1. Determine the resultant of the superposition of the parallel
waves E1 = E01 sin (ωt + ε1,) and E2 = E02 sin (ωt + ε2) when ω = 120π,
E0l = 6, E02 = 8, ε1 = 0, and ε2 = π/2. Plot each function and the
resultant. Get solution
2. Considering Section 7.1, suppose we began the analysis to find E = E1 + E2 with two cosine functions E1 = E01 cos (ωt + α1) and E2 = E02 cos (ωt + α2). To make things a little less complicated, let E01 — E02 and α1 =0. Add the two waves algebraically and make use of the familiar trigonometric identity cos θ + cos φ=2 cos ½(θ + φ) cos ½(θ-Ф) in order to show that E = E1 cos (ωt + α), where E0=2 E01 cos α2/2 and α = α2/2. Now show that these same results follow from Eqs. (7.9) and (7.10). Get solution
3. Show that when the two waves of Eq. (7.5) are in-phase, the resulting amplitude squared is a maximum equal to (E01 + E02)2, and when they are out-of-phase it is a minimum equal to (E01 – E02)2. Get solution
4. Show that the optical path length, defined as the sum of the products of the various indices times the thicknesses of media traversed by a beam, that is, ∑1n1x1 is equivalent to the length of the path in vacuum that would take the same time for that beam to negotiate. Get solution
5. Answer the following:(a) How many wavelengths of λ0 = 500 nm light will span a 1-m gap in vacuum?(b) How many waves span the gap when a glass plate 5 cm thick (n = 1.5) is inserted in the path?(c) Determine the OPD between the two situations.(d) Verify that ... corresponds to the difference between the solutions to (a) and (b) above. Get solution
6. Determine the optical path difference for the two waves A and B; both having vacuum wavelengths of 500 nm, depicted in Fig. p.7.6; the glass (n = 1.52) tank is filled with water (n=1.33). If the waves start out in-phase and all the above numbers are exact, find their relative phase difference at the finishing line.... Get solution
7. Using Eqs. (7.9), (7.10), and (7.11), show that the resultant of the two wavesE1 = E01 sin [ωt-k(x + Δx)]AndE2=E01 sin (ωt-kx) is... Get solution
8. Add the two waves of Problem 7.7 directly to find Eq. (7.17). Get solution
9. Use the complex representation to find the resultant E= E1+E2, whereE1= E0 cos (kx + ωt) and E2 =-E0 cos (kx — ωt)Describe the composite wave. Get solution
10. Consider the functions E1 = 3 cos ωt and E2 = 4 sin ωt. First prove that E2 = 4 cos (ωt — π/2). Then, using phasors and referring to Fig. P.7.10, show that E3=E1 + E2 = 5 cos (ωt-φ); determine φ. Discuss the values of E3 wherever either E1=0 or E2 = 0. Does E3 lead or lag E1 ? Explain. Get solution
11. Get solution
12. Get solution
13. Get solution
14. Considering Wiener's experiment (Fig. 7.14) in monochromatic light of wavelength 550 nm, if the film plane is angled at 1.0° to the reflecting surface, determine the number of bright bands per centimeter that will appear on it. Get solution
15. Microwaves of frequency 1010 Hz are beamed directly at a metal reflector. Neglecting the refractive index of air, determine the spacing between successive nodes in the resulting standing-wave pattern. Get solution
16. A standing wave is given by...Determine [wo waves that can be superimposed to generate it. Get solution
17. Get solution
18. Imagine that we strike two tuning forks, one with a frequency of 340 Hz, the other 342 Hz. What will we hear? Get solution
19. Use the phasor method, described in conjunction with Fig. 7.17, to explain how two equal-amplitude waves of sightly different frequencies generate the beat pattern shown in Fig. 7.16 or Fig. P.7.16a. The curve in Fig. P.7.16b is.a sketch of the phase of the resultant measured with respect to one of the constituent waves.Explain its main features. When is it zero and why? When does the phase change abruptly and why?Figure P.7.16(a). ...(b). ... Get solution
