Resonance

In Physics, there are a lot of systems that have a nature to oscillate with larger amplitudes on such frequency, and that condition is called Resonance. The frequency on which oscillation of the highest frequency is found is called Resonant Frequency. The incident of resonance is associated with all kinds of vibrations or waves. Resonance can happen with Mechanical, Sound, and Electromagnetic wave functions.

Important Things for Resonance 

There are three things needed for the incident of Resonance, and they are:

1. An Object or a System which has a natural frequency.

2. Driving Force whose frequency is the same as the natural frequency of a system.

3. The elements which can destroy the energy of the system must be least.

(Note - In a system, Friction, Viscosity, and Resistance are some elements which are responsible for the energy loss.)

Some Implications of Resonance 

Some implications of the resonant frequency are as follows:

1. It is easy to vibrate an object at its natural frequency, but it is hard to vibrate an object at the other frequency.

2. A vibrating object chooses only that frequency from complex excitation which is equal to its frequency. Thus, it works like a Filter.

3. Mostly vibrating objects have multiple Resonant frequencies. 

Examples of Resonance - Some examples of Resonance are as follows: 

1. Swing - When pushing the swing, keep in mind that the push should be given at the same interval which is the natural rotation of the swing, and then the dimension of that swing increases only. That is, every time the hammock makes more angle than its mean position. On the contrary, if it is pushed on another frequency without taking into consideration as mentioned above, its effect can be very low, zero or negative.

2. Radio and Television - There is a tuned circuit inside the radio and the TV which helps to hear or see a channel. When we rotate the 'nab' of the radio, it is virtually changing the resonance frequency of this tuned circuit. Any time the resonance frequency of this circuit matches the frequency of any station or channel, that channel we receive. 

\[I(\omega) \propto \frac{\frac{\Gamma}{2}}{(\omega - \Omega)^{2} + (\frac{\Gamma}{2})^{2}}\]

3. Laser - The Laser is an electromagnetic wave, but its special point is that it is extremely convergent, that is, the frequency of all its photons is equal to or very near one frequency. In addition, the phases of all the vibrations are also the same. The laser is also produced by using optical resonance in an optical cavity. 

4. Sound - Music instruments have special arrangements for sound resonance.

5. A Crystal Wineglass shatters when it comes into the contact with a musical tone of the right pitch

The principle of Resonance - If the resonance frequency of a linear oscillator is Ω and it is being run from a source ω frequency, then the intensity of the oscillations appears from the following equation:

f=12πΓ  Here Γ indicates the situation of damping if the system is called Linewidth. The Linewidth is directly proportional to the dampness of the System. The Intensity is directly proportional to the square of the Amplitude and the line width is inversely proportional to the Q factor, and the Q factor is a measure of the sharpness of Resonance

Types of Resonance 

There are many types of Resonance which are as follows:

Mechanical and Acoustic Resonance - In a Mechanical System, Mechanical Resonance is a nature to respond at greater amplitude, when the frequency of its oscillations matches the system's natural frequency of vibration (Resonant Frequency)

Here, m denotes the Mass and k denotes the Spring Constant. 

\[f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}\]

The Resonance Frequency for small distance is calculated by the formula given below: 

\[f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}\]

Here, g denotes the acceleration due to gravity, and L denotes the length.

Acoustic resonance is an important thing for musical instruments because resonators are used in mostly acoustic musical instruments such as the strings and the body of a violin, and the length of a tube in a flute. Apart from Acoustic musical instruments, Acoustic Resonance is an important thing for hearing too. Acoustic Resonance helps us in hearing.

Electrical Resonance - In an Electrical circuit, Electrical Resonance occurs when the inductive reactance and the capacitive reactance are equal in the magnitude. In some electrical circuits, Electrical Resonance occurs, when the impedance between the input and output of the electrical circuit is almost zero, and the transfer function of the circuit is near to one. In an electrical circuit, the series impedance of the two elements is at the minimum, and the parallel impedance is at the maximum when electrical resonance occurs.

The inductive reactance and the capacitive reactance are equal in magnitude, ωL = 1/ωC, so: 

\[\omega = \frac{1}{\sqrt{LC}}\]

Where ω = 2πf

f = frequency of Resonance, hertz

L = Inductance, Henry

C = Capacitance, Farad

The quality of Resonance is determined by the Q factor. Q factor is such a parameter in Physics which is dimensionless. How under-damped an oscillator or resonator will be described by the Q factor. It also characterizes a resonator's bandwidth relative to its center frequency. If the Q factor goes higher, the energy loss goes lower. If the Q factor goes lower, the energy loss goes higher. 

An RLC circuit consists of resistance, inductance, and capacitance which is connected in series or in parallel in electrical resonance. The RLC circuit is almost the same as the LC circuit, but there is only one difference. In the RLC circuit, a resistor is present which reduces the energy losses in a circuit. This effect of the Resistor is called Damping. Apart from this, a resistor reduces the peak resonant frequency of damped oscillation.

Optical Resonance 

Optical Resonator is a part of a Laser which consists of two mirrors, one highly reflective and one partially reflective. The optical Resonator is the main component of Lasers which surrounds the gain medium and provides feedback of the laser light. Optical Resonator is also used in optical parametric oscillators and some interferometers. The optical resonator has a large Q factor which means there is very little energy loss. 

Orbitals Resonance 

In celestial mechanics, when orbiting bodies exert a regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers, that time Orbital Resonance occurs. The orbital periods of these orbiting bodies may be related by a ratio of two small integers. The Changing Gravitational Forces of bodies are the reason behind this which go around each other. The stability of the Solar System was first examined by the great French mathematician and astronomer Laplace. As a satellite goes around a planet or two stars go around each other, the gravitational force can change slightly. This change is partly because of the ellipse shape of orbits, and the planets and stars are not spherical usually. In this condition, the forces may be unstable, so the smaller partner may change until the forces are stable, and satellites end up with one face towards their planet because that is the most stable position.

Atomic, Particle, and Molecular Resonance 

Nuclear magnetic resonance (NMR) is the main name of Atomic Resonance. Nuclear magnetic resonance is mainly used in advanced medical imaging techniques, such as in magnetic resonance imaging (MRI). Nuclear magnetic resonance is also used to study molecular physics, crystals, and non-crystalline materials. A key feature of Nuclear magnetic resonance is the resonant frequency of a substance which is directly proportional to the strength of the applied magnetic field. This key feature of Nuclear magnetic resonance is used in imaging techniques like If an object is placed into a non-uniform magnetic field then the resonant frequencies of the Object's nuclei depend on the location where they place. 

Electron paramagnetic resonance which is better known as Electron Spin Resonance (ESR) is a spectroscopic technique that is quite like Nuclear magnetic resonance, but their works are very different from each other. Electron paramagnetic resonance uses unpaired electrons. In the main formula of Resonance if Ω is replaced by the particle's mass M, and Γ is the decay rate then they can also be thought of as unstable particles. This formula comes from the particle's propagator, where its mass is replaced by the complex number M + I. By the optical theorem, this formula is further related to the particle's decay rate.