P3.15.f^Chapter 15 Ending^515^523^,,^21587^22140%
WHAT WE HAVE LEARNED
EXAM STUDY GUIDE
KEY TERMS
wave, p. 493 longitudinal wave, p. 495 transverse wave, 495 wavelength, p. 496 wave front, p. 496 wave number, p. 497 phase, p. 497 wave equation, p. 498 plane waves, p. 503 constructive interference, p. 509 destructive interference, p. 509 nodes, p. 510 antinodes, p. 510 standing wave, p. 510 resonance frequency, p. 511 harmonic, p. 511
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NEW SYMBOLS AND EQUATIONS

, wave equation

v = λf, velocity, wavelength, and frequency of a wave

κ = 2π/λ, wave number

, speed of a wave on a string

y(x,t) =A sin (κx − ωt+ φ0), one-dimensional traveling wave

y(x,t) = 2A sin (κx) cos (ωt), one-dimensional standing wave

resonance frequencies of a wave on a string

ANSWERS TO SELF-TEST OPPORTUNITIES
  • 15.1 f = 0.40 Hz.

  • 15.2 A =10 cm, κ = 0.31 cm−1, φ0 = 0.30.

  • 15.3

  • 15.4 Lowest resonance frequency:

    Wave speed: v = 2Lf1 = 2(1.5 · 10−6 m)(5.5 · 107 s−1) = 165 m/s.

PROBLEM-SOLVING PRACTICE
Problem-Solving Guidelines
  1. If a wave fits the description of a one-dimensional traveling wave, you can write a mathematical equation describing it if you know the amplitude, A, of the motion and any two of these three quantities: velocity, v, wavelength, λ, and frequency, f. (Alternatively, you need to know two of the three quantities velocity, v, wave number, κ, and angular frequency, ω.)

  2. If a wave is a one-dimensional standing wave on a string, its resonance frequencies are determined by the length, L, of the string and the speed, v, of the component waves. The speed of the wave, in turn, is determined by the tension, T, in the string and the linear mass density, μ, of the string.

SOLVED PROBLEM15.2 Traveling Wave
PROBLEM

A wave on a string is given by the function

What is the wavelength of this wave? What is its period? What is its velocity?

SOLUTION
THINK

From the given wave function, we can identify the wave number and the angular frequency, from which we can extract the wavelength and period. We can then calculate the velocity of the traveling wave from the ratio of the angular frequency to the wave number.

SKETCH

Figure 15.32 shows plots of the wave function at t = 0 (part a) and at x = 0 (part b).

 
FIGURE 15.32
(a) The wave functionat t = 0. (b) The wave function as a function of time at x = 0.
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RESEARCH

Looking at the wave function given in the problem statement, and comparing it with the solution to the wave equation for a wave traveling in the negative x-direction (see equation 15.6),

we see that κ = 78.8 m−1 and ω = 346 s−1.

SIMPLIFY

We can obtain the wavelength from the wave number,

and the period from the angular frequency,

We can then calculate the velocity of the wave from

CALCULATE

Putting in the numerical values, we get the wavelength and the period:

and

We now calculate the velocity of the traveling wave:

ROUND

We report our results to three significant figures:

DOUBLE-CHECK

To double-check our result for the wavelength, we look at Figure 15.32a. We can see that the wavelength (the distance required to complete one oscillation) is approximately 0.08 m, in agreement with our result. Looking at Figure 15.32b, we can see that the period (the time required to complete one oscillation) is approximately 0.018 s, also in agreement with our result.

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SOLVED PROBLEM15.3 Standing Wave on a String
PROBLEM

A mechanical driver is used to set up a standing wave on an elastic string, as shown in Figure 15.33a. Tension is put on the string by running it over a frictionless pulley and hanging a metal block from it (Figure 15.33b). The length of the string from the top of the pulley to the driver is 1.25 m, and the linear mass density of the string is 5.00 g/m. The frequency of the driver is 45.0 Hz. What is the mass of the metal block?

