Homework: chapter 16 Wave Motion

1. Homework: chapter 16 Wave Motion 1,2,3,4,10,13,14,17,19,21,24,27,30,32,34,36, 40, 43,46,52,59,62

2. 1. At t = 0, a transverse pulse in a wire is described by the function where x and y are in meters. Write the function y(x, t) that describes this pulse if it is traveling in the positive x direction with a speed of 4.50 m/s. Replace x by to get

3. 2. Ocean waves with a crest-to-crest distance of 10.0 m can be described by the wave function y(x, t) = (0.800 m) sin[0.628(x – vt)] where v = 1.20 m/s. (a) Sketch y(x, t) at t = 0. (b) Sketch y(x, t) at t = 2.00 s. Note how the entire wave form has shifted 2.40 m in the positive x direction in this time interval.

4. 3. A pulse moving along the x axis is described by • where x is in meters and t is in seconds. • Determine: • the direction of the wave motion, • (b) the speed of the pulse. The given function is of the form f(X+vt) It describes a wave moving to the left at .

5. 4. Two points A and B on the surface of the Earth are at the same longitude and 60.0 apart in latitude. Suppose that an earthquake at point A creates a P wave that reaches point B by traveling straight through the body of the Earth at a constant speed of 7.80 km/s. The earthquake also radiates a Rayleigh wave, which travels along the surface of the Earth in an analogous way to a surface wave on water, at 4.50 km/s. (a) Which of these two seismic waves arrives at B first? (b) What is the time difference between the arrivals of the two waves at B? Take the radius of the Earth to be 6 370 km. • The longitudinal wave is traveling a shorter distance and is faster, so it will • arrive at B first (b) The wave that travels through the Earth must travel a distance of At a speed of 7800 m/s Then, it takes The wave that travels along the Earth’s surface must travel a distance of

6. 16.4: continuation at a speed of 4 500 m/s Therefore, it takes Time difference:

7. 10. A sinusoidal wave on a string is described by y = (0.51 cm) sin(kx –t) where k = 3.10 rad/cm and  = 9.30 rad/s. How far does a wave crest move in 10.0 s? Does it move in the positive or negative x direction?

8. A sinusoidal wave train is described by • y = (0.25 m) sin(0.30x – 40t) • where x and y are in meters and t is in seconds. Determine for this wave the • (a) amplitude, • (b) angular frequency, • (c) angular wave number, • (d) wavelength, • (e) wave speed, and • (f) direction of motion. (a (b) (c ) (d) . (e) (f) The wave moves to the right,

9. 14. • Plot y versus t at x = 0 for a sinusoidal wave of the form • y = (15.0 cm) cos(0.157x – 50.3t) , where x and y are in centimeters and t is in seconds. • (b) Determine the period of vibration from this plot and compare your result with the value found in Example 16.2. This agrees with the period found in the example in the text.

10. 17. A transverse wave on a string is described by the wave function • Determine the transverse speed and acceleration at t = 0.200 s for the point on the string located at x = 1.60 m. • What are the wavelength, period, and speed of propagation of this wave? (a) : : :

11. 19. A sinusoidal wave of wavelength 2.00 m and amplitude 0.100 m travels on a string with a speed of 1.00 m/s to the right. Initially, the left end of the string is at the origin. Find: (a) the frequency and angular frequency, (b) the angular wave number, and (c) the wave function for this wave. etermine the equation of motion for (d) the left end of the string, and (e) the point on the string at x = 1.50 m to the right of the left end. (f) What is the maximum speed of any point on the string? (a) (b) (c ) e) The cosine varies between +1 and –1, so

12. 21. A telephone cord is 4.00 m long. The cord has a mass of 0.200 kg. A transverse pulse is produced by plucking one end of the taut cord. The pulse makes four trips down and back along the cord in 0.800 s. What is the tension in the cord? The down and back distance is The speed is then . Now, so

13. 24. A transverse traveling wave on a taut wire has an amplitude of 0.200 mm and a frequency of 500 Hz. It travels with a speed of 196 m/s. • Write an equation in SI units of the form y = A sin(kx – t) for this wave. • (b) The mass per unit length of this wire is 4.10 g/m. • Find the tension in the wire. b)

14. 27. Transverse waves travel with a speed of 20.0 m/s in a string under a tension of 6.00 N. What tension is required for a wave speed of 30.0 m/s in the same string? Since µ is constant, and .

