Deriving the Wave Equation Flashcards
What is the propagation of sound waves governed by?
A balance between the compressibility (restoring force) and the inertia of medium
What 3 equations is the wave equation based on?
Conservation of Mass
(medium ia compressible)
Conservation of Momentum/ Newton’s 2nd Law
(Medium has inertia)
Constitutive equation
(thermodynamic relation between the variables)
What happens to density when divergence is decreasing?
Rate of change of density increases
What does ∇ . u > 0 lead to?
divergence (vector field flows out of point)
How is the p = c_0^2 ρ derived?
- Pressure is written in terms of two thermodynamic variables (ρ, sound speed)
- For small perturbations dρ/ds goes to zero and dρ/dρ goes to c_0^2
- Then it is assumed that wave propagation is isentropic (no heat flow, no absorption)
What is the cycle of the velocity gradient?
Velocity gradient produces a compression of the fluid
Leads to a pressure gradient which provides a restoring force
Pressure gradient produces an acceleration of the fluid
Leads to a velocity gradient
When the acoustic pressure and velocity for a pulse are in phase, which direction is the pulse moving?
The wave moves in the positive x direction
When the acoustic pressure and particle velocity for a pulse are out of phase, which direction is the pulse moving?
The wave moves in the negative x direction
What are the relationships between the acoustic pressure and particle velocity?
δp/δt = - δu/δx
δu/δt = - δp/δx
How is the acoustic wave equation formed (1/c0^2 δ^2/δt^2 - ∇ ^2)p = 0 ?
Combine pressure-density relation and mass equation to get rate of change of pressure by multiplying it by δ/δt
Linearised momentum equation gives rate of change of particle velocity by multiplying it by ∇
What is the source term for the conservation of mass equation?
S_M
mass source (mass per unit volume per unit time)
What is the source term for the conservation of momentum equation?
S_F
force source (force per unit mass)
What radiates as a monopole?
A point source that is a mass source (S_co)
What radiates as a dipole?
A point source that is a force source (S_ρo)
How are medium heterogeneities re cast?
As contrast source terms
