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Technical Discussions
Marty Jones
Marty Jones
06:59 Sep-03-1996
Depth of Field

Can you tell me of a good textbook or paper that develops the equations for depth of field of focused transducers?

Robert A. Day
Milky Way Jewels, USA, Joined Nov 1998, 40

Robert A. Day

Milky Way Jewels,
Joined Nov 1998
09:59 Sep-04-1996
Re: Depth of Field
Marty -

I have good new and bad new. First the bad. There are no
equations for the depth of field of a transducer (that I
know off) and as far as I know there is no definition of
depth of field. The whole concept of depth of field in
optics is acceptable sharpness which is kind hard to get
a grasp on in NDT. I suspect every application is
different and you need to define your terms. I worked on
a program to NDT closure welds for radioactive spent
fuel containers several years ago. I defined acceptable
sharpness as 1 mm. Other applications would use other

The good news is that once you have a definition you can
calculate the diffraction field of a transducer and from
that determine the depth of field. The same rules
prevail in ultrasound as optics, i.e. you want the
diameter of the transducer (lens) to be much smaller
than the focal length (large f/number). You generally
want to get depth of field in the part and the focus
changes. Calculating the actual depth of field in the
part is much more difficult even for straight beam. The
depth of field decreases to a value similar to what
would be obtained if the focal length is entirely in the
faster material. This means if you get 2 cm of less than
1 mm focus in water the actual depth of field is about 5
mm. It does vary a bit with ratio of water path to metal
path but because the effective focal length in steel is
shorter and the wavelength increases you wind up with a
shorter depth of field and almost the same focal
diameter (usually a little larger).

Equations are available but solving them is not straight
forward. The following are good starting points:

Exact general solution for right circular piston:
D.G. Crighton, et. al., "Modern Methods in Analytical
Acoustics", 1992, Springer-Verlag, pp 530 - 536.
The talk about the high frequency limit which is the
approximation commonly used.

A more classical and less useful discussion is in
Richardson's "Ultrasonic Physics" on page 43. This is
the usual on axis only solution but he shows his work.

A more accessible but still not for focused transducers
discussion is in Timken's " Elements of Acoustics."
This has the solution in Bessel functions and is more
easily calculated.

There are many approximate solutions for focused
transducers published in the literature, and a few exact
numerical approaches. I have copies of most of them and
am planning to put together a program someday soon.
Moving my office to San Francisco has most of this in
boxes and it's not clear when I will get back to
unpacking them. As soon as I get that organized, I'll
send you a list of papers that discuss this.

Getting the depth of field from these is still not easy
since you still have to decide how to measure focal size.
The old FWHM is good but not always the most appropriate,
do you do 6 or 12dB or do you do something more
sophisticated. Once you have a diffraction code that can
do focused transducers, and have a definition if focal
size, and decide what acceptable focus is, then off
course it's easy.

Hope That helps.

: Can you tell me of a good textbook or paper that develops the equations for depth of field of focused transducers?

W. Grandia
W. Grandia
00:05 Sep-05-1996
Re: Depth of Field


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