You Can Decrease the Frequency of a Standing Wave on a String by __________.

16 Waves

16.half dozen Continuing Waves and Resonance

Learning Objectives

By the finish of this section, you lot will be able to:

Describe standing waves and explain how they are produced
Describe the modes of a standing moving ridge on a string
Provide examples of standing waves beyond the waves on a cord

Throughout this affiliate, we have been studying traveling waves, or waves that transport energy from i place to another. Under certain weather condition, waves tin bounce back and forth through a particular region, effectively becoming stationary. These are called standing waves.

Another related upshot is known as resonance. In Oscillations, we defined resonance as a phenomenon in which a pocket-size-aamplitude driving force could produce large-amplitude motility. Think of a child on a swing, which can exist modeled as a physical pendulum. Relatively pocket-size-aamplitude pushes by a parent can produce large-amplitude swings. Sometimes this resonance is good—for example, when producing music with a stringed instrument. At other times, the effects can exist devastating, such every bit the collapse of a building during an earthquake. In the case of standing waves, the relatively big amplitude standing waves are produced by the superposition of smaller aamplitude component waves.

Standing Waves

Sometimes waves do not seem to motility; rather, they just vibrate in place. You can see unmoving waves on the surface of a drinking glass of milk in a fridge, for example. Vibrations from the fridge motor create waves on the milk that oscillate upwardly and downward but practice not seem to move across the surface. (Figure) shows an experiment you tin effort at domicile. Accept a bowl of milk and place it on a common box fan. Vibrations from the fan will produce circular standing waves in the milk. The waves are visible in the photo due to the reflection from a lamp. These waves are formed by the superposition of 2 or more traveling waves, such as illustrated in (Figure) for 2 identical waves moving in opposite directions. The waves move through each other with their disturbances adding every bit they become by. If the ii waves take the same amplitude and wavelength, then they alternate between constructive and destructive interference. The resultant looks similar a wave continuing in place and, thus, is chosen a standing wave.

Photograph shows waves on the surface of a bowl of milk sitting on a box fan. — **Figure 16.25** Continuing waves are formed on the surface of a bowl of milk sitting on a box fan. The vibrations from the fan causes the surface of the milk of oscillate. The waves are visible due to the reflection of lite from a lamp.

Figure shows 8 time snapshots of two identical sine waves and a resultant wave, taken at intervals of 1 by 8 T. At t=0T and t = half T the two sine waves are in phase and the resultant wave has twice the amplitude of the two individual waves. At t = 1 by 4 T and t = 3 by 4 T, the two sine waves are opposite in phase and there is no resultant wave present. — **figure 16.26** Time snapshots of two sine waves. The ruddy moving ridge is moving in the −x-management and the blue wave is moving in the +ten-direction. The resulting wave is shown in black. Consider the resultant wave at the points
$x=0\,\text{m},3\,\text{m},6\,\text{m},9\,\text{m},12\,\text{m},15\,\text{m}$

and detect that the resultant moving ridge always equals nil at these points, no affair what the fourth dimension is. These points are known as fixed points (nodes). In between each two nodes is an antinode, a place where the medium oscillates with an aamplitude equal to the sum of the amplitudes of the individual waves.

\[x=0\,\text{m},3\,\text{m},6\,\text{m},9\,\text{m},12\,\text{m},15\,\text{m}\] — **figure 16.26** Time snapshots of two sine waves. The ruddy moving ridge is moving in the −x-management and the blue wave is moving in the +ten-direction. The resulting wave is shown in black. Consider the resultant wave at the points
$x=0\,\text{m},3\,\text{m},6\,\text{m},9\,\text{m},12\,\text{m},15\,\text{m}$

and detect that the resultant moving ridge always equals nil at these points, no affair what the fourth dimension is. These points are known as fixed points (nodes). In between each two nodes is an antinode, a place where the medium oscillates with an aamplitude equal to the sum of the amplitudes of the individual waves.

Consider two identical waves that motility in opposite directions. The first wave has a wave office of

${y}_{1}(x,t)=A\,\text{sin}(kx-\omega t)$

and the second wave has a wave office

${y}_{2}(x,t)=A\,\text{sin}(kx+\omega t)$

. The waves interfere and form a resultant wave

$\begin{array}{c}y(x,t)={y}_{1}(x,t)+{y}_{2}(x,t),\hfill \\ y(x,t)=A\,\text{sin}(kx-\omega t)+A\,\text{sin}(kx+\omega t).\hfill \end{array}$

This tin be simplified using the trigonometric identity

$\text{sin}(\alpha ±\beta )=\text{sin}\,\alpha \,\text{cos}\,\beta ±\text{cos}\,\alpha \,\text{sin}\,\beta ,$

where

$\alpha =kx$

and

$\beta =\omega t$

, giving us

$y(x,t)=A[\text{sin}(kx)\text{cos}(\omega t)-\text{cos}(kx)\text{sin}(\omega t)+\text{sin}(kx)\text{cos}(\omega t)-\text{cos}(kx)\text{sin}(\omega t)],$

which simplifies to

$y(x,t)=[2A\,\text{sin}(kx)]\text{cos}(\omega t).$

Notice that the resultant wave is a sine wave that is a function only of position, multiplied past a cosine function that is a function only of time. Graphs of y(x,t) as a part of x for various times are shown in (Figure). The red wave moves in the negative 10-direction, the blue wave moves in the positive x-direction, and the black wave is the sum of the 2 waves. As the cerise and bluish waves move through each other, they move in and out of constructive interference and destructive interference.

Initially, at time

$t=0,$

the two waves are in phase, and the result is a wave that is twice the amplitude of the individual waves. The waves are also in phase at the time

$t=\frac{T}{2}.$

In fact, the waves are in stage at any integer multiple of one-half of a menses:

$t=n\frac{T}{2}\,\text{where}\,n=0,1,2,3\text{...}.\,\text{(in phase)}.$

At other times, the two waves are

$180\text{°}(\pi \,\text{radians})$

out of phase, and the resulting wave is equal to zero. This happens at

$t=\frac{1}{4}T,\frac{3}{4}T,\frac{5}{4}T\text{,...},\frac{n}{4}T\,\text{where}\,n=1,3,5\text{...}.\,\text{(out of phase)}.$

Find that some x-positions of the resultant wave are always zero no matter what the phase relationship is. These positions are called nodes. Where do the nodes occur? Consider the solution to the sum of the 2 waves

$y(x,t)=[2A\,\text{sin}(kx)]\text{cos}(\omega t).$

Finding the positions where the sine function equals zero provides the positions of the nodes.

$\begin{array}{ccc}\hfill \text{sin}(kx)& =\hfill & 0\hfill \\ \hfill kx& =\hfill & 0,\pi ,2\pi ,3\pi \text{,...}\hfill \\ \hfill \frac{2\pi }{\lambda }x& =\hfill & 0,\pi ,2\pi ,3\pi \text{,...}\hfill \\ \hfill x& =\hfill & 0,\frac{\lambda }{2},\lambda ,\frac{3\lambda }{2}\text{,...}=n\frac{\lambda }{2}\quad n=0,1,2,3\text{,...}.\hfill \end{array}$

There are also positions where y oscillates between

$y=\text{±}A$

. These are the antinodes. We can find them by considering which values of 10 result in

$\text{sin}(kx)=\text{±}1$

$\begin{array}{ccc}\hfill \text{sin}(kx)& =\hfill & ±1\hfill \\ \hfill kx& =\hfill & \frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2}\text{,...}\hfill \\ \hfill \frac{2\pi }{\lambda }x& =\hfill & \frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2}\text{,...}\hfill \\ \hfill x& =\hfill & \frac{\lambda }{4},\frac{3\lambda }{4},\frac{5\lambda }{4}\text{,...}=n\frac{\lambda }{4}\quad n=1,3,5\text{,...}.\hfill \end{array}$

What results is a standing wave equally shown in (Figure), which shows snapshots of the resulting moving ridge of ii identical waves moving in opposite directions. The resulting moving ridge appears to exist a sine wave with nodes at integer multiples of one-half wavelengths. The antinodes oscillate between

$y=\text{±}2A$

due to the cosine term,

$\text{cos}(\omega t)$

, which oscillates between

$±1$

The resultant wave appears to be standing still, with no credible movement in the x-direction, although it is composed of 1 wave function moving in the positive, whereas the 2d wave is moving in the negative ten-direction. (Effigy) shows diverse snapshots of the resulting wave. The nodes are marked with red dots while the antinodes are marked with blueish dots.

