Page 199 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

198 CHAPTER 8 Ultrasound applications in cancer therapy

8.3 Governing equations

Despite the consistent principles of acoustic wave equations in physics, the govern-
ing equations involving acoustic waves in industry and bioengineering application
vary for each type of problem. In fact, the equations needed to analyze the problem
are extracted for each case corresponding to its fundamental and conditions. So,
first, the acoustic wave equations in general and then the ultrasonic equations, which
are acoustic waves that are so high in frequency that humans can’t hear them, are
introduced.

8.3.1 Linear ultrasonic relationships

Linear perturbations of the ultrasonic field, assuming Newtonian fluid at constant
temperature and stationary fluid, are expressed as follow [58].
In the following equations, p, v, ρ, t, c, and u are the acoustic pressure (the local
deviation from the ambient pressure), velocity, density, time, the speed of sound, and
the vector field particle velocity respectively. Also, terms with only a zero index or
a combination of multiplying variables with a zero index constitute the solution of
the current before applying the ultrasonic field. Index one indicates the linear pertur-
bations of the waves and terms two indexes represent the secondary and nonlinear
effects of the ultrasonic field.
p = p 0 + p 1
ρ =
ρ +
p=p +p ρ=ρ +ρ ν→=ν→ +v→ 1 ν = ν + v ρ 1 (8.4)

0
1
0
0
0
1
1
0
By expanding these perturbations, the continuum equation and the linearized
momentum equation for the inviscid fluid are bellow equations:
∂ ρ
∂ρ ∂t=−ρ ∇⋅ν→ 1 1 =− ρ ∇⋅ ν (8.5)
1
0
∂t 0 1

ν ∂
ρ ∂ν→ ∂t=−c ∇ρ 1 ρ 1 =− ∇ ρ (8.6)
2
2
c
0
1
0
∂t 1
Eq. (8.7) presents the equation of state.
p=p +c12ρ p = p 0 + c 1 2 ρ (8.7)
0
After several times of algebraic simplification, the linear equation of the ultra-
sonic waves in the ideal fluid is obtained as Eq. (8.8), that Feynman provides for
sound in three-dimensions.
2
2
2 2
2
∇ p =1c ∂ P ∂t 2 ∇ p = 1 ∂ P 1 (8.8)
1
1
1
c 2 ∂t 2