1. How many "yellow" lightwaves (λ = 580 nm) will fit into a
distance in space equal to the thickness of a piece of paper (0.003
in.)? How far will the same number of microwaves (v = 1010 Hz, i.e., 10
GHz, and v = 3 × 108 m/s) extend? Get solution
2. Get solution
3. Get solution
4. Get solution
5. Get solution
6. How many "yellow" lightwaves (λ = 580 nm) will fit into a distance in space equal to the thickness of a piece of paper (0.003 in.)? How far will the same number of microwaves (v = 1010 Hz, i.e., 10 GHz, and v = 3 × 108 m/s) extend? Get solution
7. The speed of light in vacuum is approximately 3 × 108 m/s. Find the wavelength of red light having a frequency of 5 × 1014 Hz. Compare this with the wavelength of a 60-Hz electromagnetic wave. Get solution
8. It is possible to generate ultrasonic waves in crystals with wavelengths similar to light (5 × 10-5 cm) but with lower frequencies (6 × 108 Hz). Compute the corresponding speed of such a wave. Get solution
9. A youngster in a boat on a lake watches waves that seem to be an endless succession of identical crests passing with a half-second interval between each. If every disturbance takes 1.5 s to sweep straight along the length of her 4.5-m-long boat, what are the frequency, period, and wavelength of the waves? Get solution
10. A vibrating hammer strikes the end of a long metal rod in such a way that a periodic compression wave with a wavelength of 4.3 m travels down the rod's length at a speed of 3.5 km/s. What was the frequency of the vibration? Get solution
11. A violin is submerged in a swimming pool at the wedding of two scuba divers. Given that the speed of compression waves in pure water is 1498 m/s, what is the wavelength of an A-note of 440 Hz played on the instrument? Get solution
12. A wavepulse travels 10 m along the length of a string in 2.0 s. A harmonic disturbance of wavelength 0.50 m is then generated on the string. What is its frequency? Get solution
13. Show that for a periodic wave to = (2п/λ)v. Get solution
14. Make up a table with columns headed by values of θ running from - п/2 to 2 п in intervals of п /4. In each column place the corresponding value of sin θ, beneath those the values of cos θ, beneath those the values of sin (θ - п /4), and similarly with the functions sin (θ - п/2), sin (θ - 3 п/4), and sin (θ + п/2). Plot each of these functions, noting the effect of the phase shift. Docs sin θ lead or lag sin (θ - п/2). In other words, does one of the functions reach a particular magnitude at a smaller value of θ than the other and therefore lead the other (as cos θ leads sin θ)? Get solution
15. Make up a table with columns headed by values of kx running from x = -λ/2 to x = +λ in intervals of x of λ/4—of course, k = 2п/λ. In each column place the corresponding values of cos (kx -п/4) and beneath that the values of cos (kx + 3 п/4). Next plot the functions 15 cos (kx - п/4) and 25 cos (kx + 3 п/4). Get solution
16. Make up a table with columns headed by values of ωt running from t = -τ/2 to t = +τ in intervals of t. of τ/4-of course, ω = 2п/τ. In each column place the corresponding values of sin ωt + п/4) and sin (п/4 - ωt and then plot these two functions. Get solution
17. The profile of a transverse harmonic wave, traveling at 1.2 m/s on a string, is given byy = (0.02 m) sin (157 m-1)xDetermine its amplitude, wavelength, frequency, and period. Get solution
18. Figure P.2.13 represents the profile (t = 0) of a transverse wave on a string traveling in the positive x-direction at a speed of, 20.0 m/s. (a) Determine its wavelength, (b) What is the frequency of the wave? (c) Write down the wavefunction for the disturbance, (d) Notice that as the wave passes any fixed point on the. x-axis the string at that location oscillates in time. Draw a graph of the ψ versus t showing how a point on the rope at x = 0 oscillates.Figure P.2.13 ... Get solution
19. Figure P.2.14 represents the profile (t = 0) of a transverse wave on a string traveling in (he positive z-direction at a speed of 100 cm/s. (a) Determine its wavelength, (b) Notice that as the wave passes any fixed point on the z-axis the string at that location oscillates in time. Draw a graph of the ψ versus t showing how a point on the rope at x = 0 oscillates, (c) What is the frequency of the wave? Get solution
20. A transverse wave on a string travels in the negative y-direction at a speed of 40.0 cm/s. Figure P.2.15 is a graph of ψ versus t showing how a point on the rope at y = 0 oscillates, (a) Determine the wave's period, (b) What is the frequency of the wave? (b) What is the wavelength of the wave? (d) Sketch the profile of the wave ((ψ versus y).Figure P.2.15... Get solution