Figure shows two sine waves with changing amplitudes that are exactly opposite in phase. Nodes marked with red dots are along the x axis at x = 0 m, 3 m, 6 m, 9 m and so on. Antinodes marked with blue dots are at the peaks and troughs of each wave. They are at x = 1.5 m, 4.5 m, 7.5 m and so on. — **Figure 16.27** When two identical waves are moving in reverse directions, the resultant wave is a continuing wave. Nodes appear at integer multiples of one-half wavelengths. Antinodes appear at odd multiples of quarter wavelengths, where they oscillate betwixt
$y=\text{±}A.$

The nodes are marked with red dots and the antinodes are marked with blue dots.

A mutual example of standing waves are the waves produced by stringed musical instruments. When the string is plucked, pulses travel along the string in opposite directions. The ends of the strings are fixed in place, so nodes appear at the ends of the strings—the purlieus conditions of the arrangement, regulating the resonant frequencies in the strings. The resonance produced on a cord instrument tin exist modeled in a physics lab using the appliance shown in (Effigy).

A string vibrator is shown on the left of the figure. A string is attached to its right. This goes over a pulley and down the side of the table. A hanging mass m is suspended from it. The pulley is frictionless. The distance between the pulley and the string vibrator is L. It is labeled mu equal to delta m by delta x equal to constant. — **Effigy 16.28** A lab setup for creating standing waves on a string. The string has a node on each end and a constant linear density. The length between the fixed boundary conditions is L. The hanging mass provides the tension in the string, and the speed of the waves on the string is proportional to the foursquare root of the tension divided by the linear mass density.

The lab setup shows a string attached to a string vibrator, which oscillates the string with an adjustable frequency f. The other end of the string passes over a frictionless caster and is tied to a hanging mass. The magnitude of the tension in the string is equal to the weight of the hanging mass. The string has a abiding linear density (mass per length)

$\mu$

and the speed at which a wave travels downwardly the cord equals

$v=\sqrt{\frac{{F}_{T}}{\mu }}=\sqrt{\frac{mg}{\mu }}$

(Figure). The symmetrical boundary weather condition (a node at each finish) dictate the possible frequencies that tin can excite standing waves. Starting from a frequency of zero and slowly increasing the frequency, the offset way

$n=1$

appears as shown in (Figure). The first mode, also called the fundamental mode or the first harmonic, shows half of a wavelength has formed, so the wavelength is equal to twice the length between the nodes

${\lambda }_{1}=2L$

. The fundamental frequency, or start harmonic frequency, that drives this mode is

${f}_{1}=\frac{v}{{\lambda }_{1}}=\frac{v}{2L},$

where the speed of the wave is

$v=\sqrt{\frac{{F}_{T}}{\mu }}.$

Keeping the tension constant and increasing the frequency leads to the second harmonic or the

$n=2$

mode. This mode is a total wavelength

${\lambda }_{2}=L$

and the frequency is twice the fundamental frequency:

${f}_{2}=\frac{v}{{\lambda }_{2}}=\frac{v}{L}=2{f}_{1}.$

Four figures of a string of length L are shown. Each has two waves. The first one has 1 node. It is labeled half lambda 1 = L, lambda 1 = 2 by 1 times L. The second figure has 2 nodes. It is labeled lambda 2 = L, lambda 2 = 2 by 2 times L. The third figure has three nodes. It is labeled 3 by 2 times lambda 3 = L, lambda 3 = 2 by 3 times L. The fourth figure has 4 nodes. It is labeled 4 by 2 times lambda 4 = L, lambda 4 = 2 by 4 times L. There is a derived formula at the bottom, lambda n equal to 2 by n times L for n = 1, 2, 3 and so on. — **Figure 16.29** Continuing waves created on a string of length L. A node occurs at each end of the string. The nodes are boundary weather condition that limit the possible frequencies that excite standing waves. (Note that the amplitudes of the oscillations accept been kept constant for visualization. The standing wave patterns possible on the cord are known every bit the normal modes. Conducting this experiment in the lab would result in a decrease in aamplitude as the frequency increases.)

The next ii modes, or the third and fourth harmonics, have wavelengths of

${\lambda }_{3}=\frac{2}{3}L$

and

${\lambda }_{4}=\frac{2}{4}L,$

driven past frequencies of

${f}_{3}=\frac{3v}{2L}=3{f}_{1}$

and

${f}_{4}=\frac{4v}{2L}=4{f}_{1}.$

All frequencies above the frequency

${f}_{1}$

are known every bit the overtones. The equations for the wavelength and the frequency tin exist summarized every bit:

${\lambda }_{n}=\frac{2}{n}L\quad n=1,2,3,4,5\text{...}$

${f}_{n}=n\frac{v}{2L}=n{f}_{1}\quad n=1,2,3,4,5\text{...}$

The standing wave patterns that are possible for a string, the start four of which are shown in (Figure), are known every bit the normal modes, with frequencies known every bit the normal frequencies. In summary, the kickoff frequency to produce a normal mode is called the primal frequency (or showtime harmonic). Any frequencies above the fundamental frequency are overtones. The second frequency of the

$n=2$

normal mode of the string is the first overtone (or second harmonic). The frequency of the

$n=3$

normal mode is the 2nd overtone (or third harmonic) then on.

The solutions shown as (Equation) and (Equation) are for a string with the boundary status of a node on each terminate. When the boundary condition on either side is the same, the arrangement is said to have symmetric boundary conditions. (Equation) and (Equation) are expert for any symmetric boundary conditions, that is, nodes at both ends or antinodes at both ends.

Example

Standing Waves on a String

Consider a string of

$L=2.00\,\text{m}.$

attached to an adaptable-frequency string vibrator as shown in (Figure). The waves produced by the vibrator travel down the cord and are reflected by the fixed boundary condition at the caster. The string, which has a linear mass density of

$\mu =0.006\,\text{kg/m,}$

is passed over a frictionless pulley of a negligible mass, and the tension is provided by a 2.00-kg hanging mass. (a) What is the velocity of the waves on the string? (b) Draw a sketch of the first 3 normal modes of the continuing waves that tin exist produced on the cord and label each with the wavelength. (c) Listing the frequencies that the cord vibrator must be tuned to in order to produce the start three normal modes of the standing waves.

Strategy

The velocity of the moving ridge can be plant using
$v=\sqrt{\frac{{F}_{T}}{\mu }}.$

The tension is provided by the weight of the hanging mass.
The standing waves volition depend on the boundary conditions. There must exist a node at each end. The first mode will exist one one-half of a wave. The 2d tin be establish past adding a one-half wavelength. That is the shortest length that will result in a node at the boundaries. For instance, adding one quarter of a wavelength will result in an antinode at the boundary and is not a mode which would satisfy the boundary conditions. This is shown in (Figure).
Since the wave speed velocity is the wavelength times the frequency, the frequency is moving ridge speed divided by the wavelength.

