Use the wave equation to find the speed of a

The following four waves are sent along strings with the same linear densities (x is in meters and t is in seconds). Rank the waves according to (a) their wave speed and (b) the tension in the strings along which they travel, greatest first: (1) Yl = (3 mm) sin(x - 3t), (3) )'3 = (1 mm) sin(4x - t), (2) Yz = (6 mm) sin(2x - t), (4) Y4 = (2 mm) sin(x - 2t).

In Fig. 16-23, wave 1 consists of a rectangular peak of height 4 units and width d, and a rectangular valley of depth 2 units and width d. The wave travels rightward along an x axis. Choices 2, 3, and 4 are similar waves, with the same heights, depths, and widths, that will travel leftward along that axis and through wave 1. Rightgoing wave 1 and one of the left-going waves will interfere as they pass through each other. With which left-going wave will the interference give, for an instant, (a) the deepest valley, (b) a flat line, and (c) a flat peak 2d wide?

Figure 16-24a gives a snapshot of a wave traveling in the direction of positive x along a string under tension. Four string elements are indicated by the lettered points. For each of those elements, determine whether, at the instant of the snapshot, the element is moving upward or downward or is momentarily at rest. (Hint: Imagine the wave as it moves through the four string elements, as if you were watching a video of the wave as it traveled rightward.) Figure 16-24b gives the displacement of a string element located at, say, x = 0 as a function of time. At the lettered times, is the element moving upward or downward or is it momentarily at rest?

Figure 16-25 shows three waves that are separately sent along a string that is stretched under a certain tension along an x axis. Rank the waves according to their (a) wavelengths, (b) speeds, and (c) angular frequencies, greatest first.

If you start with two sinusoidal waves of the same amplitude traveling in phase on a string and then somehow phase-shift one of them by 5.4 wavelengths, what type of interference will occur on the string?

The amplitudes and phase differences for four pairs of waves of equal wavelengths are (a) 2 mm, 6 mm, and 7Trad; (b) 3 mm, 5 mm, and 7Trad; (c) 7 mm, 9 mm, and 7Trad; (d) 2 mm, 2 QUESTIONS 437 mrn, and 0 rad. Each pair travels in the same direction along the same string. Without written calculation, rank the four pairs according to the amplitude of their resultant wave, greatest first. (Hint: Construct phasor diagrams.)

A sinusoidal wave is sent along a cord under tension, transporting energy at the average rate of Pavg,l' Two waves, identical to that first one, are then to be sent along the cord with a phase difference of either 0, 0.2 wavelength, or 0.5 wavelength. (a) With only mental calculation, rank those choices of according to the average rate at which the waves will transport energy, greatest first. (b) For the first choice of , what is the average rate in terms of Pavg,l?

(a) If a standing wave on a string is given by y'(t) = (3 mm) sin(5x) cos(4t), is there a node or an antinode of the oscillations of the string at x = O? (b) If the standing wave is given by y'(t) = (3 mm) sin(5x + 7T/2) cos(4t), is there a node or an antinode at x = O?

Strings A and B have identical lengths and linear densities, but string B is under greater tension than string A. Figure 16-26 shows four situations, (a) through (d), in which standing wave patterns exist on the two strings. In which situations is there the possibility that strings A and B are oscillating at the same resonant frequency?

If you set up the seventh harmonic on a string, (a) how many nodes are present, and (b) is there a node, antinode, or some intermediate state at the midpoint? If you next set up the sixth harmonic, (c) is its resonant wavelength longer or shorter than that for the seventh harmonic, and (d) is the resonant frequency higher or lower?

Figure 16-27 shows phasor diagrams for three situations in which two waves travel along the same string. All six waves have the same amplitude. Rank the situations according to the amplitude of the net wave on the string, greatest first.

The function y(x, t) = (15.0 cm) cos( 1TX - 15771), with x in meters and t in seconds, describes a wave on a taut string. What is the transverse speed for a point on the string at an instant when that point has the displacement Y = + 12.0 cm?

A sinusoidal wave of frequency 500 Hz has a speed of 350 m/s. (a) How far apart are two points that differ in phase by 1T/3 rad? (b) What is the phase difference between two displacements at a certain point at times 1.00 ms apart?

