Principle of Superposition Waves

The principle of superposition is also referred to as the superposition property. It simply states that if the input is A which produces a response equivalent to X and there is Input of B which produces response then, we can say input(A+B) will produce the response of (X+Y). 

Superposition refers to the placement of a thing above another. In physics superposition  is the combination of two different physical phenomena generally of the same kind. Eg:- spin, wavelength etc. This happened so that both of them could exist together as a part of the event. 

History of the principle of superposition 

Daniel Bernoulli was the first one to mention the principle of superposition in 1753. This principle was later rejected by Leonhard Euler and Joseph Lagrange. 

However Daniel Bernoulli continued claiming that a sonorous body has the capability to vibrate in many modes along with a proper and well defined frequency of oscillation. Euler accepted the problem of vibrating strings but he did not consider it to be a multi-modal solution. 

After the works added in by Joseph Fourier the principle of superposition was finally accepted. 

Properties of the superposition principle 

There are two properties which can define the superposition:-

• Additivity – Additivity is an additive map , Z-linear map or additive function which preserves the addition operation in algebra. 

f(x+y) = f(x) + f(y) 

• Homogeneity – Homogeneity is a function which has many variables. If a scalar quantity is multiplied by it then the value will be multiplied with a power if the scalar. This is known as degree or degree of homogeneity . 

F(ax) = aF(x) 

In this case a is the scalar quantity.

The functions which satisfy the principle of superposition are called linear functions. 

Application of superposition principle

We can see the superposition principle being applied in physics and in engineering as there are a lot of physical systems which are modelled as linear systems. Eg:- A beam is regarded as a linear system because the load put on the beam is seen as the stimulus and the deflection on the beam is the output in this case. 

Mainly the stimulus and the responses can be objects which would satisfy various axioms, such as numbers, functions, vectors etc. A superposition is regarded as a vector sum during the involvement of vector sums. Gradients, differentials or integrals can be held if the superposition holds. 

Some of the linear systems on which superposition principle can be applied on are :- 

• Equations in algebra 
• Linear differential equations
• System of equations.

Examples of the application of superposition principle are :-

• Linear circuit – This is an application in electrical engineering. So the time-varying voltage known as input has a relation with the output which is a current from the circuit. This is possible due to linear transformation. To get the superposition of the responses the superposition of the input signals are needed and Fourier analysis is used. 
• Hydrogeology – For an aquifer drawdown of more than 2  water wells are required . This requires the analytical element method . 
• Model predictive control – This is a process of control and it uses the superposition principles. 

Conclusion

Superposition and its occurrence in physics and mathematics is visible on various occasions. The principle of superposition helps one to observe the net response of multiple stimuli. These superposition principles are already being used in various different ways and places. Some modern technologies are based on the principles of superposition.