SOLUTION
THINK

The weight of the metal block is equal to the tension placed on the elastic string. We can see from Figure 15.33a that the string is vibrating at its third harmonic because three antinodes are visible. We can thus use equation 15.28 to relate the tension on the string, the harmonic, the linear mass density of the string, and the frequency. Once we have determined the tension on the string, we can calculate the mass of the metal block.

SKETCH
FIGURE 15.33
(a) A standing wave on an elastic string driven by a mechanical driver. (b) The string is put under tension by a hanging mass. (c) Free-body diagram for the hanging block.

Figure 15.33b shows the elastic string under tension due to a metal block hanging from it. The mechanical driver sets up a standing wave on the string. Here L is the length of the string from the pulley to the wave driver, μ is the linear mass density of the string, and m is the mass of the metal block. Figure 15.33c shows the free-body diagram of the hanging metal block, where T is the tension on the string and mg is the weight of the metal block.

RESEARCH

A standing wave with harmonic n and frequency fn on an elastic string of length L and linear mass density μ satisfies equation 15.28:

From the free-body diagram in Figure 15.33c, and because the metal block does not move, we can write

Therefore, the tension on the string is T = mg.

SIMPLIFY

We solve equation 15.28 for the tension on the string:

Substituting mg for T and rearranging terms, we obtain

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CALCULATE

Putting in the numerical values, we get

ROUND

We report our result to three significant figures:

DOUBLE-CHECK

As a double-check of our result, we use the fact that a half-liter bottle of water has a mass of about 0.5 kg. Even a light string could support this mass without breaking. Thus, our answer seems reasonable.

MULTIPLE-CHOICE QUESTIONS
  • 15.1 Fans at a local football stadium are so excited that their team is winning that they start “the wave” in celebration. Which of the following four statements is (are) true?

    1. This wave is a traveling wave.

    2. This wave is a transverse wave.

    3. This wave is a longitudinal wave.

    4. This wave is a combination of a longitudinal wave and a transverse wave.

      1. I and II

      2. II only

      3. III only

      4. I and IV

      5. I and III

      Answer

  • 15.2 You wish to decrease the speed of a wave traveling on a string to half its current value by changing the tension in the string. By what factor must you decrease the tension in the string?

    1. 1

    2. 2

    3. 4

    4. none of the above

  • 15.3 Suppose that the tension is doubled for a string on which a standing wave is propagated. How will the velocity of the standing wave change?

    1. It will double.

    2. It will quadruple.

    3. It will be multiplied by .

    4. It will be multiplied by .

    Answer

  • 15.4 Which of the following transverse waves has the greatest power?

    1. wave with velocity v, amplitude A, and frequency f

    2. a wave of velocity v, amplitude 2A, and frequency f/2

    3. a wave of velocity 2v, amplitude A/2, and frequency f

    4. a wave of velocity 2v, amplitude A, and frequency f/2

    5. a wave of velocity v, amplitude A/2, and frequency 2f

  • 15.5 The speed of light waves in air is greater than the speed of sound in air by about a factor of a million. Given a sound wave and a light wave of the same wavelength, both traveling through air, which statement about their frequencies is true?

    1. The frequency of the sound wave will be about a million times greater than that of the light wave.

    2. The frequency of the sound wave will be about a thousand times greater than that of the light wave.

    3. The frequency of the light wave will be about a thousand times greater than that of the sound wave.

    4. The frequency of the light wave will be about a million times greater than that of the sound wave.

    5. There is insufficient information to determine the relationship between the two frequencies.

    Answer

  • 15.6 A string is made to oscillate, and a standing wave with three antinodes is created. If the tension in the string is increased by a factor of 4,

    1. the number of antinodes increases.

    2. the number of antinodes remains the same.

    3. the number of antinodes decreases.

    4. the number of antinodes will equal the number of nodes.

  • 15.7 The different colors of light we perceive are a result of the varying frequencies (and wavelengths) of the electromagnetic radiation. Infrared radiation has lower frequencies than does visible light, and ultraviolet radiation has higher frequencies than visible light does. The primary colors are red (R), yellow (Y), and blue (B). Order these colors by their wavelength, shortest to longest.

