Chapter 16

The equation of a transverse wave traveling along a string is $$ y=(2.0 \mathrm{~mm}) \sin \left[\left(20 \mathrm{~m}^{-1}\right) x-\left(600 \mathrm{~s}^{-1}\right) t\right] . $$ 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.

Two sinusoidal waves, identical except for phase, travel in the same direction along a string, producing the net wave \(y^{\prime}(x, t)=(3.0 \mathrm{~mm}) \sin (20 x-4.0 t+0.820 \mathrm{rad})\), with \(x\) in meters and \(t\) in seconds. What are (a) the wavelength \(\lambda\) of the two waves. (b) the phase difference between them, and (c) their amplitude \(y_{m}\) ?

Three sinusoidal waves of the same frequency travel along a string in the positive direction of an \(x\) axis. Their amplitudes are \(y_{1}, y_{1} / 2\), and \(y_{1} / 3\), and their phase constants are \(0, \pi / 2\), and \(\pi\), 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.

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 \(1.00 \mathrm{~cm}\). The motion is continuous and is repeated regularly 120 times per second. The string has linear density 120 \(\mathrm{g} / \mathrm{m}\) and is kept under a tension of \(90.0 \mathrm{~N}\). Find the maximum value of (a) the transverse speed \(u\) and (b) the transverse component of the tension \(\tau\). (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?

At time \(t=0\) and at position \(x=0 \mathrm{~m}\) along a string, a traveling sinusoidal wave with an angular frequency of \(440 \mathrm{rad} / \mathrm{s}\) has displacement \(y=+4.5 \mathrm{~mm}\) and transverse velocity \(u=-0.75 \mathrm{~m} / \mathrm{s}\). If the wave has the general form \(y(x, t)=y_{m} \sin (k x-\omega t+\phi)\), what is phase constant \(\phi ?\)

(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 \(7.00 \times 10^{8} \mathrm{~N} / \mathrm{m}^{2}\). The density of steel is \(7800 \mathrm{~kg} / \mathrm{m}^{3} .\) (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 $$ y_{1}=0.050 \cos (\pi x-4 \pi t) $$ and $$ y_{2}=0.050 \cos (\pi x+4 \pi t) $$ where \(x, y_{1}\), and \(y_{2}\) 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 \(\ell\) and a mass \(m .\) When a force \(F\) is applied, the band stretches an additional length \(\Delta \ell .\) (a) What is the speed (in terms of \(m, \Delta \ell\), 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 \(1 / \sqrt{\Delta \ell}\) if \(\Delta \ell \& \ell\) and is constant if \(\Delta \ell\) ? \(\ell\).

The speed of electromagnetic waves (which include visible light, radio, and \(x\) rays) in vacuum is \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\). (a) Wavelengths of visible light waves range from about \(400 \mathrm{~nm}\) in the violet to about \(700 \mathrm{~nm}\) 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 \(300 \mathrm{MHz}\). What is the corresponding wavelength range? (c) X-ray wavelengths range from about \(5.0 \mathrm{~nm}\) to about \(1.0 \times 10^{-2} \mathrm{~nm} .\) What is the frequency range for \(\mathrm{x}\) rays?

A \(1.50 \mathrm{~m}\) wire has a mass of \(8.70 \mathrm{~g}\) and is under a tension of \(120 \mathrm{~N}\). 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 \(\mathrm{A}(440 \mathrm{~Hz}) .\) 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 \(1.00 \mathrm{~cm}\), a frequency of \(550 \mathrm{~Hz}\), and a speed of \(330 \mathrm{~m} / \mathrm{s}\). If the wave equation is of the form \(y(x, t)=y_{m} \sin (k x \pm \omega t)\), what are (a) \(y_{m},(\) b) \(\omega,(\mathrm{c}) k\), and \((\mathrm{d})\) the correct choice of sign in front of \(\omega ?\)

Two sinusoidal waves of the same wavelength travel in the same direction along a stretched string. For wave \(1, y_{m}=3.0 \mathrm{~mm}\) and \(\phi=\) \(0 ;\) for wave \(2, y_{m}=5.0 \mathrm{~mm}\) and \(\phi=70^{\circ} .\) What are the (a) amplitude and (b) phase constant of the resultant wave?

A sinusoidal transverse wave of amplitude \(y_{m}\) and wavelength \(\lambda\) 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 \(600 \mathrm{~Hz}\) tuning fork sets up standing waves in a string clamped at both ends. The wave speed for the string is \(400 \mathrm{~m} / \mathrm{s}\). The standing wave has four loops and an amplitude of \(2.0 \mathrm{~mm}\). (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 \(120 \mathrm{~cm}\) 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 cquation describing a sinusoidal transverse wave traveling on a cord in the positive direction of a \(y\) axis with an angular wave number of \(60 \mathrm{~cm}^{-1}\), a period of \(0.20 \mathrm{~s}\), and an amplitude of \(3.0 \mathrm{~mm}\). 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 $$ y(x, t)=15.0 \sin (\pi x / 8-4 \pi t), $$ where \(x\) and \(y\) are in centimeters and \(t\) is in seconds. (a) What is the transverse speed for a point on the string at \(x=6.00 \mathrm{~cm}\) when \(t=0.250 \mathrm{~s}\) ? (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 \(x=6.00 \mathrm{~cm}\) when \(t=0.250 \mathrm{~s}\) ? (d) What is the magnitude of the maximum transverse acceleration for any point on the string?

a 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 \(v_{l}\) 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. \(16-48 a\). Part of the projectile's energy goes into this motion and stretching. The transverse pulse, moving at a slower speed \(v_{t}\), 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 \(16-48 b\) 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 \(v_{s}=\) \(300 \mathrm{~m} / \mathrm{s}\) and \(t_{s}=40.0 \mu \mathrm{s}\). Take \(v_{l}=2000 \mathrm{~m} / \mathrm{s}\), and assume that the half-angle \(\theta\) of the conical dent is \(60^{\circ}\). 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 $$ y_{1}=0.30 \sin [\pi(5 x-200 t)] $$ and $$ y_{2}=0.30 \sin [\pi(5 x-200 t)+\pi / 3] $$ where \(y_{1}, y_{2}\), 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?

93 A traveling wave on a string is described by $$ y=2.0 \sin \left[2 \pi\left(\frac{t}{0.40}+\frac{x}{80}\right)\right] $$ where \(x\) and \(y\) are in centimeters and \(t\) is in seconds. (a) For \(t=0\), plot \(y\) as a function of \(x\) for \(0 \leq x \leq 160 \mathrm{~cm} .\) (b) Repeat (a) for \(t=0.05 \mathrm{~s}\) and \(t=0.10 \mathrm{~s}\). From your graphs, determine (c) the wave speed and (d) the direction in which the wave is traveling.

A continuous traveling wave with amplitude \(A\) is incident on a boundary. The continuous reflection, with a smaller amplitude \(B\), travels back through the incoming wave. The resulting interference pattern is displayed in Fig. 16-51. The standing wave ratio is defined to be \(\mathrm{SWR}=\frac{A+B}{A-B}\) The reflection coefficient \(R\) is the ratio of the power of the reflected wave to the power of the incoming wave and is thus proportional to the ratio \((B / A)^{2} .\) What is the SWR for (a) total reflection and (b) no reflection? (c) For SWR \(=1.50\), what is \(R\) expressed as a percentage?