21. Given the wavefunctions...determine in each case the values of (a) frequency, (b) wavelength, (e) period, (d) amplitude, (e) phase velocity, and (f) direction of motion. Time is in seconds and x is in meters Get solution
22. The wavefunction of a transverse wave on a string isΨ(x, t) = (30.0 cm) cos [(6.28 rad/m)x - (20.0 rad/s)t]Compute the (a) frequency, (b) wavelength, (c) period, (d) amplitude, (e) phase velocity, and (f) direction of motion. Get solution
23. Get solution
24. Show thatψ(x, t) = A sin k(x - vt)is a solution of the differential wave equation. Get solution
25. Show thatψ(x, t) = A cos (kx – ωt)is a solution of the differential wave equation. Get solution
26. Prove thatψ(x, t) = A sin (kx – ωt-п/2)is equivalent toψ(x, t) = A sin (kx – ωt) Get solution
27. Show that if the displacement of the string in Fig. 2.7 is given byy(x, t) = A sin [kx – ωt+ ε]then the hand generating the wave must be moving vertically in simple harmonic motion. Get solution
28. Write the expression for the wavefunction of a harmonic wave of amplitude 103 V/m, period 2.2 × 10-15 s, and speed 3 × 108m/s. The wave is propagating in the negative x-direction and has a value of 103 V/m at t = 0 and x = 0. Get solution
29. Consider the pulse described in terms of its displacement at t = 0 by...where C is a constant. Draw the wave profile. Write an expression for the wave, having a speed v in the negative as x-direction, as a function of time t. If v = 1 m/s, sketch the profile at t = 2 s. Get solution
30. Please determine the magnitude of the wavefunction ψ(z, t) = A cos [k(z + vt) + п] at the point z = 0, when t = τ/2 and when t = 3τ/4. Get solution
31. Does the following function, in which A is a constant,Ψ(y,t)0 = (y - vt)Arepresent a wave? Explain your reasoning. Get solution
32. Use Eq. (2.33) to calculate the speed of the wave whose representation in SI units isΨ(y,t) = A cos п(3 X 106y + 9 × 1014t) Get solution
33. Get solution
34. Beginning with the following theorem: If z = f(x, y) and x = g(t),y = h(t), then...Derive Eq. (2.34). Get solution
35. Using the results of the previous problem, show that for a harmonic wave with a phase Ф(x, t) = k(x - vt) we can determine the speed by setting dФ/dt = 0. Apply the technique to Problem 2.26 to find the speed of that wave. Get solution
36. A Gaussian wave has the form ψ(x, t) = Ae-a(bx+ct)2. Use the fact that ψ(x, t) = f(x ± vt) to determine its speed and then verify your answer using Eq. (2.34). Get solution
37. Create an expression for the profile of a harmonic wave traveling in the z-direction whose magnitude at z = -λ/12 is 0.866, at z = +λ/6 is 1/2, and at z = λ/4 is 0. Get solution
38. Which of the following expressions correspond to traveling waves? For each of those, what is the speed of the wave? The quantities a, b, and c are positive constants.... Get solution
39. Determine which of the following describe traveling waves:...Where appropriate, draw the profile and find the speed and direction of motion. Get solution
40. Given the traveling wave ψ(x, t) = 5.0 exp (-ax2 - br2 - 2√abxt), determine its direction of propagation. Calculate a few values of ψ and make a sketch of the wave at t = 0, taking a = 25 m-2 and b = 9.0 s-2. What is the speed of the wave? Get solution
41. Imagine a sound wave with a frequency of 1.10 kHz propagating with a speed of 330 m/s. Determine the phase difference in radians between any two points on the wave separated by 10.0 cm. Get solution
42. Consider a lightwave having a phase velocity of 3 × 108 m/s and a frequency of 6 × 1014 Hz. What is the shortest distance along the wave between any two points that have a phase difference of 30º? What phase shift occurs at a given point in 10-6 s, and how many waves have passed by in that lime? Get solution
43. Write an expression for the wave shown in Fig. P.2.36. Find its wavelength, velocity, frequency, and period.Figure P.2.36 A harmonic wave.... Get solution
44. Working with exponentials directly, show that the magnitude of ψ = Aeiωt is A. Then rederivc the same result using Euler's formula. Prove that eiαeiβ = ei(α+β) Get solution
45. Show that the imaginary part of a complex number ... is given by ... Get solution
46. Get solution
47. Get solution
48. Beginning with Eq. (2.51), verify that...and that α2 + β2 + γ2 = 1Draw a sketch showing all the pertinent quantities. Get solution