    1. B, Y, R

    2. B, R, Y

    3. R, Y, B

    4. R, B, Y

    Answer

  • 15.8 If transverse waves on a string travel with a velocity of 50 m/s when the string is under a tension of 20 N, what tension on the string is required for the waves to travel with a velocity of 30 m/s?

    1. 7.2 N

    2. 12 N

    3. 33 N

    4. 40 N

    5. 45 N

    6. 56 N

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QUESTIONS
  • 15.9 You and a friend are holding the two ends of a Slinky stretched out between you. How would you move your end of the Slinky to create (a) transverse waves or (b) longitudinal waves?

  • 15.10 A steel cable consists of two sections with different cross-sectional areas, A1 and A2. A sinusoidal traveling wave is sent down this cable from the thin end of the cable. What happens to the wave on encountering the A1/A 2. boundary? How do the speed, frequency and wavelength of the wave change?

  • 15.11 Noise results from the superposition of a very large number of sound waves of various frequencies (usually in a continuous spectrum), amplitudes, and phases. Can interference arise with noise produced by two sources?

  • 15.12 The 1/R2 dependency for intensity can be thought of to be due to the fact that the same power is being spread out over the surface of a larger and larger sphere. What happens to the intensity of a sound wave inside an enclosed space, say a long hallway?

  • 15.13 If two traveling waves have the same wavelength, frequency, and amplitude and are added appropriately, the result is a standing wave. Is it possible to combine two standing waves in some way to give a traveling wave?

  • 15.14 A ping-pong ball is floating in the middle of a lake and waves begin to propagate on the surface. Can you think of a situation in which the ball remains stationary? Can you think of a situation involving a single wave on the lake in which the ball remains stationary?

  • 15.15 Why do circular water waves on the surface of a pond decrease in amplitude as they travel away from the source?

  • 15.16 Consider a monochromatic wave on a string, with amplitude A and wavelength λ, traveling in one direction. Find the relationship between the maximum speed of any portion of string, vmax, and the wave speed, v.

PROBLEMS

A blue problem number indicates a worked-out solution is available in the Student Solutions Manual. One and two •• indicate increasing level of problem difficulty.

Sections 15.1 through 15.3
  • 15.17 One of the main things allowing humans to determine whether a sound is coming from the left or the right is the fact that the sound will reach one ear before the other. Given that the speed of sound in air is 343 m/s and that human ears are typically 20.0 cm apart, what is the maximum time resolution for human hearing that allows sounds coming from the left to be distinguished from sounds coming from the right? Why is it impossible for a diver to be able to tell from which direction the sound of a motor boat is coming? The speed of sound in water is 1.5 · 103 m/s.

    Answer

  • 15.18 Hiking in the mountains, you shout “hey,” wait 2.00 s and shout again. What is the distance between the sound waves you cause? If you hear the first echo after 5.00 s, what is the distance between you and the point where your voice hit a mountain?

  • 15.19 The displacement from equilibrium caused by a wave on a string is given by y(x,t) = (−0.00200 m) sin [(40.0 m−1)x − (800. s−1)t]. For this wave, what are the (a) amplitude, (b) number of waves in 1.00 m, (c) number of complete cycles in 1.00 s, (d) wavelength, and (e) speed?

    Answer

  • 15.20 A traveling wave propagating on a string is described by the following equation:

    1. Determine the minimum separation, Δx min, between two points on the string that oscillate in perfect opposition of phases (move in opposite directions at all times).

    2. Determine the separation, ΔxAB, between two points A and B on the string, if point B oscillates with a phase difference of 0.7854 rad compared to point A.

    3. Find the number of crests of the wave that pass through point A in a time interval Δt = 10.0 s and the number of troughs that pass through point B in the same interval.

    4. At what point along its trajectory should a linear driver connected to one end of the string at x = 0 start its oscillation to generate this sinusoidal traveling wave on the string?

