In order to understand local helioseismology it is crucial to understand wave propagation through generic solar models, including models with local inhomogeneities. In Section 3.1 we review the equations of motion for linear waves moving through non-magnetic backgrounds. The sources of excitation of solar oscillations are characterized in Section 3.2. In Section 3.3 we discuss methods for computing the Green’s functions for solar models. We describe the zero-order problem in Section 3.4 and the effects of weak steady perturbations in Section 3.5. Numerical tests of the Born approximation are described in Section 3.6. Some effects of magnetic fields are briefly reviewed in Section 3.7.

Throughout this section we will address two general classes of models: general models and plane-parallel models. We will not explicitly consider the case of spherically symmetric models. In general we will use the symbol to denote position in three dimensions. For plane-parallel models we decompose the position vector as , where is a two-dimensional horizontal vector and is the height coordinate. The plane-parallel approximation is valid for very high degree modes, which are routinely used in local helioseismology. Plane-parallel models are often assumed for the interpretation of data collected over a small patch of the Sun: images may be remapped onto a grid which is, locally, approximately Cartesian.

3.1 Linear waves

3.2 Wave excitation

3.3 Response to an impulsive source

3.3.1 Direct solution in plane-parallel models

3.3.2 Normal-mode summation approximation

3.3.3 Green’s functions for the observable

3.4 The zero-order problem

3.5 Effects of small steady perturbations

3.6 Tests of Born approximation for sound speed and flow perturbations

3.7 Strong perturbations: magnetic tubes and sunspots

3.2 Wave excitation

3.3 Response to an impulsive source

3.3.1 Direct solution in plane-parallel models

3.3.2 Normal-mode summation approximation

3.3.3 Green’s functions for the observable

3.4 The zero-order problem

3.5 Effects of small steady perturbations

3.6 Tests of Born approximation for sound speed and flow perturbations

3.7 Strong perturbations: magnetic tubes and sunspots

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