The equation of a transverse wave on a string is Y = (2.0 mm) sin[(20 m-1)x - (600 S-l)t]. The tension in the string is 15 N. (a) What is the wave speed? (b) Find the linear density of this string in grams per meter.

A stretched string has a mass per unit length of 5.00 g/cm and a tension of 10.0 N. A sinusoidal wave on this string has an amplitude of 0.12 mm and a frequency of 100 Hz and is traveling in the negative direction of an x axis. If the wave equation is of the form y(x, t) = YIIl sin(kx ::':: wt), what are (a) Y,m (b) Ie, (c) w, and (d) the correct choice of sign in front of w?

The speed of a transverse wave on a string is 170 m/s when the string tension is 120 N. To what value must the tension be changed to raise the wave speed to 180 rnfs?

The linear density of a string is 1.6 X 10-4 kg/m. A transverse wave on the string is described by the equation Y = (0.021 m) sin[(2.0 m-1)x + (30 S-l)t]. What are (a) the wave speed and (b) the tension in the string?

The heaviest and lightest strings on a certain violin have linear densities of 3.0 and 0.29 g/m. What is the ratio of the diameter of the heaviest string to that of the lightest string, assuming that the strings are of the same material?

What is the speed of a transverse wave in a rope of length 2.00 m and mass 60.0 g under a tension of 500 N?

The tension in a wire clamped at both ends is doubled without appreciably changing the wire's length between the clamps. What is the ratio of the new to the old wave speed for transverse waves traveling along this wire?

A 100 g wire is held under a tension of 250 N with one end at x = 0 and the other at x = 10.0 m. At time t = 0, pulse 1 is sent along the wire from the end at x = 10.0 m. At time t = 30.0 ms, pulse 2 is sent along the wire from the end at x = O. At what position x do the pulses begin to meet?

A sinusoidal wave is traveling on a string with speed 40 cm/s. The displacement of the particles of the string at x = 10 cm varies with time according to Y = (5.0 cm) sin[1.0 - (4.0 S-l)t]. The linear density of the string is 4.0 g/cm. What are (a) the frequency and (b) the wavelength of the wave? If the wave equation is of the form y(x, t) = YI11 sin(kx ::':: wt), what are (c) YIIl' (d) k, (e) w, and (f) the correct choice of sign in front of w? (g) What is the tension in the string?

A sinusoidal trans- y (em) verse wave is traveling along a string in the negative direction of an x axis. Figure 16-34 shows a plot of the displacement as a function of position at time t = 0; the scale of the Y axis is set by Ys = 4.0 cm. The string tension is 3.6 N, and its linear density is 25 g/m. Find the (a) amplitude, (b) wavelength, (c) wave speed, and (d) pe- _J -j x (em) riod of the wave. (e) Find the maximum transverse speed of a particle Fig. 16-34 Problem 23. in the string. If the wave is of the form y(x, t) = YIIl sin(kx ::':: wt + cp), what are (f) Ie, (g) w, (h) cp, and (i) the correct choice of sign in front of w?

In Fig. 16-35a, string 1 has a linear density of 3.00 g/m, and string 2 has a linear density of 5.00 g/m. They are under tension due to the hanging block of mass M = 500 g. Calculate the wave speed on (a) string 1 and (b) string 2. (Hint: When a string loops halfway around a pulley, it pulls on the pulley with a net force that is twice the tension in the string.) Next the block is divided into two blocks (with Ml + Mz = M) and the apparatus is rearranged as shown in Fig. 16-35b. Find (c) Mj and (d) Mz such that the wave speeds in the two strings are equal.

A uniform rope of mass m Fig. 16-35 Problem 24. and length L hangs from a ceiling. (a) Show that the speed of a transverse wave on the rope is a function of Y, the distance from the lower end, and is given by v = vgy. (b) Show that the time a transverse wave takes to travel the length of the rope is given by t = 2 VUii.

A string along which waves can travel is 2.70 m long and has a mass of 260 g. The tension in the string is 36.0 N. What must be the frequency of traveling waves of amplitude 7.70 mm for the average power to be 85.0 W?