20. As we've seen, Eq. (7.33) describes the beat pattern. Let's now derive a different version of that expression assuming that the two overlapping equal-amplitude cosine waves have angular spatial frequencies of kc.+ Δk and kc- Δk, and angular temporal frequencies of ωc,.+ Δω and ωc- Δω, respectively. Here kc and ωc. correspond to the central frequencies. Show that the resultant wave is thenE=2E01 cos (Δkx - ωt) cos (kcx -ωct)Explain how each term relates back to...Prove that the speed of the envelope, which is the wavelength of the envelope divided by the period of the envelope, equals the group velocity, namely, Δω/Δk. Get solution
21. FigureP.7.I8 shows a carrier of frequency ωc being amplitude-modulated by a sine wave of frequency ωm, that is,E = E0(l + a cos ωmt) cos ωctShow that this is equivalent to the superposition of three waves of frequencies ωc, ωc+ ωm, and ωc-ωm. When a number of modulating frequencies are present, we write E as a Fourier series and sum over all values of ωm. The terms ωc+ ωm constitute what is called the upper sideband, and all the ωc- ωm terms form the lower sideband.What bandwidth would you need in order to transmit the complete audible range?Figure... Get solution
22. Given the dispersion relation ω = ak2, compute both the phase and group velocities. Get solution
23. Get solution
24. Get solution
25. Get solution
26. Get solution
27. Using the relation l/vg = dk/dv, prove that... Get solution
28. In the case of light waves, show that... Get solution
29. The speed of propagation of a surface wave in a liquid of depth much greater than λ is given by...where g = acceleration of gravity, λ = wavelength, p - density, Y = surface tension. Compute the group velocity of a pulse in the long wavelength limit (these are called gravity waves), Get solution
30. Show that the group velocity can be written as... Get solution
31. Show that the group velocity can be written as... Get solution
32. With the previous problem in mind prove that... Get solution
33. Get solution
34. Get solution
35. Determine the group velocity of waves when the phase velocity varies inversely with wavelength. Get solution
36. Show that the group velocity can be written as ... Get solution
37. Get solution
38. For a wave propagating in a periodic structure for which ω(k)=2ω0 sin(Kℓ/2), determine both the phase and group velocities. Write the former as a sine function. Get solution
39. An ionized gas or plasma is a dispersive medium for EM-waves. Given that the dispersion equation isω2 = ωp2+ c2k2where ωp is the constant plasma frequency, determine expressions for both the phase and group velocities and show that vv8 = c2. Get solution
40. Using the dispersion equation,...show that the group velocity is given by...for high-frequency electromagnetic waves (e.g., X-rays). Keep in mind that since fj are the weighting factors, ∑jfj = 1. What is the phase velocity? Show that vv8 ≈ r2, Get solution
41. Analytically determine the resultant when the two functions E1 = 2E0 cos ωt and E2 = ½E0 sin 2ωt are superimposed. Draw E1, E2, and E = E1 + E2. Is the resultant periodic; if so, what is its period in terms of ω? Get solution
42. Get solution
43. Show that...where a ≠0,b≠ 0, and a and b are positive integers. Get solution
44. Get solution
45. Get solution
46. Given the function f(x) = A cos (πx/L), determine its Fourier scries. Get solution
47. Get solution
48. Take the function f(θ) = θ2 in the interval 0θ... Get solution
49. Show that the Fourier series representation of the function f(θ) = |sin θ| is... Get solution
50. Change the upper limit of Eq. (7.59) from ∞ to a and evaluate the integral. Leave the answer in terms of the so-called sine integral:...which is a function whose values are commonly tabulated. Get solution
51. Get solution
52. Get solution
53. Get solution
54. Write an expression for the transform A(ω) of the harmonic pulse of Fig. P.7.38. Check that sine u is 50% or greater for values of u roughly less than π/2. With that in mind, show that Δv Δt ≈ 1, where Δv is the bandwidth of the transform at half its maximum amplitude. Verify that Δv Δt ≈ 1 at half the maximum value of the power spectrum as well. The purpose here is to get some sense of the kind of approximations used in the discussionFigure P.7.38... Get solution
