Can we hear the thickness of ice? Prof. Dr.-Ing. Müller for MOOC@TU9

1. MOOC@TU9
Models in Civil Engineering: From Cardiology to
Fishery
Univ.- Prof. Dr.-Ing. Gerhard Müller
Dr.-Ing. Martin Buchschmid
Lehrstuhl für Baumechanik, Technische Universität München

2. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Motivation
Can we “hear” the thickness of ice?
We need information about the
following components:
- load
- ice layer
- sound field

3. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Motivation
Ice layer under jump, plate under impulse loading
acoustic fluid
plate in bending
support

4. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Can we hear the thickness of ice?
Content
1. Vibrations of beams and plates in bending
2. Wave propagation in a beam
a) Wavelength, wavenumber
b) Dispersion
3. Wave propagation in an acoustic fluid
4. Sound radiation
a) Far field
b) Evanescent field
5. Fourier analysis and spectral content of dynamic forces

5. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Vibration of beams and plate in bending
Plate in bending action (neglecting the impedance of the underlying water)
𝐵: bending stiffness
Differential equation of a plate (considering the Kirchhoff theory):
PDE of 4th order
Approach for solution
• Harmonically oscillating in space (x-,y- coordinate)
• Harmonically oscillating in time (t-coordinate)
harmonic in time
sin, cos
Excursus: Describing trigonometric functions with complex numbers (Euler‘s formula)
two conjugate complex solutions

6. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Vibration of beams and plate in bending
Plate in bending action
Approach for the computation of one part of the conjugate complex solution
(the complete solution is deduced easily)
with wavenumbers
angular frequency
taken from wikipedia.org

7. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Vibration of beams and plate in bending
Homogeneous solution
Therefore the wave propagation speed in dependence of the frequency can be calculated:
with
with
Plate in bending action

8. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Visualisation: Dependency of the wave propagation speed and wavelength compared to the frequency
Vibration patterns of a finite plate under single load with different frequencies of excitation
Vibration of beams and plate in bending
𝑓
𝑐 𝐵 𝜆 𝐵
𝑓
dispersive propagation
depending on frequency

9. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Vibration of beams and plate in bending
Computation results: Eigenmodes of an elastically supported plate
Plate in bending action

10. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Vibration of beams and plate in bending
Plate in bending action
Ω1 = 4.43 𝐻𝑧 Ω2 = 6.39 𝐻𝑧
Computation results: Response to a harmonic single load in dependency of the frequency

11. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Motivation
Ice layer under jump, plate under impulse loading
acoustic fluid
plate in bending
support

12. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Acoustic fluid
Differential equation (wave equation)
PDE of 2nd order
Approach:

13. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Acoustic fluid
Differential equation (wave equation)
PDE of 2nd order
Approach:

14. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Acoustic fluid
Case 1:
propagating wave
(far field)
Case 2:
evanescent field
Discussion of the component of the sound-field, which is orthogonal to the
x-y plane (z-coordinate)
x
z
x
z

15. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Coupling of plate and acoustic fluid
x
z
velocity of the acoustic fluid
velocity of the vibrating plate
𝜆 𝑝𝑙𝑎𝑡𝑒 > 𝜆 𝑎𝑖𝑟 in case of sound
radiation (far field)
𝜆 𝑝𝑙𝑎𝑡𝑒 < 𝜆 𝑎𝑖𝑟 in case of
evanescent field (near field)
limit case: 𝜆 𝑝𝑙𝑎𝑡𝑒 = 𝜆 𝑎𝑖𝑟
𝑐 𝑝𝑙𝑎𝑡𝑒 = 𝑐 𝑎𝑖𝑟
⇒ 𝑓𝑐𝑟𝑖𝑡

16. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Coupling of plate and acoustic fluid
Radiation of sound is (in case of infinite layers) just possible for frequencies of excitation with 𝑓 > 𝑓𝑐𝑟𝑖𝑡
Investigation of the frequency content of the load necessary
Velocity and wave length in dependency of frequency
𝑐
𝑓 𝑓
𝜆
𝑓𝑐𝑟𝑖𝑡 𝑓𝑐𝑟𝑖𝑡
radiation of sound
plate
air
plate ~
1
𝑓
air ~
1
𝑓

17. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Coupling of plate and acoustic fluid
Radiation of sound is (in case of infinite layers) just possible for frequencies of excitation with 𝑓 > 𝑓𝑐𝑟𝑖𝑡
Investigation of the frequency content of the load necessary
Velocity and wave length in dependency of frequency
𝑐
𝑓 𝑓
𝜆
𝑓𝑐𝑟𝑖𝑡 𝑓𝑐𝑟𝑖𝑡
radiation of sound
plate
air

18. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Coupling of plate and acoustic fluid
Radiation of sound is (in case of infinite layers) just possible for frequencies of excitation with 𝑓 > 𝑓𝑐𝑟𝑖𝑡
Investigation of the frequency content of the load necessary
Velocity and wave length in dependency of frequency
𝑐
𝑓 𝑓
𝜆
𝑓𝑐𝑟𝑖𝑡 𝑓𝑐𝑟𝑖𝑡
radiation of sound
plate
air

19. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Load
=
Remember: Fourier Series
with:
For an impulse an infinite number of cosine-members is needed
𝑝 𝑡 =
𝑘=−∞
∞
𝑐 𝑘 ∙ 𝑒
𝑖𝑘𝜋𝑡
𝑇 𝑐 𝑘 =
1
2𝑇
−𝑇
𝑇
𝑝(𝑡) ∙ 𝑒−
𝑖𝑘𝜋𝑡
𝑇 𝑑𝑡
𝑝(𝑡)
𝑡
𝑝(𝑓)
𝑓
𝑝(𝑡)
𝑡
Fourier Transformation

20. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Conclusion
• All frequencies are part of the signal.
• Just 𝑓 > 𝑓𝑐𝑟𝑖𝑡 radiate sound (and can be heard).
• High frequencies propagate faster (bending waves are dispersive) and can be perceived earlier at
different locations.
• The lowest frequency to be heard is 𝑓𝑐𝑟𝑖𝑡.
• 𝑓𝑐𝑟𝑖𝑡 is depending on the thickness of the ice layer.
We can „hear the thickness of an ice layer“.

21. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Example
Velocity in dependency of frequency
velocityinm/s
frequency

22. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Task of the Week
Given: Results of acoustic measurements for two different ice layers
Layer 1
Layer 2

23. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Task of the Week
Given: Results of acoustic measurements for two different ice layers
Layer 1
Layer 2
Time Domain Frequency Domain [0Hz; 5kHz]
0 0.2 0.4
-0.2
-0.1
0
0.1
0.2
Time [s]
SoundPressurep(t)
0 1000 2000 3000 4000 5000
0
2
4
x 10
-3
Frequency [Hz]
SoundPressurep(f)
0 0.2 0.4 0.6 0.8
-0.2
-0.1
0
0.1
0.2
Time [s]
SoundPressurep(t)
0 1000 2000 3000 4000 5000
0
1
2
3
x 10
-3
Frequency [Hz]
SoundPressurep(f)

24. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Task of the Week
 Compute the thickness of the ice layers (Would you jump on the ice?)
Use the following material parameters for ice
 Young's Modulus 𝐸 = 9.1 ⋅ 109 𝑁
𝑚2
 Density 𝜌 = 916.7
𝑘𝑔
𝑚3
 Poisson’s Ratio 𝜈 = 0.33
Use the following constant (frequency independent) wave velocity for air
 Wave velocity 𝑐 𝑎𝑖𝑟 = 340
𝑚
𝑠
 Is the applied model conservative (on the save side)? Please compare your results with the results
published by Lundmark (2001) and comment on the differences. Under what circumstances would you
trust the results in order to jump on the ice?
 Estimate the stiffness of the water using the model of an elastically supported plate. Comment on the
model of an elastic support of the plate.
Given: Results of acoustic measurements for two different ice layers

25. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery
Published Results
Lundmark, 2001
FrequencyinHz
Thickness in mm

26. MOOC@TU9: Models in Civil Engineering: From Cardiology to Fishery