Figure sixteen.31 (a) The figure represents the second style of the string that satisfies the boundary atmospheric condition of a node at each end of the string. (b)This figure could not possibly be a normal manner on the string considering information technology does not satisfy the boundary conditions. There is a node on one end, only an antinode on the other.

Solution

Begin with the velocity of a wave on a string. The tension is equal to the weight of the hanging mass. The linear mass density and mass of the hanging mass are given:
$v=\sqrt{\frac{{F}_{T}}{\mu }}=\sqrt{\frac{mg}{\mu }}=\sqrt{\frac{2\,\text{kg}(9.8\frac{\text{m}}{\text{s}})}{0.006\frac{\text{kg}}{\text{m}}}}=57.15\,\text{m/s}.$
The first normal mode that has a node on each end is a one-half wavelength. The next two modes are found by adding a half of a wavelength.
The frequencies of the kickoff three modes are found by using
$f=\frac{{v}_{w}}{\lambda }.$

$\begin{array}{}\\ {f}_{1}=\frac{{v}_{w}}{{\lambda }_{1}}=\frac{57.15\,\text{m/s}}{4.00\,\text{m}}=14.29\,\text{Hz}\hfill \\ {f}_{2}=\frac{{v}_{w}}{{\lambda }_{2}}=\frac{57.15\,\text{m/s}}{2.00\,\text{m}}=28.58\,\text{Hz}\hfill \\ {f}_{3}=\frac{{v}_{w}}{{\lambda }_{3}}=\frac{57.15\,\text{m/s}}{1.333\,\text{m}}=42.87\,\text{Hz}\hfill \end{array}$

Significance

The iii standing modes in this case were produced by maintaining the tension in the string and adjusting the driving frequency. Keeping the tension in the string abiding results in a constant velocity. The aforementioned modes could have been produced by keeping the frequency abiding and adjusting the speed of the wave in the string (by changing the hanging mass.)

Visit this simulation to play with a 1D or 2nd organisation of coupled mass-spring oscillators. Vary the number of masses, set up the initial conditions, and sentry the system evolve. Run across the spectrum of normal modes for arbitrary motion. Encounter longitudinal or transverse modes in the 1D system.

Check Your Understanding

The equations for the wavelengths and the frequencies of the modes of a moving ridge produced on a cord:

$\begin{array}{}\\ {\lambda }_{n}=\frac{2}{n}L\quad n=1,2,3,4,5\text{...}\,\text{and}\hfill \\ {f}_{n}=n\frac{v}{2L}=n{f}_{1}\quad n=1,2,3,4,5\text{...}\hfill \end{array}$

were derived by because a wave on a cord where there were symmetric boundary conditions of a node at each stop. These modes resulted from two sinusoidal waves with identical characteristics except they were moving in opposite directions, bars to a region L with nodes required at both ends. Will the same equations work if at that place were symmetric boundary conditions with antinodes at each end? What would the normal modes look like for a medium that was gratis to oscillate on each end? Don't worry for now if y'all cannot imagine such a medium, just consider two sinusoidal moving ridge functions in a region of length L, with antinodes on each end.

Yes, the equations would work as well for symmetric boundary conditions of a medium free to oscillate on each end where in that location was an antinode on each end. The normal modes of the showtime iii modes are shown beneath. The dotted line shows the equilibrium position of the medium.

Three figures of a string of length L are shown. Each has two waves. The first one has 1 node. It is labeled lambda 1 = 2 by 1 times L, f1 = vw by lambda 1 = vw by 2L. The second figure has 2 nodes. It is labeled lambda 2 = 2 by 2 times L, f2 = vw by lambda 2 = vw by L. The third figure has three nodes. It is labeled lambda 3 = 2 by 3 times L, f3 = vw by lambda 3 equal to 3 times vw by 2L.

Notation that the outset style is two quarters, or ane one-half, of a wavelength. The second way is 1 quarter of a wavelength, followed past i one-half of a wavelength, followed by i quarter of a wavelength, or one full wavelength. The third mode is one and a half wavelengths. These are the same effect as the string with a node on each end. The equations for symmetrical boundary conditions work equally well for stock-still boundary conditions and free boundary conditions. These results will exist revisited in the adjacent chapter when discussing sound wave in an open tube.

The free boundary weather condition shown in the last Bank check Your Understanding may seem hard to visualize. How can there exist a organization that is costless to oscillate on each end? In (Effigy) are shown 2 possible configuration of a metallic rods (shown in red) fastened to two supports (shown in blue). In part (a), the rod is supported at the ends, and there are stock-still boundary conditions at both ends. Given the proper frequency, the rod can be driven into resonance with a wavelength equal to length of the rod, with nodes at each terminate. In part (b), the rod is supported at positions ane quarter of the length from each end of the rod, and there are free boundary conditions at both ends. Given the proper frequency, this rod tin can also be driven into resonance with a wavelength equal to the length of the rod, but at that place are antinodes at each end. If you lot are having problem visualizing the wavelength in this figure, remember that the wavelength may be measured between any 2 nearest identical points and consider (Figure).

opposite in phase, forming nodes at the spots where the poles support the rod and antinodes at both ends of the rod. — **Figure 16.32** (a) A metallic rod of length L (red) supported by two supports (bluish) on each end. When driven at the proper frequency, the rod can resonate with a wavelength equal to the length of the rod with a node on each end. (b) The same metallic rod of length L (cherry-red) supported by two supports (blue) at a position a quarter of the length of the rod from each end. When driven at the proper frequency, the rod tin resonate with a wavelength equal to the length of the rod with an antinode on each end.

Figure shows a sinusoidal wave. Two boxes labeled a and b each mark one wavelength of the wave. Box a measures the wavelength between two closest points on the x axis where the wave starts gaining a positive value. Box b measures the wavelength between two adjoining crests of the wave. — **Effigy xvi.33** A wavelength may be measure betwixt the nearest 2 repeating points. On the wave on a string, this means the same summit and slope. (a) The wavelength is measured betwixt the two nearest points where the summit is zero and the gradient is maximum and positive. (b) The wavelength is measured between two identical points where the height is maximum and the slope is zero.

Note that the written report of standing waves tin can become quite complex. In (Figure)(a), the

$n=2$

manner of the standing wave is shown, and information technology results in a wavelength equal to L. In this configuration, the

$n=1$

fashion would besides take been possible with a standing wave equal to 2L. Is it possible to become the

$n=1$

mode for the configuration shown in office (b)? The answer is no. In this configuration, at that place are additional weather condition set beyond the boundary weather. Since the rod is mounted at a point one quarter of the length from each side, a node must exist there, and this limits the possible modes of standing waves that can be created. We leave it as an practise for the reader to consider if other modes of continuing waves are possible. Information technology should be noted that when a system is driven at a frequency that does not crusade the system to resonate, vibrations may all the same occur, but the amplitude of the vibrations will be much smaller than the amplitude at resonance.

A field of mechanical technology uses the sound produced by the vibrating parts of complex mechanical systems to troubleshoot problems with the systems. Suppose a function in an automobile is resonating at the frequency of the automobile's engine, causing unwanted vibrations in the auto. This may cause the engine to fail prematurely. The engineers use microphones to record the sound produced by the engine, then use a technique called Fourier analysis to find frequencies of sound produced with large amplitudes and then look at the parts list of the automobile to find a part that would resonate at that frequency. The solution may exist as simple as changing the composition of the cloth used or irresolute the length of the part in question.

There are other numerous examples of resonance in standing waves in the physical world. The air in a tube, such as found in a musical instrument similar a flute, tin be forced into resonance and produce a pleasant sound, every bit we discuss in Audio.

