The wave is a particular type of pulse that can propagate

If one end of a heavy rope is attached to one end of a lightweight rope, a wave can move from the heavy rope into the lighter one. (i) What happens to the speed of the wave? (a) It increases. (b) It decreases. (c) It is constant. (d) It changes unpredictably. (ii) What happens to the frequency? Choose from the same possibilities. (iii) What happens to the wavelength? Choose from the same possibilities.

If you stretch a rubber hose and pluck it, you can observe a pulse traveling up and down the hose. (i) What happens to the speed of the pulse if you stretch the hose more tightly? (a) It increases. (b) It decreases. (c) It is constant. (d) It changes unpredictably. (ii) What happens to the speed if you fill the hose with water? Choose from the same possibilities.

Rank the waves represented by the following functions from the largest to the smallest according to (i) their amplitudes, (ii) their wavelengths, (iii) their frequencies, (iv) their periods, and (v) their speeds. If the values of a quantity are equal for two waves, show them as having equal rank. For all functions, x and y are in meters and t is in seconds. (a) y 5 4 sin (3x 2 15t) (b) y 5 6 cos (3x 1 15t 2 2) (c) y 5 8 sin (2x 1 15t) (d) y 5 8 cos (4x 1 20t) (e) y 5 7 sin (6x 2 24t)

By what factor would you have to multiply the tension in a stretched string so as to double the wave speed? Assume the string does not stretch. (a) a factor of 8 (b) a factor of 4 (c) a factor of 2 (d) a factor of 0.5 (e) You could not change the speed by a predictable factor by changing the tension.

When all the strings on a guitar (Fig. OQ16.5) are stretched to the same tension, will the speed of a wave along the most massive bass string be (a) faster, (b) slower, or (c) the same as the speed of a wave on the lighter strings? Alternatively, (d) is the speed on the bass string not necessarily any of these answers?

Which of the following statements is not necessarily true regarding mechanical waves? (a) They are formed by some source of disturbance. (b) They are sinusoidal in nature. (c) They carry energy. (d) They require a medium through which to propagate. (e) The wave speed depends on the properties of the medium in which they travel.

(a) Can a wave on a string move with a wave speed that is greater than the maximum transverse speed vy,max of an element of the string? (b) Can the wave speed be much greater than the maximum element speed? (c) Can the wave speed be equal to the maximum element speed? (d) Can the wave speed be less than vy,max?

A source vibrating at constant frequency generates a sinusoidal wave on a string under constant tension. If the power delivered to the string is doubled, by what factor does the amplitude change? (a) a factor of 4 (b) a factor of 2 (c) a factor of !2 (d) a factor of 0.707 (e) cannot be predicted

The distance between two successive peaks of a sinusoidal wave traveling along a string is 2 m. If the frequency of this wave is 4 Hz, what is the speed of the wave? (a) 4 m/s (b) 1 m/s (c) 8 m/s (d) 2 m/s (e) impossible to answer from the information given

When a particular wire is vibrating with a frequency of 4.00 Hz, a transverse wave of wavelength 60.0 cm is produced. Determine the speed of waves along the wire.

The string shown in Figure P16.11 is driven at a frequency of 5.00 Hz. The amplitude of the motion is A 5 12.0 cm, and the wave speed is v 5 20.0 m/s. Furthermore, the wave is such that y 5 0 at x 5 0 and t 5 0. Determine (a) the angular frequency and (b) the wave number for this wave. (c) Write an expression for the wave function. Calculate (d) the maximum transverse speed and (e) the maximum transverse acceleration of an element of the string.

Consider the sinusoidal wave of Example 16.2 with the wave function y 5 0.150 cos (15.7x 2 50.3t) where x and y are in meters and t is in seconds. At a certain instant, let point A be at the origin and point B be the closest point to A along the x axis where the wave is 60.0 out of phase with A. What is the coordinate of B?

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. At t 5 0, the left end of the string is at the origin. For this wave, find (a) the frequency, (b) the angular frequency, (c) the angular wave number, and (d) the wave function in SI units. Determine the equation of motion in SI units for (e) the left end of the string and (f) the point on the string at x 5 1.50 m to the right of the left end. (g) What is the maximum speed of any element of the string?

(a) Plot y versus t at x 5 0 for a sinusoidal wave of the form y 5 0.150 cos (15.7x 2 50.3t), where x and y are in meters and t is in seconds. (b) Determine the period of vibration. (c) State how your result compares with the value found in Example 16.2.