  • ••15.21 Consider a linear array of n masses, each equal to m, connected by n+ 1 springs, all massless and having spring constant k, with the outer ends of the first and last springs fixed. The masses can move without friction in the linear dimension of the array.

    1. Write the equations of motion for the masses.

    2. Configurations of motion for which all parts of a system oscillate with the same angular frequency are called normal modes of the system; the corresponding angular frequencies are the system's normal-mode angular frequencies. Find the normal-mode angular frequencies of this array.

    Answer

Section 15.4
  • 15.22 Show that the function D = A ln (x+ vt) is a solution of the wave equation (equation 15.9).

  • 15.23 A wave travels along a string in the positive x-direction at 30.0 m/s. The frequency of the wave is 50.0 Hz. At x = 0 and t = 0, the wave velocity is 2.50 m/s and the vertical displacement is y = 4.00 mm. Write the function y(x,t) for the wave.

    Answer

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  • 15.24 A wave on a string has a wave function given by

    1. What is the amplitude of the wave?

    2. What is the period of the wave?

    3. What is the wavelength of the wave?

    4. What is the speed of the wave?

    5. In which direction does the wave travel?

  • •15.25 A sinusoidal wave traveling in the positive x-direction has a wavelength of 12 cm, a frequency of 10.0 Hz, and an amplitude of 10.0 cm. The part of the wave that is at the origin at t = 0 has a vertical displacement of 5.00 cm. For this wave, determine the

    1. wave number,

    2. period,

    3. angular frequency,

    4. speed,

    5. phase angle, and

    6. equation of motion.

    Answer

  • 15.26 A mass m hangs on a string that is connected to the ceiling. You pluck the string just above the mass, and a wave pulse travels up to the ceiling, reflects off the ceiling, and travels back to the mass. Compare the round-trip time for this wave pulse to that of a similar wave pulse on the same string if the attached mass is increased to 3.00 m. (Assume that the string does not stretch in either case and the contribution of the mass of the string to the tension is negligible.)

  • 15.27 Point A in the figure is 30.0 cm below the ceiling. Determine how much longer it will take for a wave pulse to travel along wire 1 than along wire 2.

    Answer

  • 15.28 A particular steel guitar string has mass per unit length of 1.93 g/m.

    1. If the tension on this string is 62.2 N, what is the wave speed on the string?

    2. For the wave speed to be increased by 1.0%, how much should the tension be changed?

  • •15.29 Bob is talking to Alice using a tin can telephone, which consists of two steel cans connected by a 20.0-m-long taut steel wire (see the figure). The wire has a linear density of 6.13 g/m, and the tension on the wire is 25.0 N. The sound waves leave Bob's mouth, are collected by the can on the left, and then create vibrations in the wire, which travel to Alice's can and are transformed back into sound waves in air. Alice hears both the sound waves that have traveled through the wire (wave 1) and those that have traveled through the air (wave 2), bypassing the wire. Do these two kinds of waves reach her at the same time? If not, which wave arrives sooner and by how much? The speed of sound in air is 343 m/s.

    Answer

  • ••15.30 A wire of uniform linear mass density hangs from the ceiling. It takes 1.00 s for a wave pulse to travel the length of the wire. How long is the wire?

  • ••15.31

    1. Starting from the general wave equation (equation 15.9), prove through direct derivation that the Gaussian wave packet described by the equation y(x,t) = (5.00m)e−0.1(x−5t)2 is indeed a traveling wave (that it satisfies the differential wave equation).

    2. If x is specified in meters and t in seconds, determine the speed of this wave. On a single graph, plot this wave as a function of x at t = 0, t = 1.00 s, t = 2.00 s, and t = 3.00 s.

    3. More generally, prove that any function f(x,t) that depends on x and t through a combined variable x ± vt is a solution of the wave equation, irrespective of the specific form of the function f.

    Answer

Section 15.5
  • 15.32 Suppose the slope of a beach underneath the ocean is 12.0 cm of dropoff for every 1.00 m of horizontal distance. A wave is moving inland, slowing down as it enters shallower water. What is its acceleration when it is 10.0 m from the shoreline?

