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Reflection of waves
Reflection of waves is the change in the direction of a wave upon striking the interface between two materials. When a wave strikes any interface between any two mediums the bouncing back of the wave is termed a reflection of waves.
The interface can be categorized into 2 types:
Open boundary: - When a wave strikes an interface in the case of open boundary it will get reflected as well as refracted.
Closed boundary or a rigid boundary: - When a wave is an incident on an interface it will completely get reflected. Example:-Wave striking wall (echo)
Reflection at the rigid boundary
Consider a string which is fixed to the wall at one end. When an incident wave hits a wall, it will exert a force on the wall.
By Newton’s third law, the wall exerts an equal and opposite force of equal magnitude on the string.
Since the wall is rigid, the wall won’t move, therefore no wave is generated at the boundary. This implies the amplitude at the boundary is 0.
As both the reflected wave and incident wave are completely out of phase at the boundary. Therefore φ=π.
Therefore, the incident wave is yi(x, t) = a sin (kx – ωt),
Reflected wave is yr(x, t) = a sin (kx + ωt + π) = – a sin (kx + ωt)
By superposition principle y= yi + yr =0
Conclusion: -
The reflection at the rigid body will take place with a phase reversal of π or 180.
Reflection at the open boundary
The reflection at an open boundary will take place without any phase change. In this case, a boundary pulse is generated. Therefore amplitude at the boundary is maximum.
- This means the reflected wave and incident wave are in phase with each other. As a result the phase difference φ=0.
- Therefore, the incident wave is yi(x, t) = a sin (kx – ωt),
- Reflected wave is yr(x, t) = a sin (kx – ωt).
- By superposition principle y= yi + yr =2a sin (kx – ωt)
Standing wave
A stationary wave is a wave which is not moving, i.e. it is at rest.
When two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere they produce a standing wave.
Conditions to have a standing wave: - Two traveling waves can produce a standing wave if the waves are moving in opposite directions and they have the same amplitude and frequency.
- At certain instances when the peaks of both the waves will overlap. Then both the peaks will add up to form the resultant wave.
- At certain instances when the peak of one wave combines with the negative of the second wave . Then the net amplitude will become 0.
- As a result, a standing wave is produced. In the case of a stationary wave, the waveform does not move.
Mathematically,
- Wave travelling towards left yl(x,t) =a sin(kx– ωt) and towards right yr(x,t) =a sin (kx + ωt)
The principle of superposition gives, for the combined wave
y (x, t) = yl(x, t) + yr(x, t) = a sin (kx – ωt) + a sin (kx + ωt)
Y(x, t) = (2a sin kx) cos ωt (By calculating and simplifying)
The above equation represents the standing wave expression.
Amplitude = 2a sin kx.
- The amplitude is dependent on the position of the particle.
- The cos ωt represents the time-dependent variation or the phase of the standing wave.
Below is the difference between traveling/progressing waves and standing waves
Nodes and Antinodes of Standing Wave
The amplitude of a standing wave doesn’t remain the same throughout the wave. It keeps on changing as it is a function of x.
At certain positions the value of the amplitude is maximum and at certain positions, the value of the amplitude is 0.
- Nodes: - Nodes represent the positions of zero amplitude.
- Antinodes: - Antinodes represent the positions of maximum amplitude.
Characteristics of standing wave:
- There are certain points in the medium in a standing wave, which are permanently at rest; these are called nodes. The distance between two consecutive nodes is
- There are certain points in standing waves that have maximum amplitude. These are called antinodes. The distance between two consecutive antinodes is also
- The wavelength and the time period of the stationary waves is the same as that of the component waves by which it is formed.
Nodes and Antinodes: system closed at both ends
System closed at both ends means both ends are rigid boundaries.
Whenever there is a rigid body there is no displacement at the boundary. This implies the boundary amplitude is always 0. Nodes are formed at the boundary.
Standing waves on a string of length L fixed at both ends have restricted wavelength. This means the wave will vibrate for certain specific values of wavelength.
At both ends, nodes will be formed, so Amplitude=0.
Expression for node x = (nλ)/2.This value is true when x is 0 and L.
When x=L:- L=(nλ)/2 =>λ=(2L)/n ; n=1,2,3,4,…..
λ cannot take any value but it can take values that satisfy λ= (2L)/n this expression.
That is why we can say that the standing wave on a string which is tied on both ends has a restricted wavelength. As the wavelength is restricted therefore wavenumber is also restricted.
Here n=1 is a fundamental mode of vibration (first harmonics)
Similarly n= 2, 3, and 4 are called second, third and fourth harmonics respectively and so on.
Frequency =velocity / wavelength
Corresponding frequencies which a standing wave can have is given as:
Where v= speed of the traveling wave. These frequencies are known as natural frequencies or modes of oscillations.
Vibrations of Air Column
The vibrating air column in organ pipes is a common example of stationary waves. An organ pipe is a cylindrical tube which may be closed (at one end) or open at both ends (open organ pipe).
If the air in the pipe at its open end is made to vibrate longitudinal waves are produced. These waves travel along the pipe towards its far end and are reflected back. Thus due to the superposition of incident and reflected waves, stationary waves are formed in the pipe.
Closed organ pipe
If one end of the pipe is closed, the reflected wave is 180 degrees out of phase with the incoming wave. This displacement of the small volume elements at the closed end must always be zero. Hence the closed end must be a displacement node.
Open organ pipe
If both ends of the pipes are open and the system of air is directed against an edge, standing longitudinal waves can be set up in the tube. The open end has displacement antinodes
Various modes of vibration of air column in an open organ pipe are shown below
Beats
This phenomenon arises from the interference of waves having nearly the same frequencies.
The periodic variation on the intensity of the sound wave caused by the superposition of two sound waves of nearly the same frequencies and amplitude traveling in the same direction are called beats.
One rise and one fall in the intensity of sound constitute one beat and the number of beats per second is called beat frequency.
The frequency of two sources or two waves should not differ by more than 10 Hz, because if it is more than rising and fall in intensity of sound due to persistence of hearing.
If f1 and f2 are the frequencies of the two waves such that (f1 > f2) the
Beat frequency f beat = f1 - f2
Doppler’s Effect
Doppler Effect is the phenomenon of motion-related frequency change.
Consider if a truck is coming from a very far-off location as it approaches our house, the sound increases and when it passes our house the sound will be maximum. And when it goes away from our house the sound decreases. This effect is known as the Doppler Effect.
In other words, the apparent change in frequency heard by the observer due to relative motion between source and the observer is known as the Doppler Effect.
A person who is observing is known as Observer and the object from which the sound wave is getting generated is known as Source.
When the observer and source come nearer to each other as result waves get compressed. Therefore wavelength decreases and frequency increases.
Doppler Effect will be analyzed under three cases.
Case 1:- Observer is stationary but the source is moving.
Case 2:-Observer is moving but the source is stationary.
Case 3:- Both the observer and the source are moving.