A sinusoidal wave is sent along a string with a linear density of 2.0 g/m. As it travels, the kinetic energies of the mass elements along the string vary. Figure 16-36a gives the rate dKldt at which kinetic energy passes through the string elements at a particular instant, plotted as a function of distance x along the string. Figure 16-36b is similar except that it gives the rate at which kinetic energy passes through a particular mass element (at a particular location), plotted as a function of time t. For both figures, the scale on the vertical (rate) axis is set by Rs = 10 W. What is the amplitude of the wave?

Use the wave equation to find the speed of a wave given by y(x, t) = (3.00 mm) sin[( 4.00 m-I)x - (7.00 S-I)t].

Use the wave equation to find the speed of a wave given by y(x, t) = (2.00 mm)[(20 m-I)x - (4.0 S-I)t]O.5.

Use the wave equation to find the speed of a wave given in terms of the general function hex, t): y(x, t) = (4.00 mm) h[(30 m-I)x + (6.0 S-I)t].

Two identical traveling waves, moving in the same direction, are out of phase by 1T12 rad. What is the amplitude of the resultant wave in terms of the common amplitude Ym of the two combining waves?

What phase difference between two identical traveling waves, moving in the same direction along a stretched string, results in the combined wave having an amplitude 1.50 times that of the common amplitude of the two combining waves? Express your answer in (a) degrees, (b) radians, and (c) wavelengths.

Two sinusoidal waves with the same amplitude of 9.00 mm and the same wavelength travel together along a string that is stretched along an x axis. Their resultant wave is shown twice in Fig. 16-37, as valley A travels in the negative direction of the x axis by distance d = 56.0 cm in 8.0 ms. The tick marks along the axis are separated by 10 cm, and Fig. 16-37 Problem 33. height His 8.0 mm. Let the equation for one wave be of the form y(x, t) = YIIl sin(kx wt + 4>1)' where 4>1 = 0 and you must choose the correct sign in front of w. For the equation for the other wave, what are (a) YIIl' (b) k, (c) w, (d) 4>2> and (e) the sign in front of w?

A sinusoidal wave of angular frequency 1200 rad/s and amplitude 3.00 mm is sent along a cord with linear density 2.00 g/m and tension 1200 N. (a) What is the average rate at which energy is transported by the wave to the opposite end of the cord? (b) If, simultaneously, an identical wave travels along an adjacent, identical cord, what is the total average rate at which energy is transported to the opposite ends of the two cords by the waves? If, instead, those two waves are sent along the same cord simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) 0, (d) O.41Trad, and (e) 1Trad?

Two sinusoidal waves of the same frequency travel in the same direction along a string. If YlIll = 3.0 cm, YIIl2 = 4.0 cm, 4>1 = 0, and 2 = 1T12 rad, what is the amplitude of the resultant wave?

Four waves are to be sent along the same string, in the same direction: YI (x, t) = (4.00 mm) sin(21Tx - 4001Tt) Y2(X, t) = (4.00 mm) sin(21Tx - 4001Tt + 0.71T) Y3(X, t) = (4.00 mm) sin(21Tx - 4001Tt + 1T) Y4(X, t) = (4.00 mm) sin(21Tx - 4001Tt + 1.71T). What is the amplitude of the resultant wave?

These two waves travel along the same string: Yl (x, t) = (4.60 mm) sin(21Tx - 4001Tt) heX, t) = (5.60 mm) sin(21Tx - 4001Tt + 0.801Trad). What are (a) the amplitude and (b) the phase angle (relative to wave 1) of the resultant wave? (c) If a third wave of amplitude 5.00 mm is also to be sent along the string in the same direction as the first two waves, what should be its phase angle in order to maximize the amplitude of the new resultant wave?

Two sinusoidal waves of the same frequency are to be sent in the same direction along a taut string. One wave has an amplitude of 5.0 mm, the other 8.0 mm. (a) What phase difference 4>l between the two waves results in the smallest amplitude of the resultant wave? (b) What is that smallest amplitude? (c) What phase difference 4>2 results in the largest amplitude of the resultant wave? (d) What is that largest amplitude? (e) What is the resultant amplitude if the phase angle is (4)1 - 2)/2?

Two sinusoidal waves of the same period, with amplitudes of 5.0 and 7.0 mm, travel in the same direction along a stretched string; they produce a resultant wave with an amplitude of 9.0 mm. The phase constant of the 5.0 mm wave is O. What is the phase constant of the 7.0 mm wave?

