# Amplitude of superposition of two waves

Search titles only. Search Advanced search…. Log in. Support PF! Buy your school textbooks, materials and every day products Here! JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding. Superposition: Adding two waves together -- amplitude help. Thread starter Toon Start date Sep 12, Tags standing waves. Homework Statement " Two traveling waves are generated on the same taut string. Individually, the two traveling waves can be described by the following two equations:.

RUber Homework Helper. Do you need to find a complicated answer using your k and phi terms? Is it not reasonable to assume that regardless of the terms, there is a max that the sine function will take? Homework Helper. Insights Author. Gold Member. Are the two waves travelling in the same, or in opposite directions? Can a pair of peaks - one from each wave - ever coincide?

Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. As the theory of superposition of waves express the amplitudes of the interfering waves do algebraically sum up. But when we sum up the the total energies of a particle doing a harmonic motion due to each wave, we get a different result from the value of energy that we get when when we calculate it with the resultant amplitude of the interfered wave.

How can we combine these two expressions we get? I don't think that energy considerations are particularly useful if you want to know the amplitude of the resultant wave.

For example for a standing wave, depending on the location of the particle it can get different potential energies. The correct way is as you suggest, to sum up the waves. So all that is happening is that the energy is being channel along preferred directions rather that travelling uniformly in all directions.

An equivalent analysis can be done if the amplitudes of the waves from the two sources is not the same. I think the question is in regards to energy conservation as a function of position. The power time average of the amplitude squared of each individual wave:.

For instance, with counter propagating waves, it is always ZERO at nodes. One must then average over position to recover all the energy and none more.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. When two waves interfere, how to calculate the amplitude of the wave?

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Asked 3 years, 10 months ago. Active 2 years ago. Viewed 7k times. Arun Wi Arun Wi 11 1 1 silver badge 2 2 bronze badges. Active Oldest Votes. Kyle Kanos 23k 11 11 gold badges 54 54 silver badges bronze badges.

Hans Hans 7 7 bronze badges. Farcher Farcher Thus, an interference pattern moves energy around spatially, while still conserving it globally. ChoMedit 1 1 silver badge 9 9 bronze badges.What happens when two waves meet while they travel through the same medium? What effect will the meeting of the waves have upon the appearance of the medium?

Will the two waves bounce off each other upon meeting much like two billiard balls would or will the two waves pass through each other? These questions involving the meeting of two or more waves along the same medium pertain to the topic of wave interference. Wave interference is the phenomenon that occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape that results from the net effect of the two individual waves upon the particles of the medium.

To begin our exploration of wave interference, consider two pulses of the same amplitude traveling in different directions along the same medium. Let's suppose that each displaced upward 1 unit at its crest and has the shape of a sine wave. As the sine pulses move towards each other, there will eventually be a moment in time when they are completely overlapped.

At that moment, the resulting shape of the medium would be an upward displaced sine pulse with an amplitude of 2 units.

### Wave interference

The diagrams below depict the before and during interference snapshots of the medium for two such pulses. The individual sine pulses are drawn in red and blue and the resulting displacement of the medium is drawn in green. This type of interference is sometimes called constructive interference. Constructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the same direction.

In this case, both waves have an upward displacement; consequently, the medium has an upward displacement that is greater than the displacement of the two interfering pulses. Constructive interference is observed at any location where the two interfering waves are displaced upward. But it is also observed when both interfering waves are displaced downward.

This is shown in the diagram below for two downward displaced pulses. In this case, a sine pulse with a maximum displacement of -1 unit negative means a downward displacement interferes with a sine pulse with a maximum displacement of -1 unit.

These two pulses are drawn in red and blue. The resulting shape of the medium is a sine pulse with a maximum displacement of -2 units. Destructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction.

This is depicted in the diagram below. In the diagram above, the interfering pulses have the same maximum displacement but in opposite directions. The result is that the two pulses completely destroy each other when they are completely overlapped.

At the instant of complete overlap, there is no resulting displacement of the particles of the medium. This "destruction" is not a permanent condition. In fact, to say that the two waves destroy each other can be partially misleading. When it is said that the two pulses destroy each otherwhat is meant is that when overlapped, the effect of one of the pulses on the displacement of a given particle of the medium is destroyed or canceled by the effect of the other pulse.

Recall from Lesson 1 that waves transport energy through a medium by means of each individual particle pulling upon its nearest neighbor. When two pulses with opposite displacements i. Once the two pulses pass through each other, there is still an upward displaced pulse and a downward displaced pulse heading in the same direction that they were heading before the interference.Russell and may not used in other web pages or reports without permission.