  • •15.33 An earthquake generates three kinds of waves: surface waves (L waves), which are the slowest and weakest; shear (S) waves, which are transverse waves and carry most of the energy; and pressure (P) waves, which are longitudinal waves and travel the fastest. The speed of P waves is approximately 7.0 km/s, and that of S waves is about 4.0 km/s. Animals seem to feel the P waves. If a dog senses the arrival of P waves and starts barking 30.0 s before an earthquake is felt by humans, approximately how far is the dog from the earthquake's epicenter?

    Answer

Section 15.6
  • 15.34 A string with a mass of 30.0 g and a length of 2.00 m is stretched under a tension of 70.0 N. How much power must be supplied to the string to generate a traveling wave that has a frequency of 50.0 Hz and an amplitude of 4.00 cm?

  • 15.35 A string with linear mass density of 0.100 kg/m is under a tension of 100. N. How much power must be supplied to the string to generate a sinusoidal wave of amplitude 2.00 cm and frequency 120. Hz?

    Answer

  • 15.36 A sinusoidal wave on a string is described by the equation y = (0.100 m) sin (0.75x − 40t), where x and y are in meters and t is in seconds. If the linear mass density of the string is 10 g/m, determine (a) the phase constant, (b) the phase of the wave at x = 2.00 cm and t = 0.100 s, (c) the speed of the wave, (d) the wavelength, (e) the frequency, and (f) the power transmitted by the wave.

Section 15.7 and 15.8
  • 15.37 In an acoustics experiment, a piano string with a mass of 5.00 g and a length of 70.0 cm is held under tension by running the string over a frictionless pulley and hanging a 250.-kg weight from it. The whole system is placed in an elevator.

    1. What is the fundamental frequency of oscillation for the string when the elevator is at rest?

    2. With what acceleration and in what direction (up or down) should the elevator move for the string to produce the proper frequency of 440. Hz, corresponding to middle A?

    Answer

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  • 15.38 A string is 35.0 cm long and has a mass per unit length of 5.51 · 10−4 kg/m. What tension must be applied to the string so that it vibrates at the fundamental frequency of 660 Hz?

  • 15.39 A 2.00-m-long string of mass 10.0 g is clamped at both ends. The tension in the string is 150 N.

    1. What is the speed of a wave on this string?

    2. The string is plucked so that it oscillates. What is the wavelength and frequency of the resulting wave if it produces a standing wave with two antinodes?

    Answer

  • 15.40 Write the equation for a standing wave that has three antinodes of amplitude 2.00 cm on a 3.00-m-long string that is fixed at both ends and vibrates 15.0 times a second. The time t = 0 is chosen to be an instant when the string is flat. If a wave pulse were propagated along this string, how fast would it travel?

  • •15.41 A 3.00-m-long string, fixed at both ends, has a mass of 6.00 g. If you want to set up a standing wave in this string having a frequency of 300. Hz and three antinodes, what tension should you put the string under?

    Answer

  • 15.42 A cowboy walks at a pace of about two steps per second, holding a glass of diameter 10.0 cm that contains milk. The milk sloshes higher and higher in the glass until it eventually starts to spill over the top. Determine the maximum speed of the waves in the milk.

  • 15.43 Students in a lab produce standing waves on stretched strings connected to vibration generators. One such wave is described by the wave function where y is the transverse displacement of the string, x is the position along the string, and t is time. Rewrite this wave function in the form for a right-moving and a left-moving wave: y(x,t) = f (xvt)+ g(x+vt); that is, find the functions f and g and the speed, v.

    Answer

  • 15.44 An array of wave emitters, as shown in the figure, emits a wave of wavelength λ that is to be detected at a distance L directly above the rightmost emitter. The distance between adjacent wave emitters is d.

    1. Show that when Ld, the wave from the nth emitter (counting from right to left with n = 0 being the rightmost emitter) has to travel an extra distance of Δs = n2 (d2/2L).