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string with a speed of 10 cm/s. If the time interval between instants when the string is flat is 0.50 s, what is the wavelength of the waves?

A string fixed at both ends is 8.40 m long and has a mass of 0.120 kg. It is subjected to a tension of 96.0 N and set oscillating. (a) What is the speed of the waves on the string? (b) What is the longest possible wavelength for a standing wave? (c) Give the frequency of that wave.

A string under tension Ti oscillates in the third harmonic at frequency /3, and the waves on the string have wavelength A3 If the tension is increased to T! = 47) and the string is again made to oscillate in the third harmonic, what then are (a) the frequency of oscillation in terms of /3 and (b) the wavelength of the waves in terms of A3?

What are (a) the lowest frequency, (b) the second lowest frequency, and (c) the third lowest frequency for standing waves on a wire that is 10.0 m long, has a mass of 100 g, and is stretched under a tension of 250 N?

A 125 cm length of string has mass 2.00 g and tension 7.00 N. (a) What is the wave speed for this string? (b) What is the lowest resonant frequency of this string?

A string that is stretched between fixed supports separated by 75.0 cm has resonant frequencies of 420 and 315 Hz, with no intermediate resonant frequencies. What are (a) the lowest resonant frequency and (b) the wave speed?

String A is stretched between two clamps separated by distance L. String B, with the same linear density and under the same tension as string A, is stretched between two clamps separated by distance 4L. Consider the first eight harmonics of string B. For which of these eight harmonics of B (if any) does the frequency match the frequency of (a) A's first harmonic, (b) A's second harmonic, and (c) A's third harmonic?

One of the harmonic frequencies for a particular string under tension is 325 Hz. The next higher harmonic frequency is 390 Hz. What harmonic frequency is next higher after the harmonic frequency 195 Hz?

If a transmission line in a cold climate collects ice, the increased diameter tends to cause vortex formation in a passing wind. The air pressure variations in the vortexes tend to cause the line to oscillate (gallop), especially if the frequency of the variations matches a resonant frequency of the line. In long lines, the resonant frequencies are so close that almost any wind speed can set up a resonant mode vigorous enough to pull down support towers or cause the line to short out with an adjacent line. If a transmission line has a length of 347 m, a linear density of 3.35 kglm, and a tension of 65.2 MN, what are (a) the frequency of the fundamental mode and (b) the frequency difference between successive modes?

A nylon guitar string has a linear density of 7.20 g/m and is under a tension of 150 N. The fixed supports are distance D = 90.0 cm apart. The string is oscillating in the standing Fig. 16-38 Problem 49. wave pattern shown in Fig. 16-38. Calculate the (a) speed, (b) wavelength, and (c) frequency of the traveling waves whose superposition gives this standing wave.

For a certain transverse standing wave on a long string, an antinode is at x = 0 and an adjacent node is at x = 0.10 m. The displacement yet) of the string particle at x = 0 is shown in Fig. 16-39, where the scale of the y axis is set by Ys = 4.0 cm. When t = 0.50 s, what is S the displacement of the string particle ~ at (a) x = 0.20 m and (b) x = 0.30 m? "" What is the transverse velocity of the string particle at x = 0.20 m at ( c) t = Ys 0.50 s and (d) t = 1.0 s? (e) Sketch Fig. 1 6-39 Problem 50. the standing wave at t = 0.50 s for the range x = 0 to x = 0.40 m.

Two waves are generated on a string of length 3.0 m to produce a three-loop standing wave with an amplitude of 1.0 cm. The wave speed is 100 m/s. Let the equation for one of the waves be of the form y(x, t) = YIIl sin(kx + wt). In the equation for the other wave, what are (a) YIII' (b) k, (c) w, and (d) the sign in front of w?

A rope, under a tension of 200 N and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by Y = (0.10 m)(sin m:12) sin 121Tt, where x = 0 at one end of the rope, x is in meters, and t is in seconds. What are (a) the length of the rope, (b) the speed of the waves on the rope, and (c) the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

A string oscillates according to the equation Y' = (0.50 cm) sin[ ( ; cm-1)x J cOS[(401TS-1)t]. What are the (a) amplitude and (b) speed of the two waves (identical except for direction of travel) whose superposition gives this oscillation? (c) What is the distance between nodes? (d) What is the transverse speed of a particle of the string at the position x = 1.5 cm when t = ~ s?

