1. c. The figure shows the two waves $t=6\text{s}$ and their superposition. The superposition is the point-by-point addition of the displacements of the two individual waves.

2. a. The allowed standing-wave frequencies are $f_m=m(v/2L)$, so the mode number of a standing wave of frequency f is $m=2Lf/v$. Quadrupling $T_{\text{s}}$ increases the wave speed v by a factor of 2. The initial mode number was 2, so the new mode number is 1.

3. b. 300 Hz and 400 Hz are allowed standing waves, but they are not $f_1\text{and }f{}_2$ because $400\text{Hz}\ne 2\times 300\text{Hz}$. Because there’s a 100 Hz difference between them, these must be $f_3=3\times 100\text{Hz}$ and $f_4=4\times 100\text{Hz}$, with a fundamental frequency $f_1=100\text{Hz}$. Thus the second harmonic is $f_2=2\times 100\text{Hz}=200\text{Hz}$.

4. c. Shifting the top wave 0.5 m to the left aligns crest with crest and trough with trough.

5. a. $r_1=0.5\text{λ}$ and $r_2=2.5\text{λ}$, so $\text{Δ}r=2.0\text{λ}$. This is the condition for maximum constructive interference.

6. Maximum constructive. The path-length difference is $\text{Δ}r=1.0\text{m}=\text{λ}$. For identical sources, interference is constructive when $\text{Δ}r$ is an integer multiple of $\text{λ}$.

7. f. The beat frequency is the difference between the two frequencies.

1.

3. 4 s

5. 40 Hz

7.

1. 8

2. 200 Hz

9.

• a. 12 Hz, 24 m/s

• b.

  

  Full Alternative Text

11. 12 kg

13. $10.60\mu \text{m}$, $2.828\times {10}^{13}\text{Hz}$

15.

1. 2.42 m, 1.21 m, 0.807 m

2. 4.84 m, 1.61 m, 0.968 m

17. 2180 N

19. 1.9 Hz

21. 0.34 m, 1.0 m, 1.7 m

23.

1. 25 cm

2. 25 cm

25. 0.45 ms

27.

1. In phase

2. m	$r{}_1$	$r{}_2$	$\text{Δ}r$	$\text{C}/\text{D}$
   P	$3\text{λ}$	$4\text{λ}$	$\text{λ}$	C
   Q	${\displaystyle \frac{7}{2}}\text{λ}$	$2\text{λ}$	${\displaystyle \frac{5}{2}}\text{λ}$	D
   R	${\displaystyle \frac{5}{2}}\text{λ}$	${\displaystyle \frac{7}{2}}\text{λ}$	$\text{λ}$	C

29. Perfect destructive interference

31. Maximum destructive interference

33. 203 Hz

35. 10

37. 0.62 cm, 1.2 cm, 1.6 cm, 1.9 cm, 2.0 cm

39. $2.4\times {10}^{-4}\text{N}$

41. 28.4 cm

43. $8.50\text{m}\text{/}{\text{s}}^2$

45. 6.1 cm

47. 54 Hz

49. 13 Hz

51.

1. $\sqrt[12]{2}$

2. 466 Hz

53. 328 m/s

55. 4.0 cm, 35 cm, 65 cm

57. 450 N

59. 85 cm

61.

1. 80 cm

2. ${\displaystyle \frac{3\pi }{4}}$

3. 0.77a

63.

1. 473 nm

2. 406 nm, 568 nm

3. Reflected light is blue; transmitted light is yellowish green.

65. 1.4

67. 679 nm, 428 nm

69. 3.0 m, 5.6 m, 18 m

71.

1. a

2. 1.0 m

3. 9

73.

1. 5

2. 4.6 mm

75. 7.0 m/s

77. 24

79. 2.0 kg

81.

• c. 2.09 cm/s

• d. 2.2 mm