A transverse wave on a string is described by the wave function y 5 0.120 sin a p 8 x 1 4ptb where x and y are in meters and t is in seconds. Determine (a) the transverse speed and (b) the transverse acceleration at t 5 0.200 s for an element of the string located at x 5 1.60 m. What are (c) the wavelength, (d) the period, and (e) the speed of propagation of this wave?

A wave on a string is described by the wave function y 5 0.100 sin (0.50x 2 20t), where x and y are in meters and t is in seconds. (a) Show that an element of the string at x 5 2.00 m executes harmonic motion. (b) Determine the frequency of oscillation of this particular element.

A sinusoidal wave is described by the wave function y 5 0.25 sin (0.30x 2 40t) where x and y are in meters and t is in seconds. Determine for this wave (a) the amplitude, (b) the angular frequency, (c) the angular wave number, (d) the wavelength, (e) the wave speed, and (f) the direction of motion.

A sinusoidal wave traveling in the negative x direction (to the left) has an amplitude of 20.0 cm, a wavelength of 35.0 cm, and a frequency of 12.0 Hz. The transverse position of an element of the medium at t 5 0, x 5 0 is y 5 23.00 cm, and the element has a positive velocity here. We wish to find an expression for the wave function describing this wave. (a) Sketch the wave at t 5 0. (b) Find the angular wave number k from the wavelength. (c) Find the period T from the frequency. Find (d) the angular frequency v and (e) the wave speed v. (f) From the information about t 5 0, find the phase constant f. (g) Write an expression for the wave function y(x, t).

(a) Write the expression for y as a function of x and t in SI units for a sinusoidal wave traveling along a rope in the negative x direction with the following characteristics: A 5 8.00 cm, l 5 80.0 cm, f 5 3.00 Hz, and y(0, t) 5 0 at t 5 0. (b) What If? Write the expression for y as a function of x and t for the wave in part (a) assuming y(x, 0) 5 0 at the point x 5 10.0 cm.

A transverse sinusoidal wave on a string has a period T 5 25.0 ms and travels in the negative x direction with a speed of 30.0 m/s. At t 5 0, an element of the string at x 5 0 has a transverse position of 2.00 cm and is traveling downward with a speed of 2.00 m/s. (a) What is the amplitude of the wave? (b) What is the initial phase angle? (c) What is the maximum transverse speed of an element of the string? (d) Write the wave function for the wave.

Review. The elastic limit of a steel wire is 2.70 3 108 Pa. What is the maximum speed at which transverse wave pulses can propagate along this wire without exceeding this stress? (The density of steel is 7.86 3 103 kg/m3.)

A piano string having a mass per unit length equal to 5.00 3 1023 kg/m is under a tension of 1 350 N. Find the speed with which a wave travels on this string.

Transverse waves travel with a speed of 20.0 m/s on a string under a tension of 6.00 N. What tension is required for a wave speed of 30.0 m/s on the same string?

A student taking a quiz finds on a reference sheet the two equations f 5 1 T and v 5 T m She has forgotten what T represents in each equation. (a) Use dimensional analysis to determine the units required for T in each equation. (b) Explain how you can identify the physical quantity each T represents from the units.

An Ethernet cable is 4.00 m long. The cable has a mass of 0.200 kg. A transverse pulse is produced by plucking one end of the taut cable. The pulse makes four trips down and back along the cable in 0.800 s. What is the tension in the cable?

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. (a) Write an equation in SI units of the form y 5 A sin (kx 2 vt) for this wave. (b) The mass per unit length of this wire is 4.10 g/m. Find the tension in the wire.

A steel wire of length 30.0 m and a copper wire of length 20.0 m, both with 1.00-mm diameters, are connected end to end and stretched to a tension of 150 N. During what time interval will a transverse wave travel the entire length of the two wires?

Why is the following situation impossible? An astronaut on the Moon is studying wave motion using the apparatus discussed in Example 16.3 and shown in Figure 16.12. He measures the time interval for pulses to travel along the horizontal wire. Assume the horizontal wire has a mass of 4.00 g and a length of 1.60 m and assume a 3.00-kg object is suspended from its extension around the pulley. The astronaut finds that a pulse requires 26.1 ms to traverse the length of the wire.

Tension is maintained in a string as in Figure P16.29. The observed wave speed is v 5 24.0 m/s when the suspended mass is m 5 3.00 kg. (a) What is the mass per unit length of the string? (b) What is the wave speed when the suspended mass is m 5 2.00 kg?