Two sinusoidal waves with the same amplitude and wavelength travel through each other along a string that is stretched along an x axis. Their resultant wave is shown twice in Fig. 16-40, as the antinode A travels from an extreme upward displacement to an extreme downward displacement in 6.0 ms. The tick marks along the axis are separated by 10 cm; height H is 1.80 cm. Let the equation for one of the two waves be of the form y(x, t) = YIIl sin(kx + wt). In the equation for the other wave, what are (a) YIIl' (b) k, (c) w, and (d) the sign in front of w?

The following two waves are sent in opposite directions on a horizontal string so as to create a standing wave in a vertical plane: Yl(X, t) = (6.00 mm) sin(4.00m: - 4001Tf) Yz(x, t) = (6.00 mm) sin(4.00m: + 4001Tf), with x in meters and t in seconds. An antinode is located at point A. In the time interval that point takes to move from maximum upward displacement to maximum downward displacement, how far does each wave move along the string?

A standing wave pattern on a string is described by y(x, t) = 0.040 (sin 51Tx)(cos 401Tt), where x and yare in meters and t is in seconds. For x 2: 0, what is the location of the node with the (a) smallest, (b) second smallest, and ( c) third smallest value of x? (d) What is the period of the oscillatory motion of any (nonnode) point? What are the (e) speed and (f) amplitude of the two traveling waves that interfere to produce this wave? For t 2: 0, what are the (g) first, (h) second, and (i) third time that all points on the string have zero transverse velocity?

A generator at one end of a very long string creates a wave given by 1T Y = (6.0 cm) cos 2 [(2.00 m-1)x + (8.00 S-l)t], and a generator at the other end creates the wave Y = (6.0 cm) cos ; [(2.00 m-1)x - (8.00 S-l)t]. Calculate the (a) frequency, (b) wavelength, and (c) speed of each wave. For x 2: 0, what is the location of the node having the (d) smallest, (e) second smallest, and (f) third smallest value of x? For x 2: 0, what is the location of the antinode having the (g) smallest, (h) second smallest, and (i) third smallest value of x?

In Fig. 16-41, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation L = 1.20 m, linear density f.L = 1.6 glm, and the oscillator frequency f = 120 Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q. (a) What mass m allows the oscillator to set up the fourth harmonic on the string? (b) What standing wave mode, if any, can be set up if m = 1.00 kg?

In Fig. 16-42, an aluminum wire, of length Ll = 60.0 cm, cross-sectional area 1.00 X 10-2 cm2, and density 2.60 g/cm3, is joined to a steel wire, of density 7.80 g/cm3 and the same cross-sectional area. The compound wire, loaded with a block of mass m = 10.0 kg, is arranged so that the distance L2 from the joint to the supporting pulley is 86.6 cm. Transverse waves are set up on the wire by an external source of variable frequency; a node is located at the pUlley. (a) Find the lowest frequency that generates a standing wave having the joint as one of the nodes. (b) How many nodes are observed at this frequency?

In Fig. 16-41, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. The separation L between P and Q is 1.20 m, and the frequency f of the oscillator is fixed at 120 Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q. A standing wave appears when the mass of the hanging block is 286.1 g or 447.0 g, but not for any intermediate mass. What is the linear density of the string?

In an experiment on standing waves, a string 90 cm long is attached to the prong of an electrically driven tuning fork that oscillates perpendicular to the length of the string at a frequency of 60 Hz. The mass of the string is 0.044 kg. What tension must the string be under (weights are attached to the other end) if it is to oscillate in four loops?

A sinusoidal transverse wave traveling in the positive direction of an x axis has an amplitude of 2.0 cm, a wavelength of 10 cm, and a frequency of 400 Hz. If the wave equation is of the form y(x, t) = Ym sin(kx wt), what are (a) Ym' (b) k, (c) w, and (d) the correct choice of sign in front of w? What are (e) the maximum transverse speed of a point on the cord and (f) the speed of the wave?

A wave has a speed of 240 mls and a wavelength of 3.2 ill. What are the (a) frequency and (b) period of the wave?

