Helmholtz Equation Questions

The Helmholtz equation is a linear partial differential equation; it is also associated with the laplacian, wavenumber and amplitude. Because of this, we can call the Helmholtz equation as an eigenvalue equation. If k represents the wavenumber, A represents the amplitude, and βˆ‡2 represents laplacian, then the Helmholtz equation’s mathematical formula can be written as,

\(\begin{array}{l}\bigtriangledown^{2} +k^{2}A=0\end{array} \)

We can solve the Helmholtz differential equation using separate variables in only 11 coordinate systems.

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Important Questions with Answers

1. What are the applications of the Helmholtz equation?

Using the Helmholtz equation, the physics-related problems based on the concept of acoustic, electromagnetic radiation and, most importantly, seismology can be solved.

2. What is seismology?

Seismology can be described as the study of earthquakes and their propagating elastic waves. Seismic waves are classified into two types: P-waves or primary waves and S-waves or secondary or shear waves.

3. What is electromagnetic radiation?

Electromagnetic radiation is energy produced by moving electrically charged particles travelling through a vacuum or matter or by oscillating magnetic and electric disturbances.

4. Derive the Helmholtz equation.

By considering the equation of wave, the Helmholtz equation can be solved.

The equation of the wave is,

\(\begin{array}{l}\left ( \bigtriangledown ^{2}-\frac{1}{c^{2}}\frac{\partial^2 }{\partial t^2} \right)u(r,t)=0\end{array} \)

Here, let’s assume the wave function u(r, t) is equal to the separation variable.

That is,

u(r,t) =A(r)T(t)

After substituting this value in the wave equation and simplifying, we wet,

\(\begin{array}{l}\frac{\bigtriangledown^{2}A}{A}=\frac{1}{c^{2}T}\frac{\mathrm{d}^{2}T}{\mathrm{d} t^{2}}\end{array} \)

Here, the left-hand side of the above equation depends only on r. And the right-hand side of the above equation only depends on t.

Generally, this equation is valid only if both sides of the equation are equal to the same constant value. This reasoning is crucial in solving linear partial differential equations by variable separation.

We can derive an equation from this observation, one for A(r) and another for T(t).

\(\begin{array}{l}\frac{\bigtriangledown^{2}A }{A} = -k^{2}\end{array} \)

So here,

\(\begin{array}{l}\frac{1}{c^{2}T}\frac{\mathrm{d} ^{2}T}{\mathrm{d} t^{2}} = -k^{2}\end{array} \)

We have chosen the expression -k2 for the value of the constant without loss of generality.

Here, any constant k can be used as the separation constant, and k2 is selected only for convenience in the resulting answers.

We get the Helmholtz equation by rearranging the first equation:

\(\begin{array}{l}\bigtriangledown^{2}A+k^{2}A=\left(\bigtriangledown ^{2}+k^{2} A\right )=0\end{array} \)

In this way, the Helmholtz equation can be solved.

5. The Helmholtz equation is elliptic.

  1. True
  2. False

Answer: a. True

Explanation: The common boundary value problems that Dirichlet, Neumann, and others presented for the Helmholtz equation in a bounded domain are elliptic.

6. The thermodynamic function of a system equals the difference between internal energy and the product of temperature, and entropy is the definition of _____________.

  1. Helmholtz equation
  2. Helmholtz function
  3. Gibbs free energy
  4. Helmholtz free energy

Answer: b. Helmholtz function

Explanation: The Helmholtz function can be defined as the thermodynamic function of a system that equals the difference between the product of temperature, entropy, and internal energy.

7. Write the Helmholtz equation.

If k represents the wavenumber, A represents the amplitude, and βˆ‡2 represents laplacian, then the Helmholtz equation’s mathematical formula can be written as,

\(\begin{array}{l}\bigtriangledown^{2} +k^{2}A=0\end{array} \)

Helmholtz’s equation is used to solve problems in physics such as seismology, acoustics, and electromagnetic radiation.

8. What is Gibbs’s free energy?

A Gibbs free energy, also known as a Gibbs function or free enthalpy, is a number used to estimate the maximum amount of work done in a thermodynamic system with consistent temperature and pressure. Usually, Gibbs’s free energy is denoted by the symbol ‘G’.

9. What is a partial differential equation?

A partial differential equation is a dependent variable that depends on one or more independent variables. A partial differential equation explanation is any function that helps the problem exactly. A general solution includes an arbitrary number of independent functions equal to the order of the equation.

10. What is the origin of the Helmholtz equation?

The Helmholtz equation is named after Hermann von Helmholtz. It is a linear partial differential equation; the equation is also associated with the laplacian, wavenumber and amplitude.

Practice Questions

  1. What is the Gibbs and Helmholtz equation used for?
  2. What is the distinction between Gibbs and Helmholtz free energy?
  3. What exactly is acoustics, and why is it important?
  4. What is the formula of Helmholtz free energy?
  5. Who invented the Helmholtz equation?

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