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Three Dimensional Wave Equation

Thewave equation is second order linear as well as partial differential equation which does exists for the waves in free space as well as in themedium. The wave equation is mostly used for the waves like the sound waves and the light waves and the water waves. The wave equation is required in the fields like that of acoustics and the electromagnetic fields and they are also required sometimes in the fluid dynamics. The problem of the vibrating string can be solved using the three dimensional wave equations.


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Three dimensional wave equation is given by:


3-Dwaveequation

Whereq = q(x, y, z, t) and x and y and z are nothing but the standard Cartesian coordinates. This three dimensional wave equation can be used to describe the propagation of the sound waves in the fluid i.e. in a medium. If the displacement denoted by the symbol q is a longitudinal displacement then we say that the wave is propagated through the fluid.


Solution of Three Dimensional Wave Equation


The solution of the three dimensional wave equation can be written as:


 


wherethe plus solution shows the wave which is incident or the solution in the case of the incident wave and the negative or minus solution shows the solution of the three dimensional wave equation when the wave is reflected back. In the above solution the kx ,ky and kz are the wave vectors in x and y and z directions respectively. The ω isthe frequency of the wave and it is connected with the wave vectors as:


wavefrequency




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Solution set for the three Dimensional Wave Equation


The equation is known as the dispersion relation of the three dimensional wave.If we define:



as the three dimensional wave vector and define the three dimensional position vector as:



then the solution of the three dimensional wave equation are written as:


 

Thesesolutions are valid for almost all waves which are traveling in the space with some speed and a frequency of ω. This wave equation and its solution are mostly valid for the sound waves.