The equation of a transverse wave traveling along a string is Y = 0.15 sin(0.79x - 13t), in which x and yare in meters and t is in seconds. (a) What is the displacement Y at x = 2.3 m, t = 0.16 s? A second wave is to be added to the first wave to produce standing waves on the string. If the wave equation for the second wave is of the form y(x, t) = Yill sin(kx wt), what are (b) YIIl> (c) k, (d) w, and (e) the correct choice of sign in front of w for this second wave? (f) What is the displacement of the resultant standing wave at x = 2.3 m, t = 0.16 s?

The equation of a transverse wave traveling along a string is Y = (2.0 mm) sin[(20 m-l)x - (600 S-l)t]. Find the (a) amplitude, (b) frequency, (c) velocity (including sign), and (d) wavelength of the wave. (e) Find the maximum transverse speed of a particle in the string.

Figure 16-43 shows the displacement Y versus time t of the point on a string at x = 0, as a wave passes through that point. The scale of the Y axis is set by Ys = 6.0 mrn. The wave is given by y(x, t) = YIIl sin(la: - wt + ). What is ? (Caution: A calculator does not always give the proper y(mm) ~, - -)'" Fig. 16-43 Problem 66. inverse trig function, so check your answer by substituting it and an assumed value of winto y(x, t) and then plotting the function.)

Two sinusoidal waves, identical except for phase, travel in the same direction along a string, producing the net wave y' (x, t) = (3.0 mm) sin(20x - 4.0t + 0.820 rad), with x in meters and t in seconds. What are (a) the wavelength A of the two waves, (b) the phase difference between them, and (c) their amplitude Ym?

A single pulse, given by h(x - 5.0t), is shown in Fig. 16-44 for t = O. The scale of the vertical axis is set by hs = 2. Here x is in centimeters and t is in seconds. " ~ What are the (a) speed and (b) dix rection of travel of the pulse? (c) Plot h(x - 5t) as a function of x for t = 2 s. (d) Plot h(x - 5t) as a function of t for x = 10 cm.

Three sinusoidal waves of the same frequency travel along a string in the positive direction of an x axis. Their amplitudes are Yl> y1/2, and y 1/3, and their phase constants are 0, 7T/2, and 7T, respectively. What are the (a) amplitude and (b) phase constant of the resultant wave? (c) Plot the wave form of the resultant wave at t = 0, and discuss its behavior as t increases.

Figure 16-45 shows transverse acceleration ay versus time t of the point on a string at x = 0, as a wave in the form of y(x, t) = Ym sin(kx - wt + ) passes through that point. The scale of the vertical axis is set by as = 400 mls2 What is ? (Caution: A calculator does not always give the proper inverse trig function, so check your answer by substituting it and an assumed value of w into y(x, t) and then plotting the function.)

A transverse sinusoidal wave is generated at one end of a long, horizontal string by a bar that moves up and down through a distance of 1.00 cm. The motion is continuous and is repeated regularly 120 times per second. The string has linear density 120 glm and is kept under a tension of 90.0 N. Find the maximum value of (a) the transverse speed It and (b) the transverse component of the tension 7. (c) Show that the two maximum values calculated above occur at the same phase values for the wave. What is the transverse displacement Y of the string at these phases? (d) What is the maximum rate of energy transfer along the string? (e) What is the transverse displacement Y when this maximum transfer occurs? (f) What is the minimum rate of energy transfer along the string? (g) What is the transverse displacement Y when this minimum transfer occurs?

Two sinusoidal 120 Hz waves, of the same frequency and amplitude, are to be sent in the positive direction of an x axis that is directed along a cord under tension. The waves can be s )" s s 0 1-------'1<-----1 sent in phase, or they can be Shift distance (cm) phase-shifted. Figure 16-46 shows Fig. 16-46 Problem 72. the amplitude y' of the resulting wave versus the distance of the shift (how far one wave is shifted from the other wave). The scale of the vertical axis is set by y~ = 6.0 mm. If the equations for the two waves are of the form y(x, t) = Ym sin(kx wt), what are (a) YII/' (b) k, (c) w, and (d) the correct choice of sign in front of w?

At time t = 0 and at position x = 0 m along a string, a traveling sinusoidal wave with an angular frequency of 440 rad/s has displacement Y = +4.5 mm and transverse velocity u = -0.75 m/s. If the wave has the general form y(x, t) = YII/ sin(kx - wt + ), what is phase constant ?

