1. 1 FOURIER TRANSFORM
Spectral solution of the 2D Homogeneous
acoustical wave equation
Author: Ing. Jhonatan Andr´es Amado Valderrama.
The concept of Fourier transform comes from the idea that a signal can be represented
as the sum of many harmonic signals of different frequencies.
1 Fourier Transform
Consider a ricker source f(t) for t < T given by:
1 def r i c k e r ( t , fq ) :
2 arg=(np . pi ∗∗2) ∗( fq ∗∗2) ∗( t ∗∗2)
3 r i c k e r =(1.0 −2.0∗ arg ) ∗np . exp(−arg )
As mentioned before, the ricker source can decompose as a sum of individual sinusoidal
waves, each with different amplitude, frequency and phases.
y (t) =
n
An sin (ωnt + φn) (1)
A Fourier transform is a simple way of finding the numbers An, ωn, φn. In general having
a set of N points, it is possible to write 1 as,
yn =
N−1
k=0
ˆyke2πikn/N
(2)
where ˆyk now contains information about the amplitude and the phase of the k-th
harmonic mode.
These are some properties of the Fourier transform if the signal is real.
1

2. 3 WAVE EQUATION
ˆyN−k =ˆy∗
k (3)
ˆyN+k =ˆyk (4)
2 Fast Fourier Transform
The problem of calculating Fourier Transform is that this problems requires O(N2) which
becomes huge when N increase As seismic problems as FWI and RTM. There is a family of
algorithms designed to calculate Fourier coefficients in O(N log N) known as Fast Fourier
Transform Methods. An implementation can be found in python with numpy.fft.fft and
numpy.fft.ifft
3 Wave Equation
Let consider a wave in two dimensions generated at the location r0 = (x0, z0), the per-
turbation at P(x, z, t) follow the expression in 5 assumed that the source is point-like in
space
∂2P
∂x2
+
∂2P
∂z2
−
1
c2
∂2P
∂t2
= δ(x, z)S(t) (5)
Taking the Fourier transform of P(r, t) = P(x, z, t) the equation 5 can be written as
∂2 ˆP
∂x2
+
∂2 ˆP
∂z2
−
ω2
c2
ˆP = δ(r) ˆS(ω) (6)
It can be shown that if this is the case, the solution can be written as:
ˆP(r, ω) =G(r − r0, ω) ∗ ˆS(ω) (7)
G(r, ω) =
e−iωr/c
4πr
(8)
r = |r| (9)
The solution of this problem is just to calculate the Fourier transform of the signal,
multiply it by G and calculate the inverse Fourier transform.
2

3. 4 SIMULATION
4 Simulation
The next figures represent the simulation of a ricker source at source(500, 500). Some par-
ticular propertie is that this kind of solution of the wave equation does not need boundary
condition, because with the Fourier transform all the field is known in each time in all the
homogeneous media.
(a) Field solution at t = 0.1[s]. (b) Field solution at t = 0.6[s].
(c) Field solution at t = 0.9[s].
Figure 1: Field solution at different times. Notice that in 1c
there are not reflection at the boundarys
3