The content of this page was originally posted on July 25, The principle of superposition may be applied to waves whenever two or more waves travelling through the same medium at the same time. The waves pass through each other without being disturbed. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements.

This is true of waves which are finite in length wave pulses or which are continuous sine waves. The animation at left shows two Gaussian wave pulses are travelling in the same medium but in opposite directions.

The two waves pass through each other without being disturbed, and the net displacement is the sum of the two individual displacements. It should also be mentioned that this medium is nondispersive all frequencies travel at the same speed since the Gaussian wave pulses do not change their shape as they propagate. If the medium was dispersivethen the waves would change their shape. Solitons are examples of nonlinear waves that do not obey the principle of superposition when they interact with each other.

Two waves with the same amplitude, frequency, and wavelength are travelling in the same direction. Using the principle of superposition, the resulting wave displacement may be written as:. The animation at left shows how two sinusoidal waves with the same amplitude and frequency can add either destructively or constructively depending on their relative phase.

NOTE: this animation does not depict the propagation of actual waves in a medium - it only serves to illustrate the effect of changing the phase shift between two waves and the resulting constructive or destructive interference.

The phase difference between the two waves increases with time so that the effects of both constructive and destructive interference may be seen. When the two individual waves are exactly in phase the result is large amplitude. When the two gray waves become exactly out of phase the sum wave is zero.

A travelling wave moves from one place to another, whereas a standing wave appears to stand still, vibrating in place. In this animation, two waves with the same amplitude, frequency, and wavelength are travelling in opposite directions.Up to now, we have been studying mechanical waves that propagate continuously through a medium, but we have not discussed what happens when waves encounter the boundary of the medium or what happens when a wave encounters another wave propagating through the same medium.

Waves do interact with boundaries of the medium, and all or part of the wave can be reflected. For example, when you stand some distance from a rigid cliff face and yell, you can hear the sound waves reflect off the rigid surface as an echo.

Waves can also interact with other waves propagating in the same medium. If you throw two rocks into a pond some distance from one another, the circular ripples that result from the two stones seem to pass through one another as they propagate out from where the stones entered the water.

This phenomenon is known as interference. In this section, we examine what happens to waves encountering a boundary of a medium or another wave propagating in the same medium. We will see that their behavior is quite different from the behavior of particles and rigid bodies. Later, when we study modern physics, we will see that only at the scale of atoms do we see similarities in the properties of waves and particles.

When a wave propagates through a medium, it reflects when it encounters the boundary of the medium. The wave before hitting the boundary is known as the incident wave. The wave after encountering the boundary is known as the reflected wave. How the wave is reflected at the boundary of the medium depends on the boundary conditions; waves will react differently if the boundary of the medium is fixed in place or free to move Figure A fixed boundary condition exists when the medium at a boundary is fixed in place so it cannot move.

A free boundary condition exists when the medium at the boundary is free to move. Here, one end of the string is fixed to a wall so the end of the string is fixed in place and the medium the string at the boundary cannot move. As the incident wave encounters the wall, the string exerts an upward force on the wall and the wall reacts by exerting an equal and opposite force on the string.

The reflection at a fixed boundary is inverted. Note that the figure shows a crest of the incident wave reflected as a trough. If the incident wave were a trough, the reflected wave would be a crest.

Here, one end of the string is tied to a solid ring of negligible mass on a frictionless pole, so the end of the string is free to move up and down. As the incident wave encounters the boundary of the medium, it is also reflected. In the case of a free boundary condition, the reflected wave is in phase with respect to the incident wave. In this case, the wave encounters the free boundary applying an upward force on the ring, accelerating the ring up.

The ring travels up to the maximum height equal to the amplitude of the wave and then accelerates down towards the equilibrium position due to the tension in the string. The figure shows the crest of an incident wave being reflected in phase with respect to the incident wave as a crest. If the incident wave were a trough, the reflected wave would also be a trough. The amplitude of the reflected wave would be equal to the amplitude of the incident wave. In some situations, the boundary of the medium is neither fixed nor free.

In this case, the reflected wave is out of phase with respect to the incident wave. There is also a transmitted wave that is in phase with respect to the incident wave. Both the transmitted and reflected waves have amplitudes less than the amplitude of the incident wave. If the tension is the same in both strings, the wave speed is higher in the string with the lower linear mass density. In this case, the reflected wave is in phase with respect to the incident wave. Both the incident and the reflected waves have amplitudes less than the amplitude of the incident wave.