Energy is transmitted at rate PI by a wave of frequency fl on a string under tension 71' What is the new energy transmission rate P2 in terms of PI (a) if the tension is increased to 72 = 471 and (b) if, instead, the frequency is decreased to 12 = fIl2?

(a) What is the fastest transverse wave that can be sent along a steel wire? For safety reasons, the maximum tensile stress to which steel wires should be subjected is 7.00 X 108 N/m2 The density of steel is 7800 kg/m3. (b) Does your answer depend on the diameter of the wire?

A standing wave results from the sum of two transverse traveling waves given by YI = 0.050 cos( 1TX 41Tt) and Y2 = 0.050 cos( 1TX + 4171), where x, YI> and Y2 are in meters and t is in seconds. (a) What is the smallest positive value of x that corresponds to a node? Beginning at t = 0, what is the value of the (b) first, (c) second, and (d) third time the particle at x = 0 has zero velocity?

The type of rubber band used inside some baseballs and golf balls obeys Hooke's law over a wide range of elongation of the band. A segment of this material has an unstretched length e and a mass m. When a force Fis applied, the band stretches an additional PROBLEMS 443 length ile. (a) What is the speed (in terms of m, ile, and the spring constant k) of transverse waves on this stretched rubber band? (b) Using your answer to (a), show that the time required for a transverse pulse to travel the length of the rubber band is proportional to l/ill if M <{ e and is constant if M ~ e.

The speed of electromagnetic waves (which include visible light, radio, and x rays) in vacuum is 3.0 X 108 m/s. (a) Wavelengths of visible light waves range from about 400 nm in the violet to about 700 nm in the red. What is the range of frequencies of these waves? (b) The range of frequencies for shortwave radio (for example, FM radio and VHF television) is 1.5 to 300 MHz. What is the corresponding wavelength range? (c) X-ray wavelengths range from about 5.0 nm to about 1.0 X 10-2 nm. What is the frequency range for x rays?

ALSO m wire has a mass of 8.70 g and is under a tension of 120 N. The wire is held rigidly at both ends and set into oscillation. (a) What is the speed of waves on the wire? What is the wavelength of the waves that produce (b) one-loop and (c) twoloop standing waves? What is the frequency of the waves that produce (d) one-loop and (e) two-loop standing waves?

When played in a certain manner, the lowest resonant frequency of a certain violin string is concert A (440 Hz). What is the frequency of the (a) second and (b) third harmonic of the string?

A sinusoidal transverse wave traveling in the negative direction of an x axis has an amplitude of 1.00 cm, a frequency of 550 Hz, and a speed of 330 mls. If the wave equation is of the form y(x, t) = Ym sin(kx wt), what are (a) YII/' (b) w, (c) k, and (d) the correct choice of sign in front of w?

Two sinusoidal waves of the same wavelength travel in the same direction along a stretched string. For wave 1, YII/ = 3.0 mm and = 0; for wave 2, Ym = 5.0 mm and = 70. What are the (a) amplitude and (b) phase constant of the resultant wave?

A sinusoidal transverse wave of amplitude Ym and wavelength ;\ travels on a stretched cord. (a) Find the ratio of the maximum particle speed (the speed with which a single particle in the cord moves transverse to the wave) to the wave speed. (b) Does this ratio depend on the material of which the cord is made?

Oscillation of a 600 Hz tuning fork sets up standing waves in a string clamped at both ends. The wave speed for the string is 400 mls. The standing wave has four loops and an amplitude of 2.0 mm. (a) What is the length of the string? (b) Write an equation for the displacement of the string as a function of position and time.

A 120 cm length of string is stretched between fixed supports. What are the (a) longest, (b) second longest, and (c) third longest wavelength for waves traveling on the string if standing waves are to be set up? (d) Sketch those standing waves.

(a) Write an equation describing a sinusoidal transverse wave traveling on a cord in the positive direction of a y axis with an angular wave number of 60 cm-I , a period of 0.20 s, and an amplitude of 3.0 mm. Take the transverse direction to be the z direction. (b) What is the maximum transverse speed of a point on the cord?

