1. Research Methods in Acoustics Lecture 5: Reflection and Horn Equation
Jonas Braasch

2. Repetition: Wave Equation
The wave equation was derived from the following 3 equations:
Ideal Gas Law (I): provides the relationship between pressure and density of an ideal gas. Was derived from the adiabatic law.
Euler’s Equation (II): equals Newton’s 2. Law: F=m·a for gases and fluids (Derivation replacing F and m with gas parameters p, V and r).
Continuity Equation (III): states the preservation of mass. The function was derived by observing the amount of gas entering and leaving a defined volume and setting it equal with the amount of gas increase/decrease in the defined Volume.

3. Repetition: Wave Equation
Each of the 3 equations is dependant on 2 unknown parameters:
Ideal Gas Law (I): Dynamic pressure p≈, and density r ≈.
Euler’s Equation (II): Dynamic pressure p≈, and velocity v.
Continuity Equation (III): Dynamic density r≈, and velocity v.

4. Repetition: Wave Equation
If two of the three equation would depend on the same two
Unknown variables we could easily eliminate either of the two
variables. Since this is not the case, we will have to eliminate
two variables using all three functions:
1.) resolve the ideal gas law to r and then insert it into the
continuity equation:
Equation I+III (Ideal gas law,
continuity)
2.) resolve the Euler Equation to dv:
and then insert it into the Equation I+III to eliminate v:

5. Wave Equation for Pressure
combined Equation I+III
Euler’s Equation (II)
Eliminate
velocity v

6. Wave Equation for Pressure

7. Solution for the Wave Equation
We have three oscillating terms:
forward propagating wave
backward propagating wave
oscillating function in time
Theoretically we also have a second oscillating term in time, oscillating backwards! In our environment, we can exclude this solution.

8. Solution for the Wave Equation
forward propagating wave
backward propagating wave

9. Reflection v0 p0 l=−x l=x x=0 l=0
Now let us see what happens, if a forward propagating wave meets an obstacle:
resistance v0
d<<l p0
l=−x
l=x
x=0
l=0

10. Resistance
If we apply a period sound pressure p to an air molecule, the latter will start to oscillate. The maximum displacement xmax of the air molecules depends on their resistance to deviate from their balanced condition.
Here, one can imagine that the air molecules are connected through springs. The tighter the spring, the more force or pressure we must apply to achieve the same degree of displacement.
The particle velocity v(t) is proportional to the displacement x(t):
Obviously, the remarks we made above for the displacement also apply to the particle velocity v(t).

11. Force of the Spring Hooke’s Law x-x0 x0 x x0 x With g being a constant
Note that this concept is very similar to Hooke’s Law:
Hooke’s Law
Note that the Force of the spring is proportional to the deviation of the spring from the balanced position x0.
x-x0 x0 x x0 x
With g being a constant

12. Impedance
We could define the ratio between p and v as resistance, but since this ratio might be complex, we rather use the term impedance. The impedance is defined as:
For a given pressure p, the velocity increases with decreasing impedance

13. Acoustic Impedance of a Plane Wave
Using the relationship Z=p/v, we can determine the impedance of a plane wave Zw:

14. Acoustic Impedance of a Plane Wave
We can utilize the Euler Equation, which provides a relationship between p and v to determine Zw

15. Wave Equation for Pressure
forward propagating wave
backward propagating wave L p0 v0
Z0= v0 p0 x
x=0
L−x
x=L

16. Wave Equation for Pressure
Let us insert the solution for p into Euler Equation:

17. Impedance
Now, we can determine the impedance of the occluder which is located at x=L. Hence, exp(k(L−x))=0!

18. Impedance
… let’s continue:

19. Impedance The impedance for the plane wave is calculated as follows:
Note that we can set the phase of p+ to zero, because we can
choose the starting point of our plane wave ourselves.

20. Impedance
The real part of the impedance is called resistance, the imaginery part reactance: …

21. Reflection coefficient
Reflectance/
reflection
coefficient

22. Reflection coefficient
How can we measure the wall impedance Z0
experimentally? Both the initial and the reflected wave are overlapping (superposition).

23. Standing Wave Ratio
An important step is the definition of the standing
wave ratio:
Sinusoidal Standing Wave

24. Examples of absorption coefficients
Materials Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
Concrete (painted)
Brick (painted)
Plywood (10mm)
Drapery (612 g/m2)
Fiberglass board (25mm )
Fiberglass board (50mm)
Fiberglass board (75mm)
Fiberglass board (100mm)

25. Reflection Coefficient
The reflection coefficient is defined as:
The absorption coefficient as:
Note that we can
measure the SWR
in a standing
wave apparatus

26. The Phase of our Wall Impedance
Note that the standing wave is meeting the wall material at a
different phase. We now know the magnitude of the reflection
through the reflection coefficient. Let us now determine the phase.
How can we achieve this?
Let us determine the relationship between the phase j and the
the first minima.

27. The Phase of our Wall Impedance
The minimum occurs when both the forward propagated wave and
the reflected wave are out of phase, e.g., have a phase difference
of p.

28. The Phase of our Wall Impedance
We can now determine φ experimentally in our standing wave
tube. The remaining minima are found a multiples of 2p. The
general solutions are:

29. Example absorption coefficient
Let us assume that we measured a standing wave ratio of
SWR=3 and the first node at 1/4 of the wave length. We find
for the reflection coefficient r:
and for the absorption coefficient a:

30. Example Phase
and the phase:

31. Continuity Equation for Webster’s Equation
m=r·A·dx
A(x)+dA/dx·dx
A(x) v
v+dv/dx·dx x
x+dx
If we want to determine the wave equation for slowly varying cross section areas, we have to modify the continuity equation. However, we can keep the Euler Equation and Ideal Gas Law in its original form.

32. Pressure wave in organ pipe

33. Continuity Equation

34. Continuity Equation

35. Continuity Equation

36. Continuity Equation
Alternatively, we can determine the change of mass via the change of density over time:
We can now combine this equation with our mass change equation from the last slide:
By replacing the variables dm:

37. Continuity Equation
Now we can remove all terms that appear on the left as well as on the right side of the equation, namely A·dx·dt :
Continuity Equation (III) for variable cross sections
Continuity Equation for constant cross sections

38. Webster’s Differential Equation
Now we can derive the modified Wave Equation using the modified Continuity Equation. The result is the Webster’s Differential Equation.
Webster’s
Differential
Equation
1-D Wave

39. Conical Horn
A(x0)=A0
x=0 x0
Cross section area of the conical horn:

40. Conical Horn Insert the solution into the Webster’s Eq. Webster’s Eq.

41. Conical Horn Solution Webster’s Eq.
The equation can be transformed into the following one:
Solution

42. p1 p2 A1 A2 q1 q2 I II

43. Acoustic Parameters p pascals sound pressure f hertz frequency ρ kg/m3
density of air c
m/s
speed of sound v
particle velocity
ω = 2 · π · f
radians/s
angular frequency ξ
meters
Particle displacement
Z = c • ρ
N·s/m³
acoustic impedance a
m/s²
Particle acceleration J
W/m²
sound intensity E
W·s/mm³
sound energy density
Pac
watts
sound power or acoustic power A m²
Area

44. References
T.D. Rossing: The Science of Sound, Addison Wesley; 1st edition (1982) ISBN:
Jens Blauert, Script Communication Acoustics I (wave equation derivation), The script is currently translated by Ning into English
Daniel A. Russell: Absorption Coefficients and Impedance