At other times, resonance can cause serious problems. A closer await at earthquakes provides evidence for conditions appropriate for resonance, standing waves, and constructive and subversive interference. A edifice may vibrate for several seconds with a driving frequency matching that of the natural frequency of vibration of the edifice—producing a resonance resulting in ane building collapsing while neighboring buildings do non. Ofttimes, buildings of a certain height are devastated while other taller buildings remain intact. The building height matches the status for setting up a standing wave for that detail height. The span of the roof is also important. Often it is seen that gymnasiums, supermarkets, and churches endure impairment when individual homes endure far less impairment. The roofs with large surface areas supported merely at the edges resonate at the frequencies of the earthquakes, causing them to collapse. As the convulsion waves travel along the surface of Earth and reverberate off denser rocks, constructive interference occurs at certain points. Often areas closer to the epicenter are not damaged, while areas farther away are damaged.

Summary

A standing moving ridge is the superposition of two waves which produces a wave that varies in amplitude but does non propagate.
Nodes are points of no motion in standing waves.
An antinode is the location of maximum amplitude of a standing wave.
Normal modes of a moving ridge on a string are the possible continuing wave patterns. The lowest frequency that will produce a standing wave is known as the cardinal frequency. The higher frequencies which produce standing waves are chosen overtones.

Key Equations

Wave speed	$v=\frac{\lambda }{T}=\lambda f$
Linear mass density	$\mu =\frac{\text{mass of the string}}{\text{length of the string}}$
Speed of a wave or pulse on a cord under tension	$\|v\|=\sqrt{\frac{{F}_{T}}{\mu }}$
Speed of a compression wave in a fluid	$v=\sqrt{\frac{Β}{\rho }}$
Resultant wave from superposition of two sinusoidal waves that are identical except for a phase shift	${y}_{R}(x,t)=[2A\,\text{cos}(\frac{\varphi }{2})]\text{sin}(kx-\omega t+\frac{\varphi }{2})$
Wave number	$k\equiv \frac{2\pi }{\lambda }$
Wave speed	$v=\frac{\omega }{k}$
A periodic wave	$y(x,t)=A\,\text{sin}(kx\mp \omega t+\varphi )$
Stage of a moving ridge	$kx\mp \omega t+\varphi$
The linear moving ridge equation	$\frac{{\partial }^{2}y(x,t)}{\partial {x}^{2}}=\frac{1}{{v}_{w}^{2}}\,\frac{{\partial }^{2}y(x,t)}{\partial {t}^{2}}$
Power in a moving ridge for one wavelength	${P}_{\text{ave}}=\frac{{E}_{\lambda }}{T}=\frac{1}{2}\mu {A}^{2}{\omega }^{2}\frac{\lambda }{T}=\frac{1}{2}\mu {A}^{2}{\omega }^{2}v$
Intensity	$I=\frac{P}{A}$
Intensity for a spherical wave	$I=\frac{P}{4\pi {r}^{2}}$
Equation of a continuing wave	$y(x,t)=[2A\,\text{sin}(kx)]\text{cos}(\omega t)$
Wavelength for symmetric purlieus conditions	${\lambda }_{n}=\frac{2}{n}L,\quad n=1,2,3,4,5\text{...}$
Frequency for symmetric boundary conditions	${f}_{n}=n\frac{v}{2L}=n{f}_{1},\quad n=1,2,3,4,5\text{...}$

Conceptual Questions

A truck manufacturer finds that a strut in the engine is failing prematurely. A sound engineer determines that the strut resonates at the frequency of the engine and suspects that this could be the trouble. What are two possible characteristics of the strut tin can be modified to right the problem?

[reveal-answer q="fs-id1165036894567″]Show Solution[/reveal-respond]

[hidden-answer a="fs-id1165036894567″]

It may exist as easy as changing the length and/or the density a pocket-sized corporeality so that the parts do not resonate at the frequency of the motor.

[/hidden-answer]

Why practise roofs of gymnasiums and churches seem to fail more than family unit homes when an earthquake occurs?

Wine spectacles can be prepare into resonance past moistening your finger and rubbing it around the rim of the glass. Why?

[reveal-answer q="fs-id1165038342348″]Show Solution[/reveal-reply]

[hidden-answer a="fs-id1165038342348″]

Energy is supplied to the drinking glass past the work washed past the force of your finger on the glass. When supplied at the right frequency, standing waves form. The glass resonates and the vibrations produce audio.

[/hidden-answer]

Air conditioning units are sometimes placed on the roof of homes in the city. Occasionally, the air conditioners cause an undesirable hum throughout the upper floors of the homes. Why does this happen? What can be done to reduce the hum?

Consider a standing wave modeled as

$y(x,t)=4.00\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x)\text{cos}(4\,{\text{s}}^{-1}t).$

Is there a node or an antinode at

$x=0.00\,\text{m}?$

What well-nigh a continuing wave modeled as

$y(x,t)=4.00\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x+\frac{\pi }{2})\text{cos}(4\,{\text{s}}^{-1}t)?$

Is there a node or an antinode at the

$x=0.00\,\text{m}$

position?

[reveal-respond q="fs-id1165038295126″]Show Solution[/reveal-answer]

[hidden-answer a="fs-id1165038295126″]

For the equation

$y(x,t)=4.00\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x)\text{cos}(4\,{\text{s}}^{-1}t),$

at that place is a node because when

$x=0.00\,\text{m}$

$\text{sin}(3\,{\text{m}}^{-1}(0.00\,\text{m}))=0.00,$

then

$y(0.00\,\text{m},t)=0.00\,\text{m}$

for all fourth dimension. For the equation

$y(x,t)=4.00\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x+\frac{\pi }{2})\text{cos}(4\,{\text{s}}^{-1}t),$

there is an antinode because when

$x=0.00\,\text{m}$

$\text{sin}(3\,{\text{m}}^{-1}(0.00\,\text{m})+\frac{\pi }{2})=+1.00$

, then

$y(0.00\,\text{m},t)$

oscillates between +A and −A as the cosine term oscillates betwixt +ane and -1.
[/hidden-answer]

Problems

A moving ridge traveling on a Slinky® that is stretched to 4 m takes ii.4 south to travel the length of the Slinky and back again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?

A two-1000 long cord is stretched between two supports with a tension that produces a wave speed equal to

${v}_{w}=50.00\,\text{m/s}.$

What are the wavelength and frequency of the first three modes that resonate on the string?

[reveal-answer q="fs-id1165037003942″]Show Solution[/reveal-answer]

[hidden-answer a="fs-id1165037003942″]

$\begin{array}{cc} {\lambda }_{n}=\frac{2.00}{n}L,\quad {f}_{n}=\frac{v}{{\lambda }_{n}}\hfill \\ {\lambda }_{1}=4.00\,\text{m},\quad {f}_{1}=12.5\,\text{Hz}\hfill \\ {\lambda }_{2}=2.00\,\text{m},\quad {f}_{2}=25.00\,\text{Hz}\hfill \\ {\lambda }_{3}=1.33\,\text{m},\quad {f}_{3}=37.59\,\text{Hz}\hfill \end{array}$

[/hidden-answer]

Consider the experimental setup shown beneath. The length of the string between the string vibrator and the pulley is

$L=1.00\,\text{m}.$

The linear density of the string is

$\mu =0.006\,\text{kg/m}.$

The string vibrator tin can oscillate at whatsoever frequency. The hanging mass is 2.00 kg. (a)What are the wavelength and frequency of

$n=6$

mode? (b) The cord oscillates the air around the string. What is the wavelength of the sound if the speed of the sound is

${v}_{s}=343.00\,\text{m/s?}$

A string vibrator is shown on the left of the figure. A string is attached to its right. This goes over a pulley and down the side of the table. A hanging mass m is suspended from it. The pulley is frictionless. The distance between the pulley and the string vibrator is L. It is labeled mu equal to dm by dx equal to constant.

