# Superposition And Standing Waves Pdf

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## Standing wave

In physics , a standing wave , also known as a stationary wave , is a wave which oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with time, and the oscillations at different points throughout the wave are in phase.

The locations at which the absolute value of the amplitude is minimum are called nodes , and the locations where the absolute value of the amplitude is maximum are called antinodes. Standing waves were first noticed by Michael Faraday in Faraday observed standing waves on the surface of a liquid in a vibrating container.

This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions.

The most common cause of standing waves is the phenomenon of resonance , in which standing waves occur inside a resonator due to interference between waves reflected back and forth at the resonator's resonant frequency. For waves of equal amplitude traveling in opposing directions, there is on average no net propagation of energy. As an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the lee of mountain ranges.

Such waves are often exploited by glider pilots. Standing waves and hydraulic jumps also form on fast flowing river rapids and tidal currents such as the Saltstraumen maelstrom. Many standing river waves are popular river surfing breaks. Standing wave in stationary medium.

The red dots represent the wave nodes. A standing wave black depicted as the sum of two propagating waves traveling in opposite directions red and blue. A standing wave on a circular membrane , an example of standing waves in two dimensions. This is the fundamental mode. A higher harmonic standing wave on a disk with two nodal lines crossing at the center. As an example of the second type, a standing wave in a transmission line is a wave in which the distribution of current , voltage , or field strength is formed by the superposition of two waves of the same frequency propagating in opposite directions.

The effect is a series of nodes zero displacement and anti-nodes maximum displacement at fixed points along the transmission line. Such a standing wave may be formed when a wave is transmitted into one end of a transmission line and is reflected from the other end by an impedance mismatch , i.

In practice, losses in the transmission line and other components mean that a perfect reflection and a pure standing wave are never achieved. The result is a partial standing wave , which is a superposition of a standing wave and a traveling wave. The degree to which the wave resembles either a pure standing wave or a pure traveling wave is measured by the standing wave ratio SWR. Another example is standing waves in the open ocean formed by waves with the same wave period moving in opposite directions.

These may form near storm centres, or from reflection of a swell at the shore, and are the source of microbaroms and microseisms. This section considers representative one- and two-dimensional cases of standing waves. First, an example of an infinite length string shows how identical waves traveling in opposite directions interfere to produce standing waves. Next, two finite length string examples with different boundary conditions demonstrate how the boundary conditions restrict the frequencies that can form standing waves.

Next, the example of sound waves in a pipe demonstrates how the same principles can be applied to longitudinal waves with analogous boundary conditions. Standing waves can also occur in two- or three-dimensional resonators. With standing waves on two-dimensional membranes such as drumheads , illustrated in the animations above, the nodes become nodal lines, lines on the surface at which there is no movement, that separate regions vibrating with opposite phase.

These nodal line patterns are called Chladni figures. In three-dimensional resonators, such as musical instrument sound boxes and microwave cavity resonators , there are nodal surfaces. This section includes a two-dimensional standing wave example with a rectangular boundary to illustrate how to extend the concept to higher dimensions. To begin, consider a string of infinite length along the x -axis that is free to be stretched transversely in the y direction. For a harmonic wave traveling to the right along the string, the string's displacement in the y direction as a function of position x and time t is [9].

The displacement in the y -direction for an identical harmonic wave traveling to the left is. For identical right- and left-traveling waves on the same string, the total displacement of the string is the sum of y R and y L ,. Note that Equation 1 does not describe a traveling wave. As the left-traveling blue wave and right-traveling green wave interfere, they form the standing red wave that does not travel and instead oscillates in place.

Because the string is of infinite length, it has no boundary condition for its displacement at any point along the x -axis. As a result, a standing wave can form at any frequency. These locations are called nodes. At locations on the x -axis that are odd multiples of a quarter wavelength.

These locations are called anti-nodes. The string will have some damping as it is stretched by traveling waves, but assume the damping is very small. In this situation, the driving force produces a right-traveling wave. That wave reflects off the right fixed end and travels back to the left, reflects again off the left fixed end and travels back to the right, and so on. Eventually, a steady state is reached where the string has identical right- and left-traveling waves as in the infinite-length case and the power dissipated by damping in the string equals the power supplied by the driving force so the waves have constant amplitude.

Checking the values of y at the two ends,. L is given, so the boundary condition restricts the wavelength of the standing waves to [10]. Waves can only form standing waves on this string if they have a wavelength that satisfies this relationship with L. If waves travel with speed v along the string, then equivalently the frequency of the standing waves is restricted to [10] [11]. Higher integer values of n correspond to modes of oscillation called harmonics or overtones.

To compare this example's nodes to the description of nodes for standing waves in the infinite length string, note that Equation 2 can be rewritten as. In this variation of the expression for the wavelength, n must be even. Cross multiplying we see that because L is a node, it is an even multiple of a quarter wavelength,. This example demonstrates a type of resonance and the frequencies that produce standing waves can be referred to as resonant frequencies.

This leads to a different set of wavelengths than in the two-fixed-ends example. Here, the wavelength of the standing waves is restricted to. Note that in this example n only takes odd values. Because L is an anti-node, it is an odd multiple of a quarter wavelength. This example also demonstrates a type of resonance and the frequencies that produce standing waves are called resonant frequencies. Consider a standing wave in a pipe of length L.