55. Derive an expression for the coherence length (in vacuum) of a wavetrain that has a frequency bandwidth Δv; express your answer in terms of the line width Δλ0 and the mean wavelength λ0 of the train. Get solution
56. Get solution
57. Consider a photon in the visible region of the spectrum emitted during an atomic transition of about 10-8 s. How long is the wave packet? Keeping in mind the results of the previous problem (if you've done it), estimate the line width of the packet (λ0 = 500 nm). What can you say about its monochromaticity, as indicated by the frequency stability? Get solution
58. The first* experiment directly measuring the bandwidth of a laser (in this ease a continuous-wave Pb088Sn012 Te diode laser) was carried out in 1969. The laser, operating at λ0 = 10 600 nm, was heterodyned with a C02 laser, and bandwidths as narrow as 54 kHz were observed. Compute the corresponding frequency stability and coherence length for the lead-lin-telluridc laser. Get solution
59. A magnetic-field technique for stabilizing a He-Ne laser to 2 parts in 1010 has been patented. At 632.8 nm, what would be the coherence length of a laser with such a frequency stability? Get solution
60. Imagine that we chop a continuous laserbeam (assumed to be monochromatic at λ0 = 632.8 nm) into 0.1 -ns pulses, using some son of shutter. Compute the resultant line width Δλ, bandwidth, and coherence length. Find the bandwidth and line width that would result if we could chop at 1015 Hz. Get solution
61. Suppose that we have a filter with a pass band of 1.0 Ä centered at 600 nm, and we illuminate it with sunlight. Compute the coherence length of the emerging wave. Get solution
62. A filter passes light with a mean wavelength of ... If the emerging wavetrains are roughly 20λ0 long, what is the frequency bandwidth of the exiting light? Get solution
63. Suppose we spread white light out into a fan of wavelengths by means of a diffraction grating and then pass a small select region of that spectrum out through a slit. Because of the width of the slit, a band of wavelengths 1.2 nm wide centered on 500 nm emerges. Determine the frequency bandwidth and the coherence length of this light. Get solution
2. Considering Section 7.1, suppose we began the analysis to find E = E1 + E2 with two cosine functions E1 = E01 cos (ωt + α1) and E2 = E02 cos (ωt + α2). To make things a little less complicated, let E01 — E02 and α1 =0. Add the two waves algebraically and make use of the familiar trigonometric identity cos θ + cos φ=2 cos ½(θ + φ) cos ½(θ-Ф) in order to show that E = E1 cos (ωt + α), where E0=2 E01 cos α2/2 and α = α2/2. Now show that these same results follow from Eqs. (7.9) and (7.10). Get solution
3. Show that when the two waves of Eq. (7.5) are in-phase, the resulting amplitude squared is a maximum equal to (E01 + E02)2, and when they are out-of-phase it is a minimum equal to (E01 – E02)2. Get solution
4. Show that the optical path length, defined as the sum of the products of the various indices times the thicknesses of media traversed by a beam, that is, ∑1n1x1 is equivalent to the length of the path in vacuum that would take the same time for that beam to negotiate. Get solution
5. Answer the following:(a) How many wavelengths of λ0 = 500 nm light will span a 1-m gap in vacuum?(b) How many waves span the gap when a glass plate 5 cm thick (n = 1.5) is inserted in the path?(c) Determine the OPD between the two situations.(d) Verify that ... corresponds to the difference between the solutions to (a) and (b) above. Get solution
6. Determine the optical path difference for the two waves A and B; both having vacuum wavelengths of 500 nm, depicted in Fig. p.7.6; the glass (n = 1.52) tank is filled with water (n=1.33). If the waves start out in-phase and all the above numbers are exact, find their relative phase difference at the finishing line.... Get solution
7. Using Eqs. (7.9), (7.10), and (7.11), show that the resultant of the two wavesE1 = E01 sin [ωt-k(x + Δx)]AndE2=E01 sin (ωt-kx) is... Get solution
8. Add the two waves of Problem 7.7 directly to find Eq. (7.17). Get solution
