1 ME 660 Intermediate Acoustics Instructor:Dr. Joseph Vignola Office:Pangborn G43...

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ME 660 Intermediate Acoustics

Instructor: Dr. Joseph VignolaOffice: Pangborn G43Phone: 202-319-6132, e-mail: vignola@cua.eduTime (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/

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.

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)

p̂ r,θ,t( ) =

j2ρocUoka

1r 'ej ωt−kr '( ) dx

−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)

p̂ r,θ,t( ) =

j2ρocUoka

1rej ωt−kr( ) ejkxsinθ dx

−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

The integral at the end of this expression can be found in tables of integrals

e jkx sinθ dx−L / 2

∫ =Lsin

12kLsinθ

⎝⎜⎞

⎠⎟

12kLsinθ

This part (without the first L) is referred to as the directivity

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

Only now that kL>=1Do we see any directivity

directivity =sin

12kLsinθ

⎝⎜⎞

⎠⎟

12kLsinθ

For kL>=1 we see that the radiated pressure is not reduced in the direction perpendicular to the line source

…but is starting to be reduced along the axial direction of the line source

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 )

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.

directivity =sin

12kLsinθ

⎝⎜⎞

⎠⎟

12kLsinθ

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

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

…and the main lobe gets narrower and narrower.

When 2π > ka > 4π we will expect 2 lobes.

directivity =sin

12kLsinθ

⎝⎜⎞

⎠⎟

12kLsinθ

When 2π > ka > 4π we will expect 2 lobes.

The second lobe only reaches it full amplitude when ka > 3π

directivity =sin

12kLsinθ

⎝⎜⎞

⎠⎟

12kLsinθ

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

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

1 ME 660 Intermediate Acoustics Instructor:Dr. Joseph Vignola Office:Pangborn G43...

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