The air inside the pipe serves as the medium for longitudinal sound waves traveling to the right or left through the pipe. While the transverse waves on the string from the previous examples vary in their displacement perpendicular to the direction of wave motion, the waves traveling through the air in the pipe vary in terms of their pressure and longitudinal displacement along the direction of wave motion. The wave propagates by alternately compressing and expanding air in segments of the pipe, which displaces the air slightly from its rest position and transfers energy to neighboring segments through the forces exerted by the alternating high and low air pressures.

If identical right- and left-traveling waves travel through the pipe, the resulting superposition is described by the sum. Note that this formula for the pressure is of the same form as Equation 1 , so a stationary pressure wave forms that is fixed in space and oscillates in time. If the end of a pipe is closed, the pressure is maximal since the closed end of the pipe exerts a force that restricts the movement of air.

This corresponds to a pressure anti-node. If the end of the pipe is open, the pressure variations are very small, corresponding to a pressure node. In terms of reflections, open ends partially reflect waves back into the pipe, allowing some energy to be released into the outside air.

Ideally, closed ends reflect the entire wave back in the other direction. First consider a pipe that is open at both ends, for example an open organ pipe or a recorder. Given that the pressure must be zero at both open ends, the boundary conditions are analogous to the string with two fixed ends,. Examples include a bottle and a clarinet. This pipe has boundary conditions analogous to the string with only one fixed end. Its standing waves have wavelengths restricted to [17]. Note that for the case where one end is closed, n only takes odd values just like in the case of the string fixed at only one end.

So far, the wave has been written in terms of its pressure as a function of position x and time. Alternatively, the wave can be written in terms of its longitudinal displacement of air, where air in a segment of the pipe moves back and forth slightly in the x -direction as the pressure varies and waves travel in either or both directions.

In terms of longitudinal displacement, closed ends of pipes correspond to nodes since air movement is restricted and open ends correspond to anti-nodes since the air is free to move. We can also consider a pipe that is closed at both ends. In this case, both ends will be pressure anti-nodes or equivalently both ends will be displacement nodes.

For example, the longest wavelength that resonates—the fundamental mode—is again twice the length of the pipe, except that the ends of the pipe have pressure anti-nodes instead of pressure nodes. Between the ends there is one pressure node. In the case of two closed ends, the wavelength is again restricted to. A Rubens' tube provides a way to visualize the pressure variations of the standing waves in a tube with two closed ends. Next, consider transverse waves that can move along a two dimensional surface within a rectangular boundary of length L x in the x -direction and length L y in the y -direction.

Examples of this type of wave are water waves in a pool or waves on a rectangular sheet that has been pulled taut. In two dimensions and Cartesian coordinates, the wave equation is. To solve this differential equation, let's first solve for its Fourier transform , with. This is an eigenvalue problem where the frequencies correspond to eigenvalues that then correspond to frequency-specific modes or eigenfunctions. Specifically, this is a form of the Helmholtz equation and it can be solved using separation of variables.

This leads to two coupled ordinary differential equations. The x term equals a constant with respect to x that we can define as. This x -dependence is sinusoidal—recalling Euler's formula —with constants A k x and B k x determined by the boundary conditions.

## Standing wave

Most waves do not look very simple. They look more like the waves in Figure Most waves appear complex because they result from two or more simple waves that combine as they come together at the same place at the same time—a phenomenon called superposition. Waves superimpose by adding their disturbances; each disturbance corresponds to a force, and all the forces add. If the disturbances are along the same line, then the resulting wave is a simple addition of the disturbances of the individual waves, that is, their amplitudes add. The two special cases of superposition that produce the simplest results are pure constructive interference and pure destructive interference. Pure constructive interference occurs when two identical waves arrive at the same point exactly in phase.

As we saw in the last section, when waves have the same frequency, it is possible for them to interfere completely, either destructively or constructively. Waves of the same frequency that interfere can be generated by propagating waves along a string, as the reflected waves from the end of the string will have the same frequency as, and interfere with, the original waves. The standing wave is named this way because it does not appear to propagate along the string. Instead, each point on the string will oscillate with an amplitude that depends on where the point is located along on the string. In contrast, for a traveling wave, all of the points oscillate with the same amplitude.

## 16.7: Standing Waves and Resonance

An Interactive JAVA environment to allow one to look at the linear superposition of up to four ocean waves of different amplitudes and periods. Direction of travel right or left and the local depth of water can be specified. Students learn definitions of wavelength, wave speed, wave amplitude, and wave period. Waves reflection and standing waves can also be explored. Here is a beginning sample activity that can be used with this Applet.

Standing wave , also called stationary wave , combination of two waves moving in opposite directions, each having the same amplitude and frequency. The phenomenon is the result of interference; that is, when waves are superimposed, their energies are either added together or canceled out. In the case of waves moving in the same direction, interference produces a traveling wave.

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#### Standing Waves

In physics , a standing wave , also known as a stationary wave , is a wave which oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with time, and the oscillations at different points throughout the wave are in phase. The locations at which the absolute value of the amplitude is minimum are called nodes , and the locations where the absolute value of the amplitude is maximum are called antinodes. Standing waves were first noticed by Michael Faraday in Faraday observed standing waves on the surface of a liquid in a vibrating container.

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