9. Use the complex representation to find the resultant E= E1+E2, whereE1= E0 cos (kx + ωt) and E2 =-E0 cos (kx — ωt)Describe the composite wave. Get solution
10. Consider the functions E1 = 3 cos ωt and E2 = 4 sin ωt. First prove that E2 = 4 cos (ωt — π/2). Then, using phasors and referring to Fig. P.7.10, show that E3=E1 + E2 = 5 cos (ωt-φ); determine φ. Discuss the values of E3 wherever either E1=0 or E2 = 0. Does E3 lead or lag E1 ? Explain. Get solution
11. Get solution
12. Get solution
13. Get solution
14. Considering Wiener's experiment (Fig. 7.14) in monochromatic light of wavelength 550 nm, if the film plane is angled at 1.0° to the reflecting surface, determine the number of bright bands per centimeter that will appear on it. Get solution
15. Microwaves of frequency 1010 Hz are beamed directly at a metal reflector. Neglecting the refractive index of air, determine the spacing between successive nodes in the resulting standing-wave pattern. Get solution
16. A standing wave is given by...Determine [wo waves that can be superimposed to generate it. Get solution
17. Get solution
18. Imagine that we strike two tuning forks, one with a frequency of 340 Hz, the other 342 Hz. What will we hear? Get solution
19. Use the phasor method, described in conjunction with Fig. 7.17, to explain how two equal-amplitude waves of sightly different frequencies generate the beat pattern shown in Fig. 7.16 or Fig. P.7.16a. The curve in Fig. P.7.16b is.a sketch of the phase of the resultant measured with respect to one of the constituent waves.Explain its main features. When is it zero and why? When does the phase change abruptly and why?Figure P.7.16(a). ...(b). ... Get solution
20. As we've seen, Eq. (7.33) describes the beat pattern. Let's now derive a different version of that expression assuming that the two overlapping equal-amplitude cosine waves have angular spatial frequencies of kc.+ Δk and kc- Δk, and angular temporal frequencies of ωc,.+ Δω and ωc- Δω, respectively. Here kc and ωc. correspond to the central frequencies. Show that the resultant wave is thenE=2E01 cos (Δkx - ωt) cos (kcx -ωct)Explain how each term relates back to...Prove that the speed of the envelope, which is the wavelength of the envelope divided by the period of the envelope, equals the group velocity, namely, Δω/Δk. Get solution
21. FigureP.7.I8 shows a carrier of frequency ωc being amplitude-modulated by a sine wave of frequency ωm, that is,E = E0(l + a cos ωmt) cos ωctShow that this is equivalent to the superposition of three waves of frequencies ωc, ωc+ ωm, and ωc-ωm. When a number of modulating frequencies are present, we write E as a Fourier series and sum over all values of ωm. The terms ωc+ ωm constitute what is called the upper sideband, and all the ωc- ωm terms form the lower sideband.What bandwidth would you need in order to transmit the complete audible range?Figure... Get solution
22. Given the dispersion relation ω = ak2, compute both the phase and group velocities. Get solution
23. Get solution
24. Get solution
25. Get solution
26. Get solution
27. Using the relation l/vg = dk/dv, prove that... Get solution
28. In the case of light waves, show that... Get solution
29. The speed of propagation of a surface wave in a liquid of depth much greater than λ is given by...where g = acceleration of gravity, λ = wavelength, p - density, Y = surface tension. Compute the group velocity of a pulse in the long wavelength limit (these are called gravity waves), Get solution
30. Show that the group velocity can be written as... Get solution
31. Show that the group velocity can be written as... Get solution
32. With the previous problem in mind prove that... Get solution
33. Get solution
34. Get solution
35. Determine the group velocity of waves when the phase velocity varies inversely with wavelength. Get solution
36. Show that the group velocity can be written as ... Get solution
37. Get solution
38. For a wave propagating in a periodic structure for which ω(k)=2ω0 sin(Kℓ/2), determine both the phase and group velocities. Write the former as a sine function. Get solution