    2. If λ = d2/2L, will the interference at the detector be constructive or destructive?

    3. If λ = d2/2L = 10−3 m and L = 1.00 · 103 m, what is d, the distance between adjacent emitters?

  • ••15.45 A small ball floats in the center of a circular pool that has a radius of 5.00 m. Three wave generators are placed at the edge of the pool, separated by 120°. The first wave generator operates at a frequency of 2.00 Hz. The second wave generator operates at a frequency of 3.00 Hz. The third wave generator operates at a frequency of 4.00 Hz. If the speed of each water wave is 5.00 m/s, and the amplitude of the waves is the same, sketch the height of the ball as a function of time from t = 0 to t = 2.00 s, assuming that the water surface is at zero height. Assume that all the wave generators impart a phase shift of zero. How would your answer change if one of the wave generators was moved to a different location at the edge of the pool?

    Answer

  • ••15.46 A string with linear mass density μ = 0.0250 kg/m under a tension of T = 250. N is oriented in the x-direction. Two transverse waves of equal amplitude and with a phase angle of zero (at t = 0) but with different frequencies (ω = 3000. rad/s and ω/3 = 1000. rad/s) are created in the string by an oscillator located at x = 0. The resulting waves, which travel in the positive x-direction, are reflected at a distant point, so there is a similar pair of waves traveling in the negative x-direction. Find the values of x at which the first two nodes in the standing wave are produced by these four waves.

  • ••15.47 The equation for a standing wave on a string with mass density μ is y(x,t) = 2A cos (ωt) sin (κx). Show that the average kinetic energy and potential energy over time for this wave per unit length are given by Kave(x) = μ ω2 A2 sin2 κx and U ave(x) = TA)2 (cos2 κx).

Additional Problems
  • 15.48 A sinusoidal wave traveling on a string is moving in the positive x-direction. The wave has a wavelength of 4 m, a frequency of 50.0 Hz, and an amplitude of 3.00 cm. What is the wave function for this wave?

  • 15.49 A guitar string with a mass of 10.0 g is 1.00 m long and attached to the guitar at two points separated by 65.0 cm.

    1. What is the frequency of the first harmonic of this string when it is placed under a tension of 81.0 N?

    2. If the guitar string is replaced by a heavier one that has a mass of 16.0 g and is 1.00 m long, what is the frequency of the replacement string's first harmonic?

    Answer

  • 15.50 Write the equation for a sinusoidal wave propagating in the negative x-direction with a speed of 120. m/s, if a particle in the medium in which the wave is moving is observed to swing back and forth through a 6.00-cm range in 4.00 s. Assume that t = 0 is taken to be the instant when the particle is at y = 0 and that the particle moves in the positive y-direction immediately after t = 0.

  • 15.51 Shown in the figure is a plot of the displacement, y, due a sinusoidal wave traveling along a string as a function of time, t. What are the (a) period, (b) maximum speed, and (c) maximum acceleration of this wave perpendicular to the direction that the wave is traveling?

    Answer

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  • 15.52 A 50.0-cm-long wire with a mass of 10.0 g is under a tension of 50.0 N. Both ends of the wire are held rigidly while it is plucked.

    1. What is the speed of the waves on the wire?

    2. What is the fundamental frequency of the standing wave?

    3. What is the frequency of the third harmonic?

  • 15.53 What is the wave speed along a brass wire with a radius of 0.500 mm stretched at a tension of 125 N? The density of brass is 8.60 · 103 kg/m3.

    Answer

  • 15.54 Two steel wires are stretched under the same tension. The first wire has a diameter of 0.500 mm, and the second wire has a diameter of 1.00 mm. If the speed of waves traveling along the first wire is 50.0 m/s, what is the speed of waves traveling along the second wire?

  • 15.55 The middle-C key (key 52) on a piano corresponds to a fundamental frequency of about 262 Hz, and the soprano-C key (key 64) corresponds to a fundamental frequency of 1046.5 Hz. If the strings used for both keys are identical in density and length, determine the ratio of the tensions in the two strings.

    Answer

  • 15.56 A sinusoidal wave travels along a stretched string. A point along the string has a maximum velocity of 1.00 m/s and a maximum displacement of 2.00 cm. What is the maximum acceleration at that point?

