Mechanical Waves

Question 1

What 3 things do we need to consider to derive the equation of motion for mechanical waves in a gas?

Answer 1

1. The gas moves and changes the density.
2. The change in density corresponds to a change in pressure
3. The change in pressure generates motion.

Question 2

How is the change in pressure related to the density?

Answer 2

Change in pressure in proportional to change in density.

Question 3

How does the displacement relate to the change in density?

Answer 3

dϱ/P0 = -du/dx

Question 4

What two values of pressure do we need to calculate the force of the wave?

Answer 4

Pressure at x minus pressure at x+dx = -dp/dx Δx

Question 5

What is the value of mass times acceleration for gas in a volume?

Answer 5

mass: ϱ0*Δx
acceleration: d^2 u/dt^2

Question 6

What is the wave equation including Κ?

Answer 6

1/Κ * d^2 u/dt^2 = d^ u/dx^2

Question 7

What is the bulk modulus of a gas?

Answer 7

B = -pressure change/fractional volume change = -V dp/dv = + ϱ dp/dϱ

Question 8

What is the equation for wave velocity in terms of bulk modulus?

Answer 8

v = sqrt(B/ϱ)

Question 9

What is the wave equation in terms of the bulk modulus?

Answer 9

ϱ/B * d^2 u/dt^2 = d^2 u/dx^2

Question 10

What is the equation for elastic modulus?

Answer 10

elastic modulus = stress/strain = force per unit area/fraction displacement

Question 11

What is the equation for wave speed in terms of the youngs modulus?

Answer 11

v = sqrt(Y/ϱ)