Review. 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, p. 503). 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 if the wave speed is to be 60.0m/s?

Transverse pulses travel with a speed of 200 m/s along a taut copper wire whose diameter is 1.50 mm. What is the tension in the wire? (The density of copper is 8.92 g/cm3.)

In a region far from the epicenter of an earthquake, a seismic wave can be modeled as transporting energy in a single direction without absorption, just as a string wave does. Suppose the seismic wave moves from granite into mudfill with similar density but with a much smaller bulk modulus. Assume the speed of the wave gradually drops by a factor of 25.0, with negligible reflection of the wave. (a) Explain whether the amplitude of the ground shaking will increase or decrease. (b) Does it change by a predictable factor? (This phenomenon led to the collapse of part of the Nimitz Freeway in Oakland, California, during the Loma Prieta earthquake of 1989.)

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?

Sinusoidal waves 5.00 cm in amplitude are to be transmitted along a string that has a linear mass density of 4.00 3 1022 kg/m. The source can deliver a maximum power of 300 W, and the string is under a tension of 100 N. What is the highest frequency f at which the source can operate?

A sinusoidal wave on a string is described by the wave function y 5 0.15 sin (0.80x 2 50t) where x and y are in meters and t is in seconds. The mass per unit length of this string is 12.0 g/m. Determine (a) the speed of the wave, (b) the wavelength, (c) the frequency, and (d) the power transmitted by the wave.

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 so as 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?

A long string carries a wave; a 6.00-m segment of the string contains four complete wavelengths and has a mass of 180 g. The string vibrates sinusoidally with a frequency of 50.0 Hz and a peak-to-valley displacement of 15.0 cm. (The peak-to-valley distance is the vertical distance from the farthest positive position to the farthest negative position.) (a) Write the function that describes this wave traveling in the positive x direction. (b) Determine the power being supplied to the string.

A horizontal string can transmit a maximum power P0 (without breaking) if a wave with amplitude A and angular frequency v is traveling along it. To increase this maximum power, a student folds the string and uses this double string as a medium. Assuming the tension in the two strands together is the same as the original tension in the single string and the angular frequency of the wave remains the same, determine the maximum power that can be transmitted along the double string.

The wave function for a wave on a taut string is y1x, t 2 5 0.350 sin a10pt 2 3px 1 p 4 b where x and y are in meters and t is in seconds. If the linear mass density of the string is 75.0 g/m, (a) what is the average rate at which energy is transmitted along the string? (b) What is the energy contained in each cycle of the wave?

A two-dimensional water wave spreads in circular ripples. Show that the amplitude A at a distance r from the initial disturbance is proportional to 1/!r. Suggestion: Consider the energy carried by one outwardmoving ripple.

Show that the wave function y 5 ln [b(x 2 vt)] is a solution to Equation 16.27, where b is a constant.

(a) Evaluate A in the scalar equality 4 (7 1 3) 5 A. (b) Evaluate A, B, and C in the vector equality 700 i ^ 1 3.00 k^ 5 Ai ^ 1 B j ^ 1 C k^. (c) Explain how you arrive at the answers to convince a student who thinks that you cannot solve a single equation for three different unknowns. (d) What If? The functional equality or identity A 1 B cos (Cx 1 Dt 1 E) 5 7.00 cos (3x 1 4t 1 2) is true for all values of the variables x and t, measured in meters and in seconds, respectively. Evaluate the constants A, B, C, D, and E. (e) Explain how you arrive at your answers to part (d).

Show that the wave function y 5 e b(x2vt) is a solution of the linear wave equation (Eq. 16.27), where b is a constant.

(a) Show that the function y(x, t) 5 x2 1 v2 t 2 is a solution to the wave equation. (b) Show that the function in part (a) can be written as f(x 1 vt) 1 g(x 2 vt) and determine the functional forms for f and g. (c) What If? Repeat parts (a) and (b) for the function y(x, t) 5 sin (x) cos (vt).

Motion-picture film is projected at a frequency of 24.0 frames per second. Each photograph on the film is the same height of 19.0 mm, just like each oscillation in a wave is the same length. Model the height of a frame as the wavelength of a wave. At what constant speed does the film pass into the projector?

The wave is a particular type of pulse that can propagate through a large crowd gathered at a sports arena (Fig. P16.46). The elements of the medium are the spectators, with zero position corresponding to their being seated and maximum position corresponding to their standing and raising their arms. When a large fraction of the spectators participates in the wave motion, a somewhat stable pulse shape can develop. The wave speed depends on peoples reaction time, which is typically on the order of 0.1 s. Estimate the order of magnitude, in minutes, of the time interval required for such a pulse to make one circuit around a large sports stadium. State the quantities you measure or estimate and their values.