A cable with a linear density of

$\mu =0.2\,\text{kg/m}$

is hung from telephone poles. The tension in the cablevision is 500.00 N. The distance between poles is 20 meters. The wind blows across the line, causing the cablevision resonate. A standing waves pattern is produced that has 4.five wavelengths betwixt the two poles. The air temperature is

$T=20\text{°}\text{C}.$

What are the frequency and wavelength of the hum?

[reveal-reply q="908302″]Show Answer[/reveal-answer]
[hidden-answer a="908302″]

$\begin{array}{cc} v=158.11\,\text{m/s,}\quad \lambda =4.44\,\text{m,}\quad f=35.61\,\text{Hz}\hfill \\ {\lambda }_{s}=9.63\,\text{m}\hfill \end{array}$

[/hidden-answer]

Consider a rod of length L, mounted in the center to a support. A node must exist where the rod is mounted on a back up, every bit shown beneath. Draw the first two normal modes of the rod equally information technology is driven into resonance. Label the wavelength and the frequency required to drive the rod into resonance.

Figure shows a horizontal rod of length L = 2 m supported at the centre by a pole.

Consider ii wave functions

$y(x,t)=0.30\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x-4\,{\text{s}}^{-1}t)$

and

$y(x,t)=0.30\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x+4\,{\text{s}}^{-1}t)$

. Write a moving ridge function for the resulting standing wave.

[reveal-respond q="fs-id1165038012769″]Prove Solution[/reveal-answer]

[hidden-answer a="fs-id1165038012769″]

$y(x,t)=[0.60\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x)]\text{cos}(4\,{\text{s}}^{-1}t)$

[/hidden-answer]

A ii.40-thou wire has a mass of seven.50 g and is under a tension of 160 N. The wire is held rigidly at both ends and gear up into oscillation. (a) What is the speed of waves on the wire? The string is driven into resonance by a frequency that produces a standing wave with a wavelength equal to 1.20 m. (b) What is the frequency used to drive the string into resonance?

A cord with a linear mass density of 0.0062 kg/m and a length of 3.00 m is set up into the

$n=100$

mode of resonance. The tension in the string is xx.00 North. What is the wavelength and frequency of the moving ridge?

[reveal-respond q="fs-id1165036826014″]Show Solution[/reveal-reply]

[subconscious-answer a="fs-id1165036826014″]

$\begin{array}{cc} {\lambda }_{100}=0.06\,\text{m}\hfill \\ \\ v=56.8\,\text{m/s,}\quad {f}_{n}=n{f}_{1},\quad n=1,2,3,4,5\text{...}\hfill \\ {f}_{100}=947\,\text{Hz}\hfill \end{array}$

[/subconscious-answer]

A string with a linear mass density of 0.0075 kg/1000 and a length of 6.00 m is fix into the

$n=4$

mode of resonance past driving with a frequency of 100.00 Hz. What is the tension in the string?

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions forth a string producing a standing moving ridge. The linear mass density of the string is

$\mu =0.075\,\text{kg/m}$

and the tension in the string is

${F}_{T}=5.00\,\text{N}.$

The time interval between instances of full subversive interference is

$\text{Δ}t=0.13\,\text{s}.$

What is the wavelength of the waves?

[reveal-answer q="fs-id1165037181984″]Show Solution[/reveal-answer]

[hidden-answer a="fs-id1165037181984″]

$T=2\text{Δ}t,\quad v=\frac{\lambda }{T},\quad \lambda =2.12\,\text{m}$

[/hidden-answer]

A string, fixed on both ends, is v.00 thousand long and has a mass of 0.xv kg. The tension if the cord is ninety N. The string is vibrating to produce a continuing moving ridge at the fundamental frequency of the string. (a) What is the speed of the waves on the cord? (b) What is the wavelength of the standing moving ridge produced? (c) What is the menstruum of the standing wave?

A string is fixed at both cease. The mass of the string is 0.0090 kg and the length is iii.00 m. The string is under a tension of 200.00 Northward. The string is driven past a variable frequency source to produce standing waves on the string. Notice the wavelengths and frequency of the first iv modes of standing waves.

[reveal-answer q="fs-id1165037178876″]Testify Solution[/reveal-respond]

[hidden-respond a="fs-id1165037178876″]

$\begin{array}{cc} {\lambda }_{1}=6.00\,\text{m},\quad {\lambda }_{2}=3.00\,\text{m},\quad {\lambda }_{3}=2.00\,\text{m},\quad {\lambda }_{4}=1.50\,\text{m}\hfill \\ v=258.20\,\text{m/s}=\lambda f\hfill \\ {f}_{1}=43.03\,\text{Hz},\quad {f}_{2}=86.07\,\text{Hz},\quad {f}_{3}=129.10\,\text{Hz},\quad {f}_{4}=172.13\,\text{Hz}\hfill \end{array}$

[/subconscious-answer]

The frequencies of two successive modes of standing waves on a string are 258.36 Hz and 301.42 Hz. What is the next frequency above 100.00 Hz that would produce a continuing moving ridge?

A string is stock-still at both ends to supports 3.l m autonomously and has a linear mass density of

$\mu =0.005\,\text{kg/m}.$

The cord is under a tension of ninety.00 Due north. A standing wave is produced on the cord with half dozen nodes and five antinodes. What are the wave speed, wavelength, frequency, and period of the standing wave?

[reveal-answer q="fs-id1165037077614″]Prove Solution[/reveal-answer]

[hidden-answer a="fs-id1165037077614″]

$v=134.16\,\text{ms},\lambda =1.4\,\text{m},f=95.83\,\text{Hz},T=0.0104\,\text{s}$

[/hidden-answer]

Sine waves are sent down a ane.v-m-long string fixed at both ends. The waves reverberate back in the opposite direction. The amplitude of the wave is 4.00 cm. The propagation velocity of the waves is 175 m/southward. The

$n=6$

resonance way of the cord is produced. Write an equation for the resulting standing wave.

Additional Problems

Ultrasound equipment used in the medical profession uses sound waves of a frequency above the range of homo hearing. If the frequency of the audio produced past the ultrasound machine is

$f=30\,\text{kHz,}$

what is the wavelength of the ultrasound in bone, if the speed of sound in bone is

$v=3000\,\text{m/s?}$

[reveal-reply q="fs-id1165037843797″]Show Solution[/reveal-reply]

[hidden-answer a="fs-id1165037843797″]

$\lambda =0.10\,\text{m}$

[/hidden-answer]

Shown below is the plot of a wave function that models a moving ridge at time

$t=0.00\,\text{s}$

and

$t=2.00\,\text{s}$

. The dotted line is the moving ridge function at time

$t=0.00\,\text{s}$

and the solid line is the function at fourth dimension

$t=2.00\,\text{s}$

. Estimate the amplitude, wavelength, velocity, and menstruum of the wave.
Figure shows two transverse waves on a graph whose y values vary from -3 m to 3 m. One wave is shown as a dotted line and is marked t = 0 seconds. It has crests at x approximately equal to 0.25 m and 1.25 m. The other wave is shown as a solid line and is marked t=2 seconds. It has crests at x approximately equal to 0.85 seconds and 1.85 seconds.

The speed of calorie-free in air is approximately

$v=3.00\,×\,{10}^{8}\,\text{m/s}$

and the speed of light in glass is

$v=2.00\,×\,{10}^{8}\,\text{m/s}$

. A red laser with a wavelength of

$\lambda =633.00\,\text{nm}$

shines lite incident of the glass, and some of the crimson light is transmitted to the glass. The frequency of the light is the aforementioned for the air and the glass. (a) What is the frequency of the light? (b) What is the wavelength of the light in the glass?