  • •15.57 As shown in the figure, a sinusoidal wave travels to the right at a speed of v1 along string 1, which has linear mass density μ1. This wave has frequency f1 and wavelength λ1. Since string 1 is attached to string 2 (which has linear mass density μ2 = 3μ1), the first wave will excite a new wave in string 2, which will also move to the right. What is the frequency, f2 of the wave produced in string 2? What is the speed, v2, of the wave produced in string 2? What is the wavelength, λ2, of the wave produced in string 2? Write all answers in terms of f1, v1, and λ1.

    Answer

  • 15.58 The tension in a 2.7-m-long, 1.0-cm-diameter steel cable (ρ = 7800 kg/m3) is 840 N. What is the fundamental frequency of vibration of the cable?

  • 15.59 A wave traveling on a string has the equation of motion y(x,t) = 0.02 sin (5.00x − 8.00t).

    1. Calculate the wavelength and the frequency of the wave.

    2. Calculate its velocity.

    3. If the linear mass density of the string is μ = 0.10 kg/m, what is the tension on the string?

    Answer

  • 15.60 Calvin sloshes back and forth in his bathtub, producing a standing wave. What is the frequency of such a wave if the bathtub is 150. cm long and 80.0 cm wide and contains water that is 38.0 cm deep?

  • •15.61 Consider a guitar string stretching 80.0 cm between its anchored ends. The string is tuned to play middle C, with a frequency of 256 Hz, when oscillating in its fundamental mode, that is, with one antinode between the ends. If the string is displaced 2.00 mm at its midpoint and released to produce this note, what are the wave speed, v, and the maximum speed, Vmax, of the midpoint of the string?

    Answer

  • 15.62 The largest tension that can be sustained by a stretched string of linear mass density μ, even in principle, is given by τ = μc2, where c is the speed of light in vacuum. (This is an enormous value. The breaking tensions of all ordinary materials are about 12 orders of magnitude less than this.)

    1. What is the speed of a traveling wave on a string under such tension?

    2. If a 1.000-m-long guitar string, stretched between anchored ends, were made of this hypothetical material, what frequency would its first harmonic have?

    3. If that guitar string were plucked at its midpoint and given a displacement of 2.00 mm there to produce the fundamental frequency, what would be the maximum speed attained by the midpoint of the string?

  • 15.63 A rubber band of mass 0.21 g is stretched between two fingers, putting it under a tension of 2.8 N. The overall stretched length of the band is 21.3 cm. One side of the band is plucked, setting up a vibration in 8.7 cm of the band's stretched length. What is the lowest frequency of vibration that can be set up on this part of the rubber band? Assume that the band stretches uniformly.

    Answer

  • 15.64 Two waves traveling in opposite directions along a string fixed at both ends create a standing wave described by y(x,t) = 1.00 · 10−2 sin (25x) cos (1200t). The string has a linear mass density of 0.01 kg/m, and the tension in the string is supplied by a mass hanging from one end. If the string vibrates in its third harmonic, calculate (a) the length of the string, (b) the velocity of the waves, and (c) the mass of the hanging mass.

  • 15.65 A sinusoidal transverse wave of wavelength 20.0 cm and frequency 500. Hz travels along a string in the positive z-direction. The wave oscillations take place in the xz-plane and have an amplitude of 3.00 cm. At time t =0, the displacement of the string at x = 0 is z = 3.00 cm.

    1. A photo of the wave is taken at t = 0. Make a simple sketch (including axes) of the string at this time.

    2. Determine the speed of the wave.

    3. Determine the wave's wave number.

    4. If the linear mass density of the string is 30.0 g/m, what is the tension in the string?

    5. Determine the function D(z,t) that describes the displacement x that is produced in the string by this wave.

    Answer

  • ••15.66 A heavy cable of total mass M and length L = 5.0 m has one end attached to a rigid support and the other end hanging free. A small transverse displacement is initiated at the bottom of the cable. How long does it take for the displacement to travel to the top of the cable?