Here you may notice that if the tension is the same in both strings, the wave speed is higher in the string with the lower linear mass density.

Most waves do not look very simple. Complex waves are more interesting, even beautiful, but they look formidable. Most interesting mechanical waves consist of a combination of two or more traveling waves propagating in the same medium. The principle of superposition can be used to analyze the combination of waves.Figure 1. These waves result from the superposition of several waves from different sources, producing a complex pattern.

Most waves do not look very simple. Simple waves may be created by a simple harmonic oscillation, and thus have a sinusoidal shape.

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Complex waves are more interesting, even beautiful, but they look formidable. Most waves appear complex because they result from several simple waves adding together.

Luckily, the rules for adding waves are quite simple. When two or more waves arrive at the same point, they superimpose themselves on one another.

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More specifically, the disturbances of waves are superimposed when they come together—a phenomenon called superposition. Each disturbance corresponds to a force, and 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 crests of the two waves are precisely aligned, as are the troughs. This superposition produces pure constructive interference.

Because the disturbances add, pure constructive interference produces a wave that has twice the amplitude of the individual waves, but has the same wavelength. Because the disturbances are in the opposite direction for this superposition, the resulting amplitude is zero for pure destructive interference—the waves completely cancel. Figure 2. Pure constructive interference of two identical waves produces one with twice the amplitude, but the same wavelength.

Figure 3. Pure destructive interference of two identical waves produces zero amplitude, or complete cancellation. Figure 4. Superposition of non-identical waves exhibits both constructive and destructive interference. While pure constructive and pure destructive interference do occur, they require precisely aligned identical waves.

The superposition of most waves produces a combination of constructive and destructive interference and can vary from place to place and time to time.

## Acoustics and Vibration Animations

Sound from a stereo, for example, can be loud in one spot and quiet in another. Varying loudness means the sound waves add partially constructively and partially destructively at different locations.In physicsinterference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude.

Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency.

Interference effects can be observed with all types of waves, for example, lightradioacousticsurface water wavesgravity wavesor matter waves. The resulting images or graphs are called interferograms. The principle of superposition of waves states that when two or more propagating waves of same type are incident on the same point, the resultant amplitude at that point is equal to the vector sum of the amplitudes of the individual waves.

If a crest of one wave meets a trough of another wave, then the amplitude is equal to the difference in the individual amplitudes—this is known as destructive interference. If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values.

Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, and will produce a maximum displacement. In other places, the waves will be in anti-phase, and there will be no net displacement at these points.

Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre. Interference of light is a common phenomenon that can be explained classically by the superposition of waves, however a deeper understanding of light interference requires knowledge of wave-particle duality of light which is due to quantum mechanics.

Prime examples of light interference are the famous double-slit experimentlaser speckleanti-reflective coatings and interferometers. Traditionally the classical wave model is taught as a basis for understanding optical interference, based on the Huygens—Fresnel principle.

The above can be demonstrated in one dimension by deriving the formula for the sum of two waves.

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The equation for the amplitude of a sinusoidal wave traveling to the right along the x-axis is. Suppose a second wave of the same frequency and amplitude but with a different phase is also traveling to the right.

The two waves will superpose and add: the sum of the two waves is. A simple form of interference pattern is obtained if two plane waves of the same frequency intersect at an angle. Interference is essentially an energy redistribution process. The energy which is lost at the destructive interference is regained at the constructive interference. Assuming that the two waves are in phase at the point Bthen the relative phase changes along the x -axis.

The phase difference at the point A is given by. Constructive interference occurs when the waves are in phase, and destructive interference when they are half a cycle out of phase. Thus, an interference fringe pattern is produced, where the separation of the maxima is. The fringes are observed wherever the two waves overlap and the fringe spacing is uniform throughout. A point source produces a spherical wave. If the light from two point sources overlaps, the interference pattern maps out the way in which the phase difference between the two waves varies in space.

This depends on the wavelength and on the separation of the point sources. The figure to the right shows interference between two spherical waves. The wavelength increases from top to bottom, and the distance between the sources increases from left to right.

When the plane of observation is far enough away, the fringe pattern will be a series of almost straight lines, since the waves will then be almost planar.

Interference occurs when several waves are added together provided that the phase differences between them remain constant over the observation time. It is sometimes desirable for several waves of the same frequency and amplitude to sum to zero that is, interfere destructively, cancel.