[reveal-respond q="fs-id1165037046939″]Prove Solution[/reveal-answer]

[hidden-answer a="fs-id1165037046939″]

$f=4.74\,×\,{10}^{14}\,\text{Hz;}$

$\lambda =422\,\text{nm}$

[/subconscious-answer]

A radio station broadcasts radio waves at a frequency of 101.7 MHz. The radio waves move through the air at approximately the speed of light in a vacuum. What is the wavelength of the radio waves?

A sunbather stands waist deep in the body of water and observes that six crests of periodic surface waves pass each minute. The crests are 16.00 meters apart. What is the wavelength, frequency, flow, and speed of the waves?

[reveal-respond q="fs-id1165037232478″]Show Solution[/reveal-answer]

[hidden-answer a="fs-id1165037232478″]

$\lambda =16.00\,\text{m},\quad f=0.10\,\text{Hz},\quad T=10.00\,\text{s},\quad v=1.6\,\text{m/s}$

[/hidden-answer]

A tuning fork vibrates producing sound at a frequency of 512 Hz. The speed of audio of audio in air is

$v=343.00\,\text{m/s}$

if the air is at a temperature of

$20.00\text{°}\text{C}$

. What is the wavelength of the sound?

A motorboat is traveling beyond a lake at a speed of

${v}_{b}=15.00\,\text{m/s}.$

The gunkhole bounces up and down every 0.50 s equally it travels in the aforementioned management as a moving ridge. It bounces upwardly and down every 0.30 s equally information technology travels in a management contrary the direction of the waves. What is the speed and wavelength of the wave?

[reveal-answer q="fs-id1165038376319″]Prove Solution[/reveal-respond]

[subconscious-reply a="fs-id1165038376319″]

$\lambda =({v}_{b}+v){t}_{b},\quad v=3.75\,\text{m/s,}\quad \lambda =3.00\,\text{m}$

[/subconscious-answer]

Apply the linear moving ridge equation to bear witness that the moving ridge speed of a moving ridge modeled with the moving ridge function

$y(x,t)=0.20\,\text{m}\,\text{sin}(3.00\,{\text{m}}^{-1}x+6.00\,{\text{s}}^{-1}t)$

$v=2.00\,\text{m/s}.$

What are the wavelength and the speed of the wave?

Given the wave functions

${y}_{1}(x,t)=A\,\text{sin}(kx-\omega t)$

and

${y}_{2}(x,t)=A\,\text{sin}(kx-\omega t+\varphi )$

with

$\varphi \ne \frac{\pi }{2}$

, show that

${y}_{1}(x,t)+{y}_{2}(x,t)$

is a solution to the linear wave equation with a wave velocity of

$v=\sqrt{\frac{\omega }{k}}.$

[reveal-respond q="fs-id1165036995777″]Show Solution[/reveal-answer]

[hidden-answer a="fs-id1165036995777″]

$\begin{array}{cc} \frac{{\partial }^{2}({y}_{1}+{y}_{2})}{\partial {t}^{2}}=\text{−}A{\omega }^{2}\,\text{sin}(kx-\omega t)-A{\omega }^{2}\,\text{sin}(kx-\omega t+\varphi )\hfill \\ \frac{{\partial }^{2}({y}_{1}+{y}_{2})}{\partial {x}^{2}}=\text{−}A{k}^{2}\,\text{sin}(kx-\omega t)-A{k}^{2}\,\text{sin}(kx-\omega t+\varphi )\hfill \\ \frac{{\partial }^{2}y(x,t)}{\partial {x}^{2}}=\frac{1}{{v}^{2}}\,\frac{{\partial }^{2}y(x,t)}{\partial {t}^{2}}\hfill \\ \\ -A{\omega }^{2}\,\text{sin}(kx-\omega t)-A{\omega }^{2}\,\text{sin}(kx-\omega t+\varphi )=(\frac{1}{{v}^{2}})(\text{−}A{k}^{2}\,\text{sin}(kx-\omega t)-A{k}^{2}\,\text{sin}(kx-\omega t+\varphi ))\hfill \\ v=\frac{\omega }{k}\hfill \end{array}$

[/hidden-answer]

A transverse moving ridge on a string is modeled with the wave role

$y(x,t)=0.10\,\text{m}\,\text{sin}(0.15\,{\text{m}}^{-1}x+1.50\,{\text{s}}^{-1}t+0.20)$

. (a) Notice the wave velocity. (b) Find the position in the y-management, the velocity perpendicular to the motion of the wave, and the acceleration perpendicular to the move of the wave, of a small-scale segment of the string centered at

$x=0.40\,\text{m}$

at time

$t=5.00\,\text{s}.$

A sinusoidal wave travels downwards a taut, horizontal string with a linear mass density of

$\mu =0.060\,\text{kg/m}.$

The magnitude of maximum vertical acceleration of the moving ridge is

${a}_{y\,\text{max}}=0.90\,{\text{cm/s}}^{2}$

and the aamplitude of the wave is 0.forty m. The cord is nether a tension of

${F}_{T}=600.00\,\text{N}$

. The wave moves in the negative ten-direction. Write an equation to model the wave.

[reveal-answer q="fs-id1165038355361″]Show Solution[/reveal-respond]

[hidden-answer a="fs-id1165038355361″]

$y(x,t)=0.40\,\text{m}\,\text{sin}(0.015\,{\text{m}}^{-1}x+1.5\,{\text{s}}^{-1}t)$

[/subconscious-respond]

A transverse moving ridge on a string

$(\mu =0.0030\,\text{kg/m})$

is described with the equation

$y(x,t)=0.30\,\text{m}\,\text{sin}(\frac{2\pi }{4.00\,\text{m}}(x-16.00\frac{\text{m}}{\text{s}}t)).$

What is the tension under which the string is held taut?

A transverse wave on a horizontal string

$(\mu =0.0060\,\text{kg/m})$

is described with the equation

$y(x,t)=0.30\,\text{m}\,\text{sin}(\frac{2\pi }{4.00\,\text{m}}(x-{v}_{w}t)).$

The string is under a tension of 300.00 N. What are the wave speed, moving ridge number, and angular frequency of the wave?

[reveal-answer q="fs-id1165037222679″]Prove Solution[/reveal-answer]

[subconscious-reply a="fs-id1165037222679″]

$v=223.61\,\text{m/s},\,k=1.57\,{\text{m}}^{-1},\,\omega =142.43\,{\text{s}}^{-1}$

[/hidden-respond]

A student holds an inexpensive sonic range finder and uses the range finder to find the distance to the wall. The sonic range finder emits a audio moving ridge. The sound wave reflects off the wall and returns to the range finder. The round trip takes 0.012 s. The range finder was calibrated for use at room temperature

$T=20\text{°}\text{C}$

, but the temperature in the room is actually

$T=23\text{°}\text{C}.$

Bold that the timing mechanism is perfect, what percentage of error can the educatee wait due to the calibration?

A wave on a string is driven past a string vibrator, which oscillates at a frequency of 100.00 Hz and an amplitude of 1.00 cm. The string vibrator operates at a voltage of 12.00 V and a current of 0.20 A. The power consumed by the cord vibrator is

$P=IV$

. Assume that the string vibrator is

*** QuickLaTeX cannot compile formula: \[90\text{%}\]  *** Error message: File ended while scanning utilize of \text@. Emergency stop.

efficient at converting electric energy into the free energy associated with the vibrations of the string. The string is three.00 m long, and is nether a tension of sixty.00 Due north. What is the linear mass density of the cord?