49. Show that Eqs. (2.64) and (2.65), which are plane waves of arbitrary form, satisfy the three-dimensional differential wave equation. Get solution
50. Get solution
51. Get solution
52. De BrOglie's hypothesis states that every particle has associated with it a wavelength given by Planck's constant (h = 6.6 × 10-34 j.s) divided by the particle's momentum. Compare the wavelength of a 6.0-kg stone moving at a speed of 1.0 m/s with that of light. Get solution
53. Write an expression in Cartesian coordinates for a harmonic plane wave of amplitude A and frequency ω propagating in the direction of the vector k, which in tum lies on a line drawn from the origin to the point (4, 2, I). Hint; First determine k and then dot it with ... Get solution
54. Write an expression in Cartesian coordinates for a harmonic plane wave of amplitude A and frequency ω propagating in the positive x-direelion. Get solution
55. Show that ω(... may represent a plane wave where ... is normal to the wavefront. Hint: Let ..., and ... be position vectors drawn to any two points on the plane and show that .... Get solution
56. Get solution
57. Make up a table with columns headed by values of θ running from -п/2 to 2п in intervals of п/4. In each column place the corresponding value of sin θ, and beneath those the values of 2 sin θ. Next add these, column by column, to yield the corresponding values of the function sin θ + 2 sin θ. Plot each of these three functions, noting their relative amplitudes and phases. Get solution
58. Make up a table with columns headed by values of θ running from - п/2 to 2п in intervals of п/4. In each column place the corresponding value of sin θ, and beneath those the values of sin (θ - п/2). Next add these, column by column, to yield the corresponding values of the function sin θ + sin (θ - п/2). Plot each of these three functions, noting their relative amplitudes and phases. Get solution
59. With the last two problems in mind, draw a plot of the three functions (a) sin θ, (b) sin (θ - 3п/4), and (c) sin θ + sin (θ -3п/4). Compare the amplitude of the combined function (c) in this case with that of the previous problem. Get solution
60. Make up a table with columns headed by values of kx running from x = -λ/2 to x = +λ in intervals of x of п/4. In each column place the corresponding values of cos kx and beneath that the values of cos (kx + п). Next plot the three functions cos kx, cos (kx + п), and cos kx + cos (kx + п). Get solution
2. Get solution
3. Get solution
4. Get solution
5. Get solution
6. How many "yellow" lightwaves (λ = 580 nm) will fit into a distance in space equal to the thickness of a piece of paper (0.003 in.)? How far will the same number of microwaves (v = 1010 Hz, i.e., 10 GHz, and v = 3 × 108 m/s) extend? Get solution
7. The speed of light in vacuum is approximately 3 × 108 m/s. Find the wavelength of red light having a frequency of 5 × 1014 Hz. Compare this with the wavelength of a 60-Hz electromagnetic wave. Get solution
8. It is possible to generate ultrasonic waves in crystals with wavelengths similar to light (5 × 10-5 cm) but with lower frequencies (6 × 108 Hz). Compute the corresponding speed of such a wave. Get solution
9. A youngster in a boat on a lake watches waves that seem to be an endless succession of identical crests passing with a half-second interval between each. If every disturbance takes 1.5 s to sweep straight along the length of her 4.5-m-long boat, what are the frequency, period, and wavelength of the waves? Get solution
10. A vibrating hammer strikes the end of a long metal rod in such a way that a periodic compression wave with a wavelength of 4.3 m travels down the rod's length at a speed of 3.5 km/s. What was the frequency of the vibration? Get solution
11. A violin is submerged in a swimming pool at the wedding of two scuba divers. Given that the speed of compression waves in pure water is 1498 m/s, what is the wavelength of an A-note of 440 Hz played on the instrument? Get solution
12. A wavepulse travels 10 m along the length of a string in 2.0 s. A harmonic disturbance of wavelength 0.50 m is then generated on the string. What is its frequency? Get solution
13. Show that for a periodic wave to = (2п/λ)v. Get solution
14. Make up a table with columns headed by values of θ running from - п/2 to 2 п in intervals of п /4. In each column place the corresponding value of sin θ, beneath those the values of cos θ, beneath those the values of sin (θ - п /4), and similarly with the functions sin (θ - п/2), sin (θ - 3 п/4), and sin (θ + п/2). Plot each of these functions, noting the effect of the phase shift. Docs sin θ lead or lag sin (θ - п/2). In other words, does one of the functions reach a particular magnitude at a smaller value of θ than the other and therefore lead the other (as cos θ leads sin θ)? Get solution
