Solutions to Problems: Mechanical Waves‐2 P347: 35, 46, 49; P368: 35, 39, 60 35. Suppose two linear waves of equal amplitude and frequency have a phase difference as they travel in the same medium. They can be represented by D1 = DM sin ( kx − ωt ) , D2 = DM sin ( kx − ωt + φ ) . (a) Use the trigonometric identity sin θ1 + sin θ 2 = 2sin ⎡⎣(θ1 + θ 2 ) 2 ⎤⎦ cos ⎡⎣(θ1 − θ 2 ) 2 ⎤⎦ to show that the resultant wave is given by D = ( 2 DM cos (φ 2 ) ) sin ( kx − ωt + φ 2 ) . (b) What is the amplitude of this resultant wave? Is the wave purely sinusoidal, or not? (c) Show that constructive interference occurs if φ = 0, 2π , 4π , and so on, and destructive interference occurs if φ = π ,3π ,5π , etc. (d) Describe the resultant wave, by equation and in words, if φ = π 2. Solution: (a) Omitted. (b) The amplitude reads 2 DM cos (φ 2 ) , which is a function of position. Therefore the wave is not purely sinusoidal. (c) For φ = 0, 2π , 4π , the amplitude is 2 DM cos (φ 2 ) = 2 DM , so constructive interference occurs, and similarly one can find destructively occurs if φ = π ,3π ,5π . (d) For φ = π 2, the resultant wave function is D = 2 DM sin ( kx − ωt + φ 2 ) , it is purely sinusoidal and the amplitude equals to 2 DM . 46. The displacement of a standing wave is given by D = 8.6sin ( 0.60 x ) cos ( 58t ) , where x and D are in centimeters and t is in seconds. (a) What is the distance between nodes? (b) Give the amplitude, frequency, and speed of each of the component waves. (c) Find the speed of a particle of the string at x = 3.20 cm when t = 2.5 s. Solution: The locations of the nodes satisfy the relation 0.60xn = kπ , which tells that xn = 5kπ 3, with k = 0,1,". So the distance between the nodes is dn,kl = xk − xl = 5mπ 3 cm, with m = 1,2," . (b)From the wave function, one can find DM = 8.6 cm, f = ω 2π = 9.23 Hz, v1 = v2 = ω k = 96.7 cm/s . (c)The speed of a particle at x at an instant t is v p = ∂D ∂t = −499sin ( 0.60 x ) sin ( 58t ) cm/s , So, the speed of a particle of the string at x = 3.20 cm when t = 2.5 s is v p ( 3.20 cm, 2.5 s ) = −499sin ( 0.60 x ) sin ( 58t ) cm/s = 2.19 m/s 49. A particular violin string plays at a frequency of 294 Hz. If the tension is increased 10 percent what will the new frequency be? Solution: According to the relation v = T ρ , we have v ' v = T ' T = 1.1 = 1.05 ⇒ v ' = 1.05v ⇒ f ' = 1.05 f = 308 Hz . P368: 35, 39, 60 35. The predominant frequency of a certain police car’s siren is 1550 Hz when at rest. What frequency do you detected if you move with a speed of 30.0 m/s (a) toward the car, and (b) away from the car? Solution: (a) The frequency reads f ' =
(b) The frequency reads f ' =
v + vo
f = 1686 Hz . v
v − vo
f = 1414 Hz . v
39. Two automobiles are equipped with the same single‐frequency horn. When one is at rest and the other is moving toward one observer at 15 m/s, a beat frequency of 5.5 Hz is heard. What is the frequency the horns emit? Assume T = 20 °C. Solution: The beat frequency is fbeat = f1 − f 2 , and f 2 = vf1 ( v − va ) . So one can figure out that f1 = ( v − va ) f beat va = 120 Hz . 60. The frequency of a steam train whistle as it approaches you is 538 Hz. After it passes you, its frequency is measured as 486 Hz. How fast was the train moving (assume constant velocity). Solution: According to the problem, we can get following equations: v
v
f = f1 , f = f2 . v − vS
v + vS
Which tells us that
v − vS
f
f −f
= 2 ⇒ vS = 1 2 v = 17.4 m/s . v + vS
f1
f1 + f 2