[reveal-reply q="fs-id1165037157829″]Show Solution[/reveal-answer]

[hidden-answer a="fs-id1165037157829″]

$\begin{array}{cc} P=\frac{1}{2}{A}^{2}{(2\pi f)}^{2}\sqrt{\mu {F}_{T}}\hfill \\ \mu =2.00\,×\,{10}^{-4}\,\text{kg/m}\hfill \end{array}$

[/hidden-answer]

A traveling wave on a string is modeled past the wave equation

$y(x,t)=3.00\,\text{cm}\,\text{sin}(8.00\,{\text{m}}^{-1}x+100.00\,{\text{s}}^{-1}t).$

The string is under a tension of 50.00 Northward and has a linear mass density of

$\mu =0.008\,\text{kg/m}.$

What is the average power transferred by the wave on the string?

A transverse wave on a string has a wavelength of 5.0 grand, a menstruation of 0.02 s, and an amplitude of 1.5 cm. The average power transferred by the moving ridge is 5.00 W. What is the tension in the string?

[reveal-answer q="fs-id1165037183624″]Show Solution[/reveal-answer]

[hidden-respond a="fs-id1165037183624″]

$P=\frac{1}{2}\mu {A}^{2}{\omega }^{2}\frac{\lambda }{T},\,\mu =0.0018\,\text{kg/m}$

[/hidden-answer]

(a) What is the intensity of a laser beam used to burn away cancerous tissue that, when

$90.0\text{%}$

absorbed, puts 500 J of free energy into a circular spot 2.00 mm in bore in 4.00 s? (b) Talk over how this intensity compares to the boilerplate intensity of sunlight (about) and the implications that would have if the light amplification by stimulated emission of radiation beam entered your eye. Note how your answer depends on the time elapsing of the exposure.

Consider two periodic wave functions,

${y}_{1}(x,t)=A\,\text{sin}(kx-\omega t)$

and

${y}_{2}(x,t)=A\,\text{sin}(kx-\omega t+\varphi ).$

(a) For what values of

$\varphi$

volition the wave that results from a superposition of the wave functions have an amplitude of twoA? (b) For what values of

$\varphi$

will the wave that results from a superposition of the wave functions have an amplitude of zero?

[reveal-answer q="fs-id1165037056466″]Prove Solution[/reveal-answer]

[hidden-answer a="fs-id1165037056466″]

${A}_{R}=2A\,\text{cos}(\frac{\varphi }{2}),\,\text{cos}(\frac{\varphi }{2})=1,\,\varphi =0,2\pi ,4\pi \text{,...}$

; b.

${A}_{R}=2A\,\text{cos}(\frac{\varphi }{2}),\,\text{cos}(\frac{\varphi }{2})=0,\,\varphi =0,\pi ,3\pi ,5\pi \text{...}$

[/hidden-answer]

Consider two periodic wave functions,

${y}_{1}(x,t)=A\,\text{sin}(kx-\omega t)$

and

${y}_{2}(x,t)=A\,\text{cos}(kx-\omega t+\varphi )$

. (a) For what values of

$\varphi$

will the moving ridge that results from a superposition of the wave functions have an amplitude of 2A? (b) For what values of

$\varphi$

will the wave that results from a superposition of the wave functions have an aamplitude of zero?

A trough with dimensions 10.00 meters past 0.x meters past 0.10 meters is partially filled with water. Small-amplitude surface water waves are produced from both ends of the trough past paddles aquiver in unproblematic harmonic movement. The height of the water waves are modeled with 2 sinusoidal wave equations,

${y}_{1}(x,t)=0.3\,\text{m}\,\text{sin}(4\,{\text{m}}^{-1}x-3\,{\text{s}}^{-1}t)$

and

${y}_{2}(x,t)=0.3\,\text{m}\,\text{cos}(4\,{\text{m}}^{-1}x+3\,{\text{s}}^{-1}t-\frac{\pi }{2}).$

What is the moving ridge part of the resulting wave afterward the waves reach i some other and earlier they attain the end of the trough (i.due east., presume that there are only two waves in the trough and ignore reflections)? Utilize a spreadsheet to check your results. (Hint: Use the trig identities

$\text{sin}(u±v)=\text{sin}\,u\,\text{cos}\,v±\text{cos}\,u\,\text{sin}\,v$

and

$\text{cos}(u±v)=\text{cos}\,u\,\text{cos}\,v\mp \text{sin}\,u\,\text{sin}\,v)$

[reveal-answer q="fs-id1165037163373″]Evidence Solution[/reveal-answer]

[subconscious-reply a="fs-id1165037163373″]

${y}_{R}(x,t)=0.6\,\text{m}\,\text{sin}(4\,{\text{m}}^{-1}x)\text{cos}(3\,{\text{s}}^{-1}t)$

[/subconscious-answer]

A seismograph records the Due south- and P-waves from an earthquake 20.00 s apart. If they traveled the aforementioned path at constant wave speeds of

${v}_{S}=4.00\,\text{km/s}$

and

${v}_{P}=7.50\,\text{km/s},$

how far away is the epicenter of the earthquake?

Consider what is shown below. A 20.00-kg mass rests on a frictionless ramp inclined at

$45\text{°}$

. A string with a linear mass density of

$\mu =0.025\,\text{kg/m}$

is fastened to the xx.00-kg mass. The string passes over a frictionless pulley of negligible mass and is attached to a hanging mass (m). The system is in static equilibrium. A wave is induced on the cord and travels upwards the ramp. (a) What is the mass of the hanging mass (m)? (b) At what wave speed does the wave travel up the string?
Figure shows a slope of 45 degrees going up and right. A mass of 20 kg rests on it. This is supported by a string, which goes over a pulley at the top of the slope. A mass m hangs from it on the other side. A wave is shown in the string.

[reveal-answer q="905297″]Prove Respond[/reveal-respond]
[subconscious-answer a="905297″]a.

$\begin{array}{cc} (1){F}_{T}-20.00\,\text{kg}(9.80\,{\text{m/s}}^{2})\text{cos}\,45\text{°}=0\hfill \\ (2)m(9.80\,{\text{m/s}}^{2})-{F}_{T}=0\hfill \\ m=14.14\,\text{kg}\hfill \end{array}$

; b.

$\begin{array}{cc} \hfill {F}_{T}& =\hfill & 138.57\,\text{N}\hfill \\ \hfill v& =\hfill & 67.96\,\text{m/s}\hfill \end{array}$

[/hidden-respond]

Consider the superposition of iii wave functions

$y(x,t)=3.00\,\text{cm}\,\text{sin}(2\,{\text{m}}^{-1}x-3\,{\text{s}}^{-1}t),$

$y(x,t)=3.00\,\text{cm}\,\text{sin}(6\,{\text{m}}^{-1}x+3\,{\text{s}}^{-1}t),$

and

$y(x,t)=3.00\,\text{cm}\,\text{sin}(2\,{\text{m}}^{-1}x-4\,{\text{s}}^{-1}t).$

What is the height of the resulting wave at position

$x=3.00\,\text{m}$

at time

$t=10.0\,\text{s?}$

A string has a mass of 150 grand and a length of 3.4 m. I finish of the cord is fixed to a lab stand up and the other is fastened to a bound with a spring abiding of

${k}_{s}=100\,\text{N/m}.$

The gratuitous stop of the spring is attached to some other lab pole. The tension in the string is maintained by the spring. The lab poles are separated past a altitude that stretches the leap two.00 cm. The string is plucked and a pulse travels along the string. What is the propagation speed of the pulse?

[reveal-respond q="fs-id1165036885148″]Show Solution[/reveal-answer]

[subconscious-answer a="fs-id1165036885148″]

${F}_{T}=12\,\text{N,}\,v=16.49\,\text{m/s}$

[/subconscious-answer]

A standing wave is produced on a string under a tension of 70.0 Northward by ii sinusoidal transverse waves that are identical, only moving in opposite directions. The string is stock-still at

$x=0.00\,\text{m}$

and

$x=10.00\,\text{m}.$

Nodes appear at

$x=0.00\,\text{m,}$

2.00 grand, 4.00 grand, 6.00 m, eight.00 m, and x.00 yard. The amplitude of the standing wave is 3.00 cm. It takes 0.10 s for the antinodes to make one complete oscillation. (a) What are the wave functions of the two sine waves that produce the standing wave? (b) What are the maximum velocity and acceleration of the string, perpendicular to the direction of motion of the transverse waves, at the antinodes?

