and s=z 1−z 2, both in polar form, for each of the following pairs:

   1. a.z 1=2+j 3 and z 2=1−j 2

   2. b.z 1=3 and z 2=−j 3

   3. c.z 1=3 30° and z 2=3−30°

   4. *d.z 1=3 30° and z 2=3−150°

6. 1.22 Complex numbers z 1 and z 2 are given by

   z 1=5−60°,z 2=4 45°.

   1. Determine the product z 1 z 2 in polar form.

   2. Determine the product z 1 z 2∗ in polar form.

   3. Determine the ratio z 1/z 2 in polar form.

   4. Determine the ratio z 1∗/z 2∗ in polar form.

   5. Determine z 1 in polar form.

7. *1.23 If z=3−j 5, find the value of ln(z).

8. 1.24 If z=3 e j π/6, find the value of e z.

9. 1.25 If z=3−j 4, find the value of e z.

# Section 1.6: Phasors

1. *1.26 A voltage source given by

   υ s(t)=25 cos(2 π×10 3 t−30°)(V)

   is connected to a series RC load as shown in Fig. 1-20. If R=1 MΩ and C=200 pF, obtain an expression for the voltage across the capacitor, υ c(t).

2. 1.27 Find the instantaneous time sinusoidal functions corresponding to the following phasors:

   1. a.V˜=−5 e j π/3 (V)

   2. b.V˜=j 6 e−j π/4 (V)

   3. c.I˜=(6+j 8) (A)

   4. *d.I˜=−3+j 2 (A)

   5. e.I˜=j (A)

   6. f.I˜=2 e j π/6 (A)

3. 1.28 Find the phasors of the following time functions:

   1. a.υ(t)=9 cos(ω t−π/3) (V)

   2. b.υ(t)=12 sin(ω t+π/4) (V)

   3. c.i(x,t)=5 e−3 x sin(ω t+π/6) (A)

   4. *d.i(t)=−2 cos(ω t+3 π/4) (A)

   5. e.i(t)=4 sin(ω t+π/3)+3 cos(ω t−π/6) (A)

4. 1.29 A series RLC circuit is connected to a generator with a voltage υ s(t)=V 0 cos(ω t+π/3) (V).

   1. Write the voltage loop equation in terms of the current i(t), R, L, C, and υ s(t).

   2. Obtain the corresponding phasor-domain equation.

   3. Solve the equation to obtain an expression for the phasor current I˜.

5. 1.30 The voltage source of the circuit shown in Fig. P1.30 is given by

   υ s(t)=25 cos(4×10 4 t−45°)(V).

   # Figure P1.30 Circuit for Problem 1.30.

   Figure P1.30 Full Alternative Text

   Obtain an expression for the current flowing through the inductor, i L(t).

6. 1.31 Repeat Problem 1.30, but this time determine current i R 2(t).

Test Your Understanding of Chapter 1:
http://em8e.eecs.umich.edu/all_qs/chapter1.phtml