A wave on a string is described by y(x, t) = 15.0 sine 1Tx/8 - 4171), where x and yare in centimeters and t is in seconds. (a) What is the transverse speed for a point on the string at x = 6.00 cm when t = 0.250 s? (b) What is the maximum transverse speed of any point on the string? (c) What is the magnitude of the transverse acceleration for a point on the string at x = 6.00 cm when t = 0.250 s? (d) What is the magnitude of the maximum transverse acceleration for any point on the string?

Body armor. When a high-speed projectile such as a bullet or bomb fragment strikes modern body armor, the fabric of the armor stops the projectile and prevents penetration by quickly spreading the projectile's energy over a large area. This spreading is done by longitudinal and transverse pulses that move radially from the impact point, where the projectile pushes a cone-shaped dent into the fabric. The longitudinal pulse, racing along the fibers of the fabric at speed VI ahead of the denting, causes the fibers to thin and stretch, with material flowing radially inward into the dent. One such radial fiber is shown in Fig. 16-47a. Part of the projectile's energy goes into this motion and stretching. The transverse pulse, moving at a slower speed VI' is due to the denting. As the projectile increases the dent's depth, the dent increases in radius, causing the material in the fibers to move in the same direction as the projectile (perpendicular to the transverse pulse's direction of travel). The rest of the projectile's energy goes into this motion. All the energy that does not eventually go into permanently deforming the fibers ends up as thermal energy. Figure 16-47b is a graph of speed V versus time t for a bullet of mass 10.2 g fired from a .38 Special revolver directly into body armor. The scales of the vertical and horizontal axes are set by Vs = 300 mls and ts = 40.0 f.LS. Take VI = 2000 mis, and assume that the half-angle (J of the conical dent is 60. At the end of the collision, what are the radii of (a) the thinned region and (b) the dent (assuming that the person wearing the armor remains stationary)?

Two waves are described by Yl = 0.30 sin[ 7T(5x 200)t] and Y2 = 0.30 sin[ 7T(5x - 200t) + 1T/3], where Yb Yz, and x are in meters and t is in seconds. When these two waves are combined, a traveling wave is produced. What are the (a) amplitude, (b) wave speed, and (c) wavelength of that traveling wave?

A certain transverse sinusoidal wave of wavelength 20 cm is moving in the positive direction of an x axis. The transverse velocity of the particle at x = 0 as a function of time is shown in Fig. 1 6-48 Problem 90. Fig. 16-48, where the scale of the vertical axis is set by Us = 5.0 cm/s. What are the (a) wave speed, (b) amplitude, and (c) frequency? (d) Sketch the wave between x = 0 and x = 20 cm at t = 2.0 s.

In a demonstration, a 1.2 kg horizontal rope is fixed in place at its two ends (x = 0 and x = 2.0 m) and made to oscillate up and down in the fundamental mode, at frequency 5.0 Hz. At t = 0, the point at x = 1.0 m has zero displacement and is moving upward in the positive direction of a Y axis with a transverse velocity of 5.0 m/s. What are (a) the amplitude of the motion of that point and (b) the tension in the rope? (c) Write the standing wave equation for the fundamental mode.

Two waves, Yl = (2.50 mm) sin[(25.1 rad/m)x - (440 rad/s)t] and Y2 = (1.50 mm) sin[(25.1 rad/m)x + (440 rad/s)t], travel along a stretched string. (a) Plot the resultant wave as a function of t for x = 0, A/8, Al4, 3A18, and A12, where A is the wavelength. The graphs should extend from t = 0 to a little over one period. (b) The resultant wave is the superposition of a standing wave and a traveling wave. In which direction does the traveling wave move? (c) How can you change the original waves so the resultant wave is the superposition of standing and traveling waves with the same amplitudes as before but with the traveling wave moving in the opposite direction? Next, use your graphs to find the place at which the oscillation amplitUde is (d) maximum and (e) minimum. (f) How is the maximum amplitude related to the amplitudes of the original two waves? (g) How is the minimum amplitude related to the amplitudes of the original two waves?

A traveling wave on a string is described by Y = 2.0 sin[ 21T( O.~O + ~~) J where x and yare in centimeters and t is in seconds. (a) For t = 0, plot Y as a function ofx for 0::; x::; 160 cm. (b) Repeat (a) for t = 0.05 sand t = 0.10 s. From your graphs, determine (c) the wave speed and (d) the direction in which the wave is traveling.