15. 30. Review problem. A light string with a mass per unit length of 8.00 g/m has its ends tied to two walls separated by a distance equal to three-fourths the length of the string (Fig. P16.30). An object of mass m is suspended from the center of the string, putting a tension in the string. (a) Find an expression for the transverse wave speed in the string as a function of the mass of the hanging object. (b) What should be the mass of the object suspended from the string in order to produce a wave speed of 60.0 m/s? a) Figure P16.30 and

16. 32. Review problem. A light string of mass m and length L has its ends tied to two walls that are separated by the distance D. Two objects, each of mass M, are suspended from the string as in Figure P16.32. If a wave pulse is sent from point A, how long does it take to travel to point B? Refer to the diagrams. From the free-body diagram of point A: Combining these equations to eliminate T1 gives the tension in the string connecting points A and B: Figure P16.32 The speed of transverse waves in this segment of string is then Time for the pulse to travel from A to B

17. 34. A taut rope has a mass of 0.180 kg and a length of 3.60 m. What power must be supplied to the rope in order to generate sinusoidal waves having an amplitude of 0.100 m and a wavelength of 0.500 m and traveling with a speed of 30.0 m/s? power

18. 36. Transverse waves are being generated on a rope under constant tension. By what factor is the required power increased or decreased if (a) the length of the rope is doubled and the angular frequency remains constant, (b) the amplitude is doubled and the angular frequency is halved, (c) both the wavelength and the amplitude are doubled, and (d) both the length of the rope and the wavelength are halved? T is constant (a) If L is doubled, v remains constant and (b) If A is doubled and  is halved, . . (c) If  and A are doubled, the product remains constant, So p remains constant (d) If L and  are halved, then (Changing L doesn’t affect p)

19. 40. The wave function for a wave on a taut string is y(x, t) = (0.350 m)sin(10t – 3x + /4) where x is in meters and t in seconds. (a) What is the average rate at which energy is transmitted along the string if the linear mass density is 75.0 g/m? (b) What is the energy contained in each cycle of the wave? Comparing with Then: (a) The rate of energy transport is (b) The energy per cycle is

20. 43. • Evaluate A in the scalar equality • (7+3)4=A • (b) Evaluate A, B, and C in the vector equality . • Explain how you arrive at the answers to convince a student who thinks that you cannot solve a single equation for three different unknowns. • (c) What If? The functional equality or identity • A + B cos (Cx + Dt + E) = (7.00 mm) cos(3x + 4t + 2) is true for all values of the variables x and t, which are measured in meters and in seconds, respectively. Evaluate the constants A, B, C, D, and E. Explain how you arrive at the answers.  b) A=7.0, B=0 and C=3.0 (c) In order for two functions to be identically equal, they must be equal for every value of every variable. They must have the same graphs. In

21. 16.43 continuation the equality of average values requires that The equality of maximum values requires The equality for the wavelength or periodicity as a function of x requires . The equality of period requires The equality of period requires And the equality of zero-crossings requires

22. 46. (a) Show that the function y(x,t) = x2 + v2 t2is a solution to the wave equation. (b) Show that the function in part (a) can be written as f(x + vt) + g(x – vt), and determine the functional forms for f and g. (c) What If? Repeat parts (a) and (b) for the function y(x,t) = sin(x)cos(vt). follows As required so

23. Part c) of 16.47 becomes Then: Note So which is true as required. . and

24. 52.Review problem. A block of mass M, supported by a string, rests on an incline making an angle  with the horizontal (Fig. P16.52). The length of the string is L and its mass is m <<M. Derive an expression for the time interval required for a transverse wave to travel from one end of the string to the other. Assuming the incline to be frictionless and taking the positive x-direction to be up the incline: or the tension in the string is The speed of transverse waves in the string is then Fig. 16.52 The time interval for a pulse to travel the string’s length is

25. 59. A rope of total mass m and length L is suspended vertically. Show that a transverse pulse travels the length of the rope in a time interval . (Suggestion: First find an expression for the wave speed at any point a distance x from the lower end by considering the tension in the rope as resulting from the weight of the segment below that point.) the weight of a length x, of rope. Then But

26. 62. Determine the speed and direction of propagation of each of the following sinusoidal waves, assuming that x is measured in meters and t in seconds. (a) y = 0.60 cos(3.0x – 15t + 2) (b) y = 0.40 cos(3.0x + 15t – 2) (c) y = 1.2 sin(15t + 2.0x) (d) y = 0.20 sin[(12t – (x/2) + ] a) b) c) d)