A string with a length of 4 grand is held under a constant tension. The cord has a linear mass density of

$\mu =0.006\,\text{kg/m}.$

Two resonant frequencies of the string are 400 Hz and 480 Hz. In that location are no resonant frequencies between the two frequencies. (a) What are the wavelengths of the two resonant modes? (b) What is the tension in the string?

[reveal-respond q="fs-id1165037233763″]Prove Solution[/reveal-answer]

[hidden-answer a="fs-id1165037233763″]

$\begin{array}{cc} {f}_{n}=\frac{nv}{2L},\,v=\frac{2L{f}_{n+1}}{n+1},\,\frac{n+1}{n}=\frac{2L{f}_{n+1}}{2L{f}_{n}},\,1+\frac{1}{n}=1.2,\,n=5\hfill \\ {\lambda }_{n}=\frac{2}{n}L,\,{\lambda }_{5}=1.6\,\text{m},\,{\lambda }_{6}=1.33\,\text{m}\hfill \end{array}$

; b.

${F}_{T}=245.76\,\text{N}$

[/hidden-respond]

Challenge Problems

A copper wire has a radius of

$200\,\text{μm}$

and a length of 5.0 m. The wire is placed nether a tension of 3000 Northward and the wire stretches by a minor corporeality. The wire is plucked and a pulse travels down the wire. What is the propagation speed of the pulse? (Assume the temperature does not modify:

$(\rho =8.96\frac{\text{g}}{{\text{cm}}^{3}},Y=1.1\,×\,{10}^{11}\frac{\text{N}}{\text{m}}).)$

A pulse moving along the x centrality tin be modeled as the wave role

$y(x,t)=4.00\,\text{m}{e}^{\text{−}{(\frac{x+(2.00\,\text{m/s})t}{1.00\,\text{m}})}^{2}}.$

(a)What are the direction and propagation speed of the pulse? (b) How far has the wave moved in 3.00 south? (c) Plot the pulse using a spreadsheet at time

$t=0.00\,\text{s}$

and

$t=3.00\,\text{s}$

to verify your respond in part (b).

[reveal-answer q="fs-id1165037263477″]Testify Solution[/reveal-answer]

[subconscious-answer a="fs-id1165037263477″]

a. Moves in the negative ten direction at a propagation speed of

$v=2.00\,\text{m/s}$

. b.

$\text{Δ}x=-6.00\,\text{m;}$

c.
Figure shows a graph labeled wave function versus time. Two identical pulse waves are shown on the graph. The red wave, labeled y parentheses x, 3, peaks at x = -6 m. The blue wave, labeled y parentheses x, 0, peaks at x = 0 m. The distance between the two peaks is labeled delta x = -6 m.

[/hidden-answer]

A string with a linear mass density of

$\mu =0.0085\,\text{kg/m}$

is stock-still at both ends. A 5.0-kg mass is hung from the cord, equally shown beneath. If a pulse is sent forth section A, what is the wave speed in department A and the wave speed in section B?

Consider 2 wave functions

${y}_{1}(x,t)=A\,\text{sin}(kx-\omega t)$

and

${y}_{2}(x,t)=A\,\text{sin}(kx+\omega t+\varphi )$

. What is the wave part resulting from the interference of the two wave? (Hint:

$\text{sin}(\alpha ±\beta )=\text{sin}\,\alpha \,\text{cos}\,\beta ±\text{cos}\,\alpha \,\text{sin}\,\beta$

and

$\varphi =\frac{\varphi }{2}+\frac{\varphi }{2}$

[reveal-answer q="fs-id1165038355254″]Bear witness Solution[/reveal-answer]

[hidden-answer a="fs-id1165038355254″]

$\begin{array}{cc} \text{sin}(kx-\omega t)=\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2})-\text{cos}(kx+\frac{\varphi }{2})\text{sin}(\omega t+\frac{\varphi }{2})\hfill \\ \text{sin}(kx-\omega t+\varphi )=\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2})+\text{cos}(kx+\frac{\varphi }{2})\text{sin}(\omega t+\frac{\varphi }{2})\hfill \\ \text{sin}(kx-\omega t)+\text{sin}(kx+\omega t+\varphi )=2\,\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2})\hfill \\ {y}_{R}=2\,A\,\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2})\hfill \end{array}$

[/hidden-answer]

The wave office that models a continuing wave is given as

${y}_{R}(x,t)=6.00\,\text{cm}\,\text{sin}(3.00\,{\text{m}}^{-1}x+1.20\,\text{rad})$

$\text{cos}(6.00\,{\text{s}}^{-1}t+1.20\,\text{rad})$

. What are two moving ridge functions that interfere to form this wave function? Plot the 2 wave functions and the sum of the sum of the ii moving ridge functions at

$t=1.00\,\text{s}$

to verify your reply.

Consider ii wave functions

${y}_{1}(x,t)=A\,\text{sin}(kx-\omega t)$

and

${y}_{2}(x,t)=A\,\text{sin}(kx+\omega t+\varphi )$

. The resultant wave form when you add the two functions is

${y}_{R}=2A\,\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2}).$

Consider the case where

$A=0.03\,{\text{m}}^{-1},$

$k=1.26\,{\text{m}}^{-1},$

$\omega =\pi \,{\text{s}}^{-1}$

, and

$\varphi =\frac{\pi }{10}$

. (a) Where are the kickoff iii nodes of the standing wave office starting at zero and moving in the positive x direction? (b) Using a spreadsheet, plot the two wave functions and the resulting function at fourth dimension

$t=1.00\,\text{s}$

to verify your respond.

[reveal-respond q="fs-id1165037009841″]Bear witness Solution[/reveal-respond]

[hidden-answer a="fs-id1165037009841″]

$\begin{array}{cc} \text{sin}(kx+\frac{\varphi }{2})=0,\,kx+\frac{\varphi }{2}=0,\pi ,2\pi ,\,1.26\,{\text{m}}^{-1}x+\frac{\pi }{20}=\pi ,2\pi ,3\pi \hfill \\ x=2.37\,\text{m},4.86\,\text{m},7.35\,\text{m}\hfill \end{array}$

;
Figure shows a graph with wave y1 in blue, wave y2 in red and wave y1 plus y2 in black. All three have a wavelength of 5 m. Waves y1 and y2 have the same amplitude and are slightly out of phase with each other. The amplitude of the black wave is almost twice that of the other two.

[/hidden-answer]

Glossary

antinode: location of maximum amplitude in standing waves

key frequency: everyman frequency that volition produce a standing moving ridge

node: point where the cord does not motion; more by and large, nodes are where the wave disturbance is zero in a standing moving ridge

normal style: possible standing moving ridge pattern for a standing wave on a string

overtone: frequency that produces standing waves and is higher than the cardinal frequency

standing wave: wave that can bounce back and forth through a particular region, effectively becoming stationary

chasonprather.blogspot.com

Source: https://opentextbc.ca/universityphysicsv1openstax/chapter/16-6-standing-waves-and-resonance/

You Can Decrease the Frequency of a Standing Wave on a String by __________.

16.half dozen Continuing Waves and Resonance

Learning Objectives

Standing Waves

Example

Standing Waves on a String

Solution

Significance

Check Your Understanding

Summary

Key Equations

Conceptual Questions

Problems

Additional Problems

Challenge Problems

Glossary

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