39. An ionized gas or plasma is a dispersive medium for EM-waves. Given that the dispersion equation isω2 = ωp2+ c2k2where ωp is the constant plasma frequency, determine expressions for both the phase and group velocities and show that vv8 = c2. Get solution
40. Using the dispersion equation,...show that the group velocity is given by...for high-frequency electromagnetic waves (e.g., X-rays). Keep in mind that since fj are the weighting factors, ∑jfj = 1. What is the phase velocity? Show that vv8 ≈ r2, Get solution
41. Analytically determine the resultant when the two functions E1 = 2E0 cos ωt and E2 = ½E0 sin 2ωt are superimposed. Draw E1, E2, and E = E1 + E2. Is the resultant periodic; if so, what is its period in terms of ω? Get solution
42. Get solution
43. Show that...where a ≠0,b≠ 0, and a and b are positive integers. Get solution
44. Get solution
45. Get solution
46. Given the function f(x) = A cos (πx/L), determine its Fourier scries. Get solution
47. Get solution
48. Take the function f(θ) = θ2 in the interval 0θ... Get solution
49. Show that the Fourier series representation of the function f(θ) = |sin θ| is... Get solution
50. Change the upper limit of Eq. (7.59) from ∞ to a and evaluate the integral. Leave the answer in terms of the so-called sine integral:...which is a function whose values are commonly tabulated. Get solution
51. Get solution
52. Get solution
53. Get solution
54. Write an expression for the transform A(ω) of the harmonic pulse of Fig. P.7.38. Check that sine u is 50% or greater for values of u roughly less than π/2. With that in mind, show that Δv Δt ≈ 1, where Δv is the bandwidth of the transform at half its maximum amplitude. Verify that Δv Δt ≈ 1 at half the maximum value of the power spectrum as well. The purpose here is to get some sense of the kind of approximations used in the discussionFigure P.7.38... Get solution
55. Derive an expression for the coherence length (in vacuum) of a wavetrain that has a frequency bandwidth Δv; express your answer in terms of the line width Δλ0 and the mean wavelength λ0 of the train. Get solution
56. Get solution
57. Consider a photon in the visible region of the spectrum emitted during an atomic transition of about 10-8 s. How long is the wave packet? Keeping in mind the results of the previous problem (if you've done it), estimate the line width of the packet (λ0 = 500 nm). What can you say about its monochromaticity, as indicated by the frequency stability? Get solution
58. The first* experiment directly measuring the bandwidth of a laser (in this ease a continuous-wave Pb088Sn012 Te diode laser) was carried out in 1969. The laser, operating at λ0 = 10 600 nm, was heterodyned with a C02 laser, and bandwidths as narrow as 54 kHz were observed. Compute the corresponding frequency stability and coherence length for the lead-lin-telluridc laser. Get solution
59. A magnetic-field technique for stabilizing a He-Ne laser to 2 parts in 1010 has been patented. At 632.8 nm, what would be the coherence length of a laser with such a frequency stability? Get solution
60. Imagine that we chop a continuous laserbeam (assumed to be monochromatic at λ0 = 632.8 nm) into 0.1 -ns pulses, using some son of shutter. Compute the resultant line width Δλ, bandwidth, and coherence length. Find the bandwidth and line width that would result if we could chop at 1015 Hz. Get solution
61. Suppose that we have a filter with a pass band of 1.0 Ä centered at 600 nm, and we illuminate it with sunlight. Compute the coherence length of the emerging wave. Get solution
62. A filter passes light with a mean wavelength of ... If the emerging wavetrains are roughly 20λ0 long, what is the frequency bandwidth of the exiting light? Get solution
63. Suppose we spread white light out into a fan of wavelengths by means of a diffraction grating and then pass a small select region of that spectrum out through a slit. Because of the width of the slit, a band of wavelengths 1.2 nm wide centered on 500 nm emerges. Determine the frequency bandwidth and the coherence length of this light. Get solution
