Chapter 4 : Derivation of the wave equation

Question 1

What three fundamental assumptions is the wave equation based on?

Answer 1

1. Conservation of mass - medium is compressible 2. Conservation of momentum - medium has inertia 3. Equation of state

Question 2

what two ways are there of deriving the conservation equations?

Answer 2

the physical approach (eulerian) and the mathematical approach (lagrangian)

Question 3

how does the wave equation describe propagating waves?

Answer 3

1. velocity gradient produces a compression of the fluid 2. leads to a pressure gradient which provides a restoring force 3. pressure gradient produces an acceleration of the fluid 4. leads to a velocity gradient

Question 4

the 1d wave equation has ___ solutions

Answer 4

two solutions moving in different directions.

Question 5

without losses, wave motion is considered to be _____

Answer 5

adiabatic

Question 6

Explain in words what eaxh of the terms in the linearised mass and momnentim conservation equations represents

Answer 6

Linearised mass equation: the time rate of change of the density at a point is inversely related to the divergence of the momentum density there.

Linearised momentum equation: the time rate of change of the momentum density of a fluid element is inversely related to the pressure gradient.

Question 7

if the divergence divu(x) > 0 where u is the particle velocity, is the flow converging or diverging at that point?

Answer 7

Out

Question 8

What assumptions do we make to linearise the conservation equations on our way to deriving the wave equation?

Answer 8