A sinusoidal wave in a rope is described by the wave function y 5 0.20 sin (0.75px 1 18pt) where x and y are in meters and t is in seconds. The rope has a linear mass density of 0.250 kg/m. The tension in the rope is provided by an arrangement like the one illustrated in Figure P16.29. What is the mass of the suspended object?

The ocean floor is underlain by a layer of basalt that constitutes the crust, or uppermost layer, of the Earth in that region. Below this crust is found denser periodotite rock that forms the Earths mantle. The boundary between these two layers is called the Mohorovicic discontinuity (Moho for short). If an explosive charge is set off at the surface of the basalt, it generates a seismic wave that is reflected back out at the Moho. If the speed of this wave in basalt is 6.50 km/s and the two-way travel time is 1.85 s, what is the thickness of this oceanic crust?

Review. A 2.00-kg block hangs from a rubber cord, being supported so that the cord is not stretched. The unstretched length of the cord is 0.500 m, and its mass is 5.00 g. The spring constant for the cord is 100 N/m. The block is released and stops momentarily at the lowest point. (a) Determine the tension in the cord when the block is at this lowest point. (b) What is the length of the cord in this stretched position? (c) If the block is held in this lowest position, find the speed of a transverse wave in the cord.

Review. A block of mass M hangs from a rubber cord. The block is supported so that the cord is not stretched. The unstretched length of the cord is L0, and its mass is m, much less than M. The spring constant for the cord is k. The block is released and stops momentarily at the lowest point. (a) Determine the tension in the string when the block is at this lowest point. (b) What is the length of the cord in this stretched position? (c) If the block is held in this lowest position, find the speed of a transverse wave in the cord.

A transverse wave on a string is described by the wave function y(x, t) 5 0.350 sin (1.25x 1 99.6t) where x and y are in meters and t is in seconds. Consider the element of the string at x 5 0. (a) What is the time interval between the first two instants when this element has a position of y 5 0.175 m? (b) What distance does the wave travel during the time interval found in part (a)?

A sinusoidal wave in a string is described by the wave function y 5 0.150 sin (0.800x 2 50.0t) where x and y are in meters and t is in seconds. The mass per length of the string is 12.0 g/m. (a) Find the maximum transverse acceleration of an element of this string. (b) Determine the maximum transverse force on a 1.00-cm segment of the string. (c) State how the force found in part (b) compares with the tension in the string.

Review. A block of mass M, supported by a string, rests on a frictionless incline making an angle u with the horizontal (Fig. P16.53). 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.

An undersea earthquake or a landslide can produce an ocean wave of short duration carrying great energy, called a tsunami. When its wavelength is large compared to the ocean depth d, the speed of a water wave is given approximately by v 5 !gd. Assume an earthquake occurs all along a tectonic plate boundary running north to south and produces a straight tsunami wave crest moving everywhere to the west. (a) What physical quantity can you consider to be constant in the motion of any one wave crest? (b) Explain why the amplitude of the wave increases as the wave approaches shore. (c) If the wave has amplitude 1.80 m when its speed is 200 m/s, what will be its amplitude where the water is 9.00 m deep? (d) Explain why the amplitude at the shore should be expected to be still greater, but cannot be meaningfully predicted by your model.

Review. A block of mass M 5 0.450 kg is attached to one end of a cord of mass 0.003 20 kg; the other end of the cord is attached to a fixed point. The block rotates with constant angular speed in a circle on a frictionless, horizontal table as shown in Figure P16.55. Through what angle does the block rotate in the time interval during which a transverse wave travels along the string from the center of the circle to the block?

Review. A block of mass M 5 0.450 kg is attached to one end of a cord of mass m 5 0.003 20 kg; the other end of the cord is attached to a fixed point. The block rotates with constant angular speed v 5 10.0 rad/s in a circle on a frictionless, horizontal table as shown in Figure P16.55. What time interval is required for a transverse wave to travel along the string from the center of the circle to the block?

Review. A block of mass M is attached to one end of a cord of mass m; the other end of the cord is attached to a fixed point. The block rotates with constant angular speed v in a circle on a frictionless, horizontal table as shown in Figure P16.55. What time interval is required for a transverse wave to travel along the string from the center of the circle to the block?

