Unit of the Green's function

In summary, the Green's function of the wave equation has a unit of 1/(m*s) and this is determined by the dimensional consistency of the equation and the sifting property of the delta function. This unit allows for the correct units of pressure to be obtained when convolving with the unit of the forcing function f(t).
  • #1
elgen
64
5
Is there a physical unit related to the Green's function of the wave equation?

In particular, let
[tex]\nabla^2 P -\frac{1}{c^2}\frac{\partial^2 P}{\partial t^2} = f(t)[/tex]
where P is pressure in Pa. Since the Green's function solves the PDE when f(t) is the delta function, the Green's function G has a unit of Pa.

The solution to the inhomogeneous PDE is

P = G*f(t),

where * denotes convolution. This leads to contradiction since LHS is in Pa and RHS is in Pa * Pa/m^2 * (sec m^3), where (Pa/m^2) is the unit of f(t) and (sec m^3) denotes the integration in both space and time.

Thx.
 
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  • #2
Elgen,

Recall the sifting property of the delta function:
[tex]\int_{-\infty}^{\infty} \delta(r) dr = 1. [/tex] With this in mind, we see that the units of RHS (actually both sides) change from the original wave equation once it is replaced with a delta function. This is because the delta function must have units which are the inverse of its argument's units in order to make the sifting property true. So in the example above r has units of distance, and the delta function must therefore have units of 1/distance. The delta function returns a unitless quantity after integration.
Hence, the RHS of [tex]\nabla^2 G -\frac{1}{c^2}\frac{\partial^2 G}{\partial t^2} = \delta(t)\delta(r),[/tex] where G is the Green's function and the RHS contains the necessary spatial and temporal delta functions, has units 1/(m^3*s). To figure out the units of G, we demand that this equation has dimensional consistency among all its terms. Thus, matching units between the second term of the LHS (an arbitrary choice) and the RHS demands that
[tex][-\frac{1}{c^2}]*[\frac{\partial^2 G}{\partial t^2}] = [\delta(t)\delta(r)],[/tex] dimensionally (and only dimensionally!). Thus, [tex][\frac{s^2}{m^2}]*[\frac{G}{s^2}] = [\frac{1}{m^3*s}],[/tex] and (for this example only) G has the following units: [tex][G] = [\frac{1}{m*s}]. [/tex] Finally, we check this result by performing the convolution with f(t) from the original wave equation. Remember that f(t) has units Pa/m^2 due to dimensional consistency of the wave equation. Thus, convolving G and f(t) in space and time, we get
[tex]\int [G]*[f(t)]*[dr dt] = [\frac{1}{m*s}]*[\frac{Pa}{m^2}]*[m^3*s] = Pa,[/tex] which are the correct units for pressure.
 

1. What is a "Unit of the Green's function"?

A Unit of the Green's function is a mathematical concept used in physics and engineering to describe the response of a system to a given input. It represents the ratio of the output of the system to the input, and is often used in the study of differential equations.

2. How is a Unit of the Green's function calculated?

The calculation of a Unit of the Green's function involves solving a differential equation for a specific input function. This can be done analytically or numerically, depending on the complexity of the system and the input function. Once the solution is obtained, the Unit of the Green's function is determined by taking the ratio of the output to the input.

3. What is the significance of the Unit of the Green's function?

The Unit of the Green's function is an important concept in physics and engineering as it allows us to understand how a system responds to different inputs. It can also be used to solve complex problems involving differential equations, such as in the study of heat transfer, electromagnetism, and fluid dynamics.

4. How does the Unit of the Green's function relate to the concept of impulse response?

The Unit of the Green's function and impulse response are closely related concepts. Both represent the response of a system to a specific input, although the Unit of the Green's function is more general and can be applied to a wider range of systems and inputs. The impulse response is a special case of the Green's function, where the input is a delta function.

5. Can the Unit of the Green's function be used in real-world applications?

Yes, the Unit of the Green's function has many practical applications in fields such as electrical engineering, mechanical engineering, and acoustics. It is used to model and analyze systems in these fields, and can also be used to design and optimize systems for specific tasks or functions.

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