15. Make up a table with columns headed by values of kx running from x = -λ/2 to x = +λ in intervals of x of λ/4—of course, k = 2п/λ. In each column place the corresponding values of cos (kx -п/4) and beneath that the values of cos (kx + 3 п/4). Next plot the functions 15 cos (kx - п/4) and 25 cos (kx + 3 п/4). Get solution
16. Make up a table with columns headed by values of ωt running from t = -τ/2 to t = +τ in intervals of t. of τ/4-of course, ω = 2п/τ. In each column place the corresponding values of sin ωt + п/4) and sin (п/4 - ωt and then plot these two functions. Get solution
17. The profile of a transverse harmonic wave, traveling at 1.2 m/s on a string, is given byy = (0.02 m) sin (157 m-1)xDetermine its amplitude, wavelength, frequency, and period. Get solution
18. Figure P.2.13 represents the profile (t = 0) of a transverse wave on a string traveling in the positive x-direction at a speed of, 20.0 m/s. (a) Determine its wavelength, (b) What is the frequency of the wave? (c) Write down the wavefunction for the disturbance, (d) Notice that as the wave passes any fixed point on the. x-axis the string at that location oscillates in time. Draw a graph of the ψ versus t showing how a point on the rope at x = 0 oscillates.Figure P.2.13 ... Get solution
19. Figure P.2.14 represents the profile (t = 0) of a transverse wave on a string traveling in (he positive z-direction at a speed of 100 cm/s. (a) Determine its wavelength, (b) Notice that as the wave passes any fixed point on the z-axis the string at that location oscillates in time. Draw a graph of the ψ versus t showing how a point on the rope at x = 0 oscillates, (c) What is the frequency of the wave? Get solution
20. A transverse wave on a string travels in the negative y-direction at a speed of 40.0 cm/s. Figure P.2.15 is a graph of ψ versus t showing how a point on the rope at y = 0 oscillates, (a) Determine the wave's period, (b) What is the frequency of the wave? (b) What is the wavelength of the wave? (d) Sketch the profile of the wave ((ψ versus y).Figure P.2.15... Get solution
21. Given the wavefunctions...determine in each case the values of (a) frequency, (b) wavelength, (e) period, (d) amplitude, (e) phase velocity, and (f) direction of motion. Time is in seconds and x is in meters Get solution
22. The wavefunction of a transverse wave on a string isΨ(x, t) = (30.0 cm) cos [(6.28 rad/m)x - (20.0 rad/s)t]Compute the (a) frequency, (b) wavelength, (c) period, (d) amplitude, (e) phase velocity, and (f) direction of motion. Get solution
23. Get solution
24. Show thatψ(x, t) = A sin k(x - vt)is a solution of the differential wave equation. Get solution
25. Show thatψ(x, t) = A cos (kx – ωt)is a solution of the differential wave equation. Get solution
26. Prove thatψ(x, t) = A sin (kx – ωt-п/2)is equivalent toψ(x, t) = A sin (kx – ωt) Get solution
27. Show that if the displacement of the string in Fig. 2.7 is given byy(x, t) = A sin [kx – ωt+ ε]then the hand generating the wave must be moving vertically in simple harmonic motion. Get solution
28. Write the expression for the wavefunction of a harmonic wave of amplitude 103 V/m, period 2.2 × 10-15 s, and speed 3 × 108m/s. The wave is propagating in the negative x-direction and has a value of 103 V/m at t = 0 and x = 0. Get solution
29. Consider the pulse described in terms of its displacement at t = 0 by...where C is a constant. Draw the wave profile. Write an expression for the wave, having a speed v in the negative as x-direction, as a function of time t. If v = 1 m/s, sketch the profile at t = 2 s. Get solution
30. Please determine the magnitude of the wavefunction ψ(z, t) = A cos [k(z + vt) + п] at the point z = 0, when t = τ/2 and when t = 3τ/4. Get solution
31. Does the following function, in which A is a constant,Ψ(y,t)0 = (y - vt)Arepresent a wave? Explain your reasoning. Get solution
32. Use Eq. (2.33) to calculate the speed of the wave whose representation in SI units isΨ(y,t) = A cos п(3 X 106y + 9 × 1014t) Get solution
33. Get solution
34. Beginning with the following theorem: If z = f(x, y) and x = g(t),y = h(t), then...Derive Eq. (2.34). Get solution
35. Using the results of the previous problem, show that for a harmonic wave with a phase Ф(x, t) = k(x - vt) we can determine the speed by setting dФ/dt = 0. Apply the technique to Problem 2.26 to find the speed of that wave. Get solution
