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1
ME 660 Intermediate Acoustics
Instructor: Dr. Joseph VignolaOffice: Pangborn G43Phone: 202-319-6132, e-mail: [email protected] (section 1): Tuesdays, 6:35-9:00PMTime (section 2): Thursdays, 1:35-4:00PMOffice Hours: Tuesdays 1:00 -2:00,
(or anytime I’m in my office or the lab which is most of normal business hours)Web pages: http://josephfv.googlepages.com/
http://faculty.cua.edu/vignola/
2
Line Source DirectivityNow lets think about a source that is long relative to the wavelength as shown.
This type of source is important in ultrasound and oceanographic application and it helps us build up to more complex source geometries.
A cylindrical line source, length L, that vibrates radially is shown on the x-axes of a coordinate system.
3
Line Source DirectivityNow lets think about a source that is long relative to the wavelength as shown.
Assume that the surface vibrates radially with a surface velocity dQ =Uo2πadxeach little bit of the line source has an incremental volume velocity (source strength)
4
Line Source DirectivityNow lets think about a source that is long relative to the wavelength as shown.
p̂ r,θ,t( ) =
j2ρocUoka
1r 'ej ωt−kr '( ) dx
−L / 2
L / 2
∫
Assume that the surface vibrates radially with a surface velocity dQ =Uo2πadxeach little bit of the line source has an incremental volume velocity (source strength)
5
Line Source DirectivityNow lets think about a source that is long relative to the wavelength as shown.
p̂ r,θ,t( ) =
j2ρocUoka
1rej ωt−kr( ) ejkxsinθ dx
−L / 2
L / 2
∫
Evaluation this integral is a little messy but if we assume that is and independent of x then the expressing becomes easier to work with. This is the case if is moderately large relative to a wavelength
6
Line Source DirectivityNow lets think about a source that is long relative to the wavelength as shown.
The integral at the end of this expression can be found in tables of integrals
e jkx sinθ dx−L / 2
L / 2
∫ =Lsin
12kLsinθ
⎛
⎝⎜⎞
⎠⎟
12kLsinθ
This part (without the first L) is referred to as the directivity
7
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
8
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
directivity =sin
12kLsinθ
⎛
⎝⎜⎞
⎠⎟
12kLsinθ
9
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
10
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
11
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
12
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
13
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
14
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
15
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
16
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
17
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
18
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
19
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
20
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
21
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
22
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
23
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
24
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
25
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
26
Line Source Directivity
When the kL is small,(length is smaller then the wavelength).
the line source radiates uniformly in all directions.
27
Line Source Directivity
Only now that kL>=1Do we see any directivity
directivity =sin
12kLsinθ
⎛
⎝⎜⎞
⎠⎟
12kLsinθ
28
Line Source Directivity
For kL>=1 we see that the radiated pressure is not reduced in the direction perpendicular to the line source
29
Line Source Directivity
…but is starting to be reduced along the axial direction of the line source
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
35
Line Source Directivity
Along the direction perpendicular (θ = 0°) to the line source, a receiver sees all the points as essentially the same distance away. The pressure from all point on the source add coherently (constructively )
36
Line Source Directivity
Sound rays traveling along the direction (θ = 90°) of the line source, travel different distances to get to a receiver. This means they don’t add coherently and we see some cancelations.
37
Line Source Directivity
38
Line Source Directivity
39
Line Source Directivity
40
Line Source Directivity
41
Line Source Directivity
42
Line Source Directivity
directivity =sin
12kLsinθ
⎛
⎝⎜⎞
⎠⎟
12kLsinθ
43
Line Source Directivity
When ka = 2π we have complete cancelation in axial direction (θ = 90°)
directivity =sin
122π sinθ
⎛
⎝⎜⎞
⎠⎟
122π sinθ
When θ goes from 0° to90° the agreement of the outer (red) sine function goes from 0 to π
44
Line Source Directivity
When ka > 2π we see the appearance of a additional lobe(s) because the argument of the outer (red) sine function can now be greater then π.
directivity =sin
12kLsinθ
⎛
⎝⎜⎞
⎠⎟
12kLsinθ
45
Line Source Directivity
…and the main lobe gets narrower and narrower.
46
Line Source Directivity
When 2π > ka > 4π we will expect 2 lobes.
directivity =sin
12kLsinθ
⎛
⎝⎜⎞
⎠⎟
12kLsinθ
47
Line Source Directivity
When 2π > ka > 4π we will expect 2 lobes.
The second lobe only reaches it full amplitude when ka > 3π
directivity =sin
12kLsinθ
⎛
⎝⎜⎞
⎠⎟
12kLsinθ
48
Line Source Directivity
49
Line Source Directivity
When ka = 4π we have two complete lobes and cancelation in axial direction (θ = 90°)
directivity =sin
122π sinθ
⎛
⎝⎜⎞
⎠⎟
122π sinθ
When θ goes from 0° to90° the agreement of the outer (red) sine function goes from 0 to 2π
50
Line Source Directivity
As we continue to increase ka past 4π we we see more lobes
directivity =sin
122π sinθ
⎛
⎝⎜⎞
⎠⎟
122π sinθ
…and the primary lobe get narrower
51
Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity
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Line Source Directivity