A string with linear density 0.500 g/m is held under tension 20.0 N. As a transverse sinusoidal wave propagates on the string, elements of the string move with maximum speed vy,max. (a) Determine the power transmitted by the wave as a function of vy,max. (b) State in words the proportionality between power and vy,max. (c) Find the energy contained in a section of string 3.00 m long as a function of vy,max. (d) Express the answer to part (c) in terms of the mass m of this section. (e) Find the energy that the wave carries past a point in 6.00 s.

A wire of density r is tapered so that its cross-sectional area varies with x according to A 5 1.00 3 1025 x 1 1.00 3 1026 where A is in meters squared and x is in meters. The tension in the wire is T. (a) Derive a relationship for the speed of a wave as a function of position. (b) What If? Assume the wire is aluminum and is under a tension T 5 24.0 N. Determine the wave speed at the origin and at x 5 10.0 m.

A rope of total mass m and length L is suspended vertically. Analysis shows that for short transverse pulses, the waves above a short distance from the free end of the rope can be represented to a good approximation by the linear wave equation discussed in Section 16.6. Show that a transverse pulse travels the length of the rope in a time interval that is given approximately by Dt < 2!L/g. Suggestion: First find an expression for the wave speed at any point a distance x from the lower end by considering the ropes tension as resulting from the weight of the segment below that point.

A pulse traveling along a string of linear mass density m is described by the wave function y 5 [A0e2bx] sin (kx 2 vt) where the factor in brackets is said to be the amplitude. (a) What is the power P(x) carried by this wave at a point x? (b) What is the power P(0) carried by this wave at the origin? (c) Compute the ratio P(x)/P(0).

Why is the following situation impossible? Tsunamis are ocean surface waves that have enormous wavelengths (100 to 200 km), and the propagation speed for these waves is v < !gdavg, where davg is the average depth of the water. An earthquake on the ocean floor in the Gulf of Alaska produces a tsunami that reaches Hilo, Hawaii, 4 450 km away, in a time interval of 5.88 h. (This method was used in 1856 to estimate the average depth of the Pacific Ocean long before soundings were made to give a direct determination.)

Review. An aluminum wire is held between two clamps under zero tension at room temperature. Reducing the temperature, which results in a decrease in the wires equilibrium length, increases the tension in the wire. Taking the cross-sectional area of the wire to be 5.00 3 1026 m2, the density to be 2.70 3 103 kg/m3, and Youngs modulus to be 7.00 3 1010 N/m2, what strain (DL/L) results in a transverse wave speed of 100 m/s?

Assume an object of mass M is suspended from the bottom of the rope of mass m and length L in Problem 60. (a) Show that the time interval for a transverse pulse to travel the length of the rope is Dt 5 2 L mg 1"M 1 m 2 "M2 (b) What If? Show that the expression in part (a) reduces to the result of Problem 60 when M 5 0. (c) Show that for m ,, M, the expression in part (a) reduces to Dt 5 mL Mg

A rope of total mass m and length L is suspended vertically. As shown in Problem 60, a pulse travels from the bottom to the top of the rope in an approximate time interval Dt 5 2!L/g with a speed that varies with position x measured from the bottom of the rope as v 5 !gx . Assume the linear wave equation in Section 16.6 describes waves at all locations on the rope. (a) Over what time interval does a pulse travel halfway up the rope? Give your answer as a fraction of the quantity 2!L/g. (b) A pulse starts traveling up the rope. How far has it traveled after a time interval !L/g ?

A string on a musical instrument is held under tension T and extends from the point x 5 0 to the point x 5 L. The string is overwound with wire in such a way that its mass per unit length m(x) increases uniformly from m0 at x 5 0 to mL at x 5 L. (a) Find an expression for m(x) as a function of x over the range 0 # x # L. (b) Find an expression for the time interval required for a transverse pulse to travel the length of the string.

If a loop of chain is spun at high speed, it can roll along the ground like a circular hoop without collapsing. Consider a chain of uniform linear mass density m whose center of mass travels to the right at a high speed v0 as shown in Figure P16.67. (a) Determine the tension in the chain in terms of m and v0. Assume the weight of an individual link is negligible compared to the tension. (b) If the loop rolls over a small bump, the resulting deformation of the chain causes two transverse pulses to propagate along the chain, one moving clockwise and one moving counterclockwise. What is the speed of the pulses traveling along the chain? (c) Through what angle does each pulse travel during the time interval over which the loop makes one revolution?