36. A Gaussian wave has the form ψ(x, t) = Ae-a(bx+ct)2. Use the fact that ψ(x, t) = f(x ± vt) to determine its speed and then verify your answer using Eq. (2.34). Get solution
37. Create an expression for the profile of a harmonic wave traveling in the z-direction whose magnitude at z = -λ/12 is 0.866, at z = +λ/6 is 1/2, and at z = λ/4 is 0. Get solution
38. Which of the following expressions correspond to traveling waves? For each of those, what is the speed of the wave? The quantities a, b, and c are positive constants.... Get solution
39. Determine which of the following describe traveling waves:...Where appropriate, draw the profile and find the speed and direction of motion. Get solution
40. Given the traveling wave ψ(x, t) = 5.0 exp (-ax2 - br2 - 2√abxt), determine its direction of propagation. Calculate a few values of ψ and make a sketch of the wave at t = 0, taking a = 25 m-2 and b = 9.0 s-2. What is the speed of the wave? Get solution
41. Imagine a sound wave with a frequency of 1.10 kHz propagating with a speed of 330 m/s. Determine the phase difference in radians between any two points on the wave separated by 10.0 cm. Get solution
42. Consider a lightwave having a phase velocity of 3 × 108 m/s and a frequency of 6 × 1014 Hz. What is the shortest distance along the wave between any two points that have a phase difference of 30º? What phase shift occurs at a given point in 10-6 s, and how many waves have passed by in that lime? Get solution
43. Write an expression for the wave shown in Fig. P.2.36. Find its wavelength, velocity, frequency, and period.Figure P.2.36 A harmonic wave.... Get solution
44. Working with exponentials directly, show that the magnitude of ψ = Aeiωt is A. Then rederivc the same result using Euler's formula. Prove that eiαeiβ = ei(α+β) Get solution
45. Show that the imaginary part of a complex number ... is given by ... Get solution
46. Get solution
47. Get solution
48. Beginning with Eq. (2.51), verify that...and that α2 + β2 + γ2 = 1Draw a sketch showing all the pertinent quantities. Get solution
49. Show that Eqs. (2.64) and (2.65), which are plane waves of arbitrary form, satisfy the three-dimensional differential wave equation. Get solution
50. Get solution
51. Get solution
52. De BrOglie's hypothesis states that every particle has associated with it a wavelength given by Planck's constant (h = 6.6 × 10-34 j.s) divided by the particle's momentum. Compare the wavelength of a 6.0-kg stone moving at a speed of 1.0 m/s with that of light. Get solution
53. Write an expression in Cartesian coordinates for a harmonic plane wave of amplitude A and frequency ω propagating in the direction of the vector k, which in tum lies on a line drawn from the origin to the point (4, 2, I). Hint; First determine k and then dot it with ... Get solution
54. Write an expression in Cartesian coordinates for a harmonic plane wave of amplitude A and frequency ω propagating in the positive x-direelion. Get solution
55. Show that ω(... may represent a plane wave where ... is normal to the wavefront. Hint: Let ..., and ... be position vectors drawn to any two points on the plane and show that .... Get solution
56. Get solution
57. Make up a table with columns headed by values of θ running from -п/2 to 2п in intervals of п/4. In each column place the corresponding value of sin θ, and beneath those the values of 2 sin θ. Next add these, column by column, to yield the corresponding values of the function sin θ + 2 sin θ. Plot each of these three functions, noting their relative amplitudes and phases. Get solution
58. Make up a table with columns headed by values of θ running from - п/2 to 2п in intervals of п/4. In each column place the corresponding value of sin θ, and beneath those the values of sin (θ - п/2). Next add these, column by column, to yield the corresponding values of the function sin θ + sin (θ - п/2). Plot each of these three functions, noting their relative amplitudes and phases. Get solution
59. With the last two problems in mind, draw a plot of the three functions (a) sin θ, (b) sin (θ - 3п/4), and (c) sin θ + sin (θ -3п/4). Compare the amplitude of the combined function (c) in this case with that of the previous problem. Get solution
60. Make up a table with columns headed by values of kx running from x = -λ/2 to x = +λ in intervals of x of п/4. In each column place the corresponding values of cos kx and beneath that the values of cos (kx + п). Next plot the three functions cos kx, cos (kx + п), and cos kx + cos (kx + п). Get solution
