Two different waves meet at a point are called nodes. The maximum displacement of the wave in either side from equilibrium position is called antinodes. In Chapter 8 we studied what happens when two particles meet (collide). the two ``colliding'' waves, a wide variety of different wave patterns can occur, . Similarly, the maximum displacement of the medium can be found at points such that. If the period off a wave is equal to the amount of time it takes for the wave to travel to a fixed When two waves meet the point of maximum displacement is a.

The only difference is that the two above waves are not in synch. For example, when wave one causes a particular medium particle to move up at a specific instant of time, wave two makes it move down, or perhaps move up by a smaller or larger distance. Such a difference is described mathematically by the phase constant. For example, if ,both waves are in synch and make the medium particles move in the same direction by the same amount, so they simply add up for maximum effect.

We then say that the two waves are in phase and interfere constructively. On the other hand, ifthe two waves are completely out of synch.

- Interference of Waves
- Constructive and Destructive Interference
- Superposition of two opposite direction wave pulses

When one causes a certain medium particle to move up, the other causes it to move down by the same exact amount, so the net effect is zero: Such an interference effect can also be studied mathematically using simple trigonometry. The net disturbance of the medium resulting from the interference of the two above waves is described by the wavefunction We can see what the resulting wave pattern looks like by using the simple identity which gives for the net wavefunction Recalling Chapter 13, we recognize a sinusoidal travelling wave moving toward the positive x-axis, just like the two original waves.

Its frequency and wavelength are the same as for the original waves. However, its amplitude A' is different: As you can see, by changing the difference in the phase,between the two incoming waves, we can dramatically alter the amplitude of the resulting wave. Standing waves A travelling wave transports energy and momentum in the direction of propagation.

What happens when two sinusoidal waves with the same frequency, wavelength, and amplitude travel in opposite directions?

For example, consider the interference of two waves described by the wavefunctions By using the above trigonometric identity, we find that the result is described by the wavefunction This no longer has the form of a travelling wave.

## Acoustics and Vibration Animations

Instead, it has the form of the harmonic oscillator displacement studied in Chapter 12, with an amplitude that differs for different medium particles: Contrast this to the wavefunction of a sinusoidal travelling wave. In the latter case, different particles oscillate with different phases, given by kx, but with the same amplitude.

In the former case, different particles oscillate with the same phase but with different amplitudes A', and there is no moving disturbance, no propagating pulse. We call such a disturbance a standing waves because there is no net transfer of energy or momentum. The above amplitude vanishes at medium points located at position x such that Such points in the medium never move and are called nodes. It is important to note that consequative nodes are a distance apart. Similarly, the maximum displacement of the medium can be found at points such that Such points in the medium are called antinodes.

Adjacent antinodes are separated by a distance ofwhile the distance between a node and an adjacent antinode is.

Consider now a string of length L. What kind of wave patterns might we expect when both ends are fixed tied? If we create a travelling wave on this string, it will reflect at the ends. We will then have interference between the reflected and incident waves.

### Interference of Waves

These move in opposite directions and will therefore result in a standing wave pattern. Such a restriction has in fact very important consequences. As discussed above, the distance between adjacent nodes is equal to.

If both ends of the string are nodes, then we must have that where is an integer. When the waves move away from the point where they came together, in other words, their form and motion is the same as it was before they came together. Constructive interference Constructive interference occurs whenever waves come together so that they are in phase with each other.

This means that their oscillations at a given point are in the same direction, the resulting amplitude at that point being much larger than the amplitude of an individual wave.

For two waves of equal amplitude interfering constructively, the resulting amplitude is twice as large as the amplitude of an individual wave. For waves of the same amplitude interfering constructively, the resulting amplitude is times larger than the amplitude of an individual wave. Constructive interference, then, can produce a significant increase in amplitude.

The following diagram shows two pulses coming together, interfering constructively, and then continuing to travel as if they'd never encountered each other. Another way to think of constructive interference is in terms of peaks and troughs; when waves are interfering constructively, all the peaks line up with the peaks and the troughs line up with the troughs.

Destructive interference Destructive interference occurs when waves come together in such a way that they completely cancel each other out. When two waves interfere destructively, they must have the same amplitude in opposite directions.

When there are more than two waves interfering the situation is a little more complicated; the net result, though, is that they all combine in some way to produce zero amplitude.

In general, whenever a number of waves come together the interference will not be completely constructive or completely destructive, but somewhere in between. It usually requires just the right conditions to get interference that is completely constructive or completely destructive.

The following diagram shows two pulses interfering destructively. Again, they move away from the point where they combine as if they never met each other. Reflection of waves This applies to both pulses and periodic waves, although it's easier to see for pulses. Consider what happens when a pulse reaches the end of its rope, so to speak. The wave will be reflected back along the rope.

### Wave interference - Wikipedia

If the end is free, the pulse comes back the same way it went out so no phase change. If the pulse is traveling along one rope tied to another rope, of different density, some of the energy is transmitted into the second rope and some comes back. For a pulse going from a light rope to a heavy rope, the reflection occurs as if the end is fixed. From heavy to light, the reflection is as if the end is free.

Standing waves Moving on towards musical instruments, consider a wave travelling along a string that is fixed at one end. The reflected wave will interfere with the part of the wave still moving towards the fixed end.