ap and λ are both measured in mm. Then
Dividing top and bottom by mm gives
and so the answer will be in mm.
If you want to change units, once again just write the units as if they were numbers. For example, if λ = 0.3 cm and we want to express it in mm instead, we simply replace [cm] with [10 mm] (since 1 cm = 10 mm) and then take the 10 out of the square brackets and multiply by it:
Scientific notation
When dealing with very large and very small numbers, it is useful to use scientific notation. This consists of a number between 1 and 10 multiplied by the appropriate power of 10. Working with this notation has several advantages. It is easier to read (and get right) and separates the task of keeping track of the powers of 10 from the rest of the calculation.
What do we mean when we talk about "powers of 10"?
You are probably familiar with the convention that 102 means (10 ×10) and 103 means (10 × 10 × 10). More generally 10n means 10 multiplied by itself n times, where n could be any number. Another way of talking about this is to refer to 10n as "10 raised to the power n".
As an example:
Here is another example. The following equation allows us to calculate a particular angle (θ):
where λ = 0.3 mm and ap = 1.5 cm. First we will convert λ to cm (using 1 mm = 0.1 cm) to make the units of length consistent:
Notice that the [cm] units on top and bottom of the equation cancel each other out and the answer therefore is a pure number with no units. This is true of functions such as the sine function. Now we will find what angle has this value for its sine by using the inverse sine (sin-1) function on a calculator. This gives us the final answer:
Logarithms
The logarithm is a function related to powers of 10. If
then
Thus the logarithm of a number (in this case the logarithm of x) is the power to which 10 must be raised to get that number. If y is an integer (whole number) this is straightforward:
The idea can also be extended to numbers other than exact powers of 10. For example
Logarithms can be calculated on a scientific calculator using the log function. (Note this is generally indicated by log or log10. It should not be confused with the "natural logarithm" function, written as either ln or loge, which is irrelevant to decibels.)
Logarithms have a number of useful properties which come directly from properties of powers of 10. Thus the logarithm of a product (i.e. two numbers multiplied together) is simply the sum of their logarithms, and a similar relationship exists for the logarithm of a ratio:
Decibels
The decibel (dB) unit of measurement is based on logarithms. It is always used to express the ratio of two quantities. For example, if the ultrasound power transmitted into the patient is P1 and the power at a certain depth is P2 then the attenuation in decibels is calculated as follows:
In diagnostic ultrasound, decibels are used as the units for attenuation, gain and dynamic range.
Table 1.4 Examples of power ratios and their value in decibels.
| P1/P2 | dB |
|---|---|
| 1 | 0 dB |
| 2 | 3 dB |
| 10 | 10 dB |
| 100 | 20 dB |
| 1000 | 30 dB |
| 0.1 | -10 dB |
| 0.01 | -20 dB |
Some of the advantages that come from using decibels are:
multiplication and division of numbers becomes addition and subtraction when the numbers are measured in decibels (since decibels are based on logarithms)
decibels make very large and very small numbers easier to manage
the calculation of attenuation becomes straightforward
the concept of dynamic range would be much more complex if decibels were not used.
Reality checking answers
It is easy to make a mistake when making calculations. It is therefore strongly recommended that you do "reality checks" whenever you can.
One way to do this is to look at your final answer and see if it seems right. You will learn, for example, that typical ultrasound wavelengths are in the range 0.1 - 1.0 mm. Thus, if you are calculating a wavelength and get an answer of 25 cm you will know that something is wrong and the calculation needs to be checked.
You can also check the accuracy of calculations themselves. As mentioned above, the use of scientific notation separates the task of the numerical calculation from that of keeping track of powers of 10. It is usually possible to do an approximate check of the numerical part of the calculation by mental arithmetic to compare with the answer you have calculated using
