“The source problem” — Transient waves propagating from internal sources in non-stationary media

Q: What is the main procedure to obtain the solutions of (2.1)?

The main procedure to obtain the solutions of (2.1) is the following: First, in (2.1) put k−(z, t) = 0(L1 L2 L3 L4) ( u1+(z, t) u1 −(z, t)) = ( k+(z, t)0) (3.1)and evaluate the solutions u1 ±(z, t).

Q: What is the symbol of the reaction from the medium to the impulse?

For the moment, let H1− be the notation of the sum of the leftgoing regular waves at z = z0, and let ∫ R−H1− be a symbol of the reaction from the medium in z < z0 to H1 −.

Q: what is the wave front factor for z?

For waves propagated to the right, i.e. z > z0, they are E+(z,t; z0, t0) = p +(z0, z, τ +(z0; z, t))E +(z0, τ +(z0; z, t); z0, t0) + ∫ τ+(z0;z,t) −∞ G+(z0, z, τ +(z0; z, t), t ′)p+(z0, z, t ′)E+(z0, t ′; z0, t0) dt ′E−(z,t; z0, t0) =∫ τ+(z0;z,t) −∞ G−(z0, z, τ +(z0; z, t), t ′)p+(z0, z, t ′)E+(z0, t ′; z0, t0) dt ′(3.8)with the wave front factor defined asp+(z0, z, t) = exp {∫ z z0 α(ζ, τ+(ζ; z0, t)) dζ }and for waves propagated to the left, i.e. z < z0, they are E−(z,t; z0, t0) = p −(z0, z, τ −(z0; z, t))E −(z0, τ −(z0; z, t); z0, t0) + ∫ τ−(z0;z,t) −∞ Q−(z0, z, τ −(z0; z, t), t ′)p−(z0, z, t ′)E−(z0, t ′; z0, t0) dt ′E+(z,t; z0, t0) =∫ τ−(z0;z,t) −∞ Q+(z0, z, τ −(z0; z, t), t ′)p−(z0, z, t ′)E−(z0, t ′; z0, t0) dt ′(3.9)with the wave front factor defined asp−(z0, z, t) = exp {∫ z z0 δ(ζ, τ−(ζ; z0, t)) dζ }The propagator kernels G±(z0, z, t, t ′) and Q±(z0, z, t, t ′) are generalizations of the Green functions G±(z, t, t′) in Ref. 3. They are solutions to initial-boundary value problems equivalent to those of the Green functions.

Question

1. What are the contributions in "“the source problem"—Transient waves propagating from internal sources in non-stationary media" ?

2. What is the reaction of the medium to the impulse?

3. What are the wave fronts of the two resulting waves?

4. Why is the term “Green functions” used?

Accepted Answer

As an illustration, three special problems are studied ; the inhomogeneous, second order wave equation, and source problems in homogeneous and time invariant media.

Accepted Answer

The medium to the right of z = z0, i.e. in z > z0, immediately responds to the impulse δ by sending out a leftgoing reflected wave (labeled R+) from z = z0.

Accepted Answer

The wave fronts of the two resulting wave solutions E1+(z0 + 0, t; z0, t0) and E1 −(z0 − 0, t; z0, t0) are bound to move along the characteristics τ+(z; z0, t) and7 τ−(z; z0, t), respectively.

Accepted Answer

The name “Green functions” was chosen, because the Green functions of the Krueger and Ochs method map the excitation at the boundary to the interior of the medium, whichis also a property of the conventional Green functions.

Accepted Answer

The suggested technique is based upon the concept of wave splitting, in conjunction with either an invariant imbedding or a Green functions approach.

Accepted Answer

For simplicity, the functions A(z, t, t′), B(z, t, t′), C(z, t, t′) and D(z, t, t′) are assumed continuous and bounded as functions of the variables z, t and t′ in the region t > t′, 0 < z < d.

Accepted Answer

The main procedure to obtain the solutions of (2.1) is the following: First, in (2.1) put k−(z, t) = 0(L1 L2 L3 L4) ( u1+(z, t) u1 −(z, t)) = ( k+(z, t)0) (3.1)and evaluate the solutions u1 ±(z, t).

Accepted Answer

For the moment, let H1− be the notation of the sum of the leftgoing regular waves at z = z0, and let ∫ R−H1− be a symbol of the reaction from the medium in z < z0 to H1 −.

Accepted Answer

For waves propagated to the right, i.e. z > z0, they are E+(z,t; z0, t0) = p +(z0, z, τ +(z0; z, t))E +(z0, τ +(z0; z, t); z0, t0) + ∫ τ+(z0;z,t) −∞ G+(z0, z, τ +(z0; z, t), t ′)p+(z0, z, t ′)E+(z0, t ′; z0, t0) dt ′E−(z,t; z0, t0) =∫ τ+(z0;z,t) −∞ G−(z0, z, τ +(z0; z, t), t ′)p+(z0, z, t ′)E+(z0, t ′; z0, t0) dt ′(3.8)with the wave front factor defined asp+(z0, z, t) = exp {∫ z z0 α(ζ, τ+(ζ; z0, t)) dζ }and for waves propagated to the left, i.e. z < z0, they are E−(z,t; z0, t0) = p −(z0, z, τ −(z0; z, t))E −(z0, τ −(z0; z, t); z0, t0) + ∫ τ−(z0;z,t) −∞ Q−(z0, z, τ −(z0; z, t), t ′)p−(z0, z, t ′)E−(z0, t ′; z0, t0) dt ′E+(z,t; z0, t0) =∫ τ−(z0;z,t) −∞ Q+(z0, z, τ −(z0; z, t), t ′)p−(z0, z, t ′)E−(z0, t ′; z0, t0) dt ′(3.9)with the wave front factor defined asp−(z0, z, t) = exp {∫ z z0 δ(ζ, τ−(ζ; z0, t)) dζ }The propagator kernels G±(z0, z, t, t ′) and Q±(z0, z, t, t ′) are generalizations of the Green functions G±(z, t, t′) in Ref. 3. They are solutions to initial-boundary value problems equivalent to those of the Green functions.

“The source problem” — Transient waves propagating from internal sources in non-stationary media

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Most frequently asked questions

1. What are the contributions in "“the source problem"—Transient waves propagating from internal sources in non-stationary media" ?

2. What is the reaction of the medium to the impulse?

3. What are the wave fronts of the two resulting waves?

4. Why is the term “Green functions” used?

5. What is the suggested technique for analyzing wave propagation problems in non-stationary media?

6. What is the function f(z, t, t′)?

7. What is the main procedure to obtain the solutions of (2.1)?

8. What is the symbol of the reaction from the medium to the impulse?

9. what is the wave front factor for z?

Figures

Citations

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Transient waves from internal sources in non-stationary media - Numerical implementation

Transient waves from internal sources in non-stationary media - Numerical implementation

One-way wave operators for nonstationary dielectrics

References

Methods of Theoretical Physics

Methods of theoretical physics

Methods of Mathematical Physics

Methods of Mathematical Physics

Green's functions and boundary value problems

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