In physics, a wave equation describes the behavior of waves in a given medium. A stationary wave, also known as a standing wave, is a type of wave that appears to be stationary, but is actually a combination of two waves traveling in opposite directions. The wave equation for a stationary wave can be expressed in different forms, each emphasizing different aspects of the wave's behavior.


The general wave equation describes the propagation of waves in a medium. It is given by the equation:

∂²ψ/∂t² = v²∂²ψ/∂x²

where ψ is the wave function, t is time, x is position, and v is the velocity of the wave. This equation represents a wave traveling in a medium without any reflections or interference.


The one-dimensional wave equation is a simplified form of the general wave equation, considering waves traveling in one dimension. It is expressed as:

∂²ψ/∂t² = v²∂²ψ/∂x²

This equation represents a wave traveling along a single axis, such as a wave on a string or a sound wave in a tube.


The harmonic wave equation is a specific form of the wave equation that describes harmonic waves. It is given by:

∂²ψ/∂t² = v²∂²ψ/∂x² - γ²ψ

where γ is the wave number. This equation takes into account the damping or attenuation of the wave as it propagates through a medium.


The Schrödinger wave equation is a quantum mechanical equation that describes the behavior of particles as waves. It is given by:

iħ∂ψ/∂t = -ħ²/2m∂²ψ/∂x² + Vψ

where ħ is the reduced Planck's constant, m is the mass of the particle, and V is the potential energy. This equation is used to determine the wave function of particles in quantum mechanics.


In conclusion, there are different forms of wave equations for stationary waves, each with its own specific characteristics. The general wave equation describes the propagation of waves in a medium, while the one-dimensional wave equation considers waves traveling in one dimension. The harmonic wave equation takes into account damping or attenuation, and the Schrödinger wave equation describes the behavior of particles as waves in quantum mechanics. Understanding these different forms of wave equations helps us analyze and predict the behavior of stationary waves in various physical systems.

Different Forms of Wave Equation for Stationary Wave

There are different forms of wave equations for stationary waves, which describe the behavior of waves that do not propagate but rather oscillate in a fixed position. These equations differ in terms of their mathematical representation and the variables they consider. Let's explore each form in detail:

1. One-Dimensional Wave Equation:
This form of the wave equation describes the behavior of waves in one dimension. It is given by the equation:

∂²ψ/∂x² = (1/v²) * ∂²ψ/∂t²

Here, ψ represents the displacement of the wave, x denotes the spatial coordinate, t represents time, and v is the velocity of the wave. This equation relates the second derivative of the displacement with respect to position (x) to the second derivative of the displacement with respect to time (t). It is commonly used to describe waves on a string or in a medium with one-dimensional motion.

2. Two-Dimensional Wave Equation:
The two-dimensional wave equation is an extension of the one-dimensional wave equation to describe wave behavior in two dimensions. It is given by the equation:

∂²ψ/∂x² + ∂²ψ/∂y² = (1/v²) * ∂²ψ/∂t²

Here, ψ represents the displacement of the wave, x and y represent the spatial coordinates in the two-dimensional plane, t denotes time, and v is the velocity of the wave. This equation relates the second derivatives of the displacement in both x and y directions to the second derivative of the displacement with respect to time.

3. Three-Dimensional Wave Equation:
The three-dimensional wave equation further extends the wave equation to describe wave behavior in three dimensions. It is given by the equation:

∂²ψ/∂x² + ∂²ψ/∂y² + ∂²ψ/∂z² = (1/v²) * ∂²ψ/∂t²

Here, ψ represents the displacement of the wave, x, y, and z represent the spatial coordinates in the three-dimensional space, t denotes time, and v is the velocity of the wave. This equation relates the second derivatives of the displacement in all three spatial dimensions to the second derivative of the displacement with respect to time.

4. Complex Wave Equation:
In some cases, it is mathematically convenient to represent stationary waves using complex numbers. The complex wave equation is given by:

∇²ψ - (1/v²) * (∂²ψ/∂t²) = 0

Here, ψ represents the complex wave function, ∇²ψ denotes the Laplace operator applied to ψ, and (∂²ψ/∂t²) represents the second derivative of the wave function with respect to time. This equation relates the spatial Laplacian of the wave function to its temporal second derivative.

In summary, the different forms of wave equations for stationary waves differ in terms of the dimensions they consider and the mathematical representations used. Each equation provides a unique perspective on the behavior of stationary waves, allowing scientists and researchers to study and analyze various physical phenomena associated with these waves in different contexts.