The intensity I of a sound wave is measured in watts per metre squared ( ). The lowest intensity that the average human ear can detect, i.e. the threshold of hearing, is denoted by , where . The loudness of sound, i.e. its intensity level , is measured in decibels (dB), where . From this function a specific relationship between and can be drawn that holds true for any increase in intensity. By knowing the value of beta ( ), the value of can be found via manipulation of the logarithmic function, and by knowing the value of beta can by found by just taking the log of and multiplying it by ten. The intensity level of ordinary conversation is 65 dB. In order to find the intensity of normal conversation on must set beta to 65 dB and to . Afterwards, via using the division property of equality one joins like terms. Once the logarithm is alone, one can apply the properties of logarithms and separate the logarithm into two logarithms. The quotient rule for logarithms is applied to this equation, , where and . By the definition of the logarithmic function, if and only if , one knows that in order to cancel out the logarithm one must exponentiate the log to ten. When one does this one must also keep in mind that equality must be kept on both sides of the equation, so the -5.5 becomes the exponent of ten. After doing this, one knows that .
If one wanted to find the intensity of the sound inside and automobile travelling at that has an intensity level of 75 dB one would follow the same procedure mentioned previously to find intensity. In addition to this method one can use a graphing calculator in order to make the finding of I simpler. By following the procedure one can use a Ti-82 graphing calculator to find the intensity I of a sound by knowing the intensity level b. The function used is just all the steps followed above summarized into one function.
By the use of the calculator, one finds the correct response in a quicker fashion than by working the problem out. However, in order to prove that the answer provided by the calculator is correct one should always work out the problem by hand. dB
Source of Sound
Intensity Level (dB)
Intensity ( )
Jet plane at 30m
Threshold of pain
Loud indoor rock concert
Siren at 30m
Busy street traffic
The chart above is provided in order to assist in determining the relationship between Intensity Level (dB) and Intensity ( ). The values denoted by an * are values that were inserted during the investigation for this project. The process to find the values of Intensity with a given Intensity Level was already mentioned, but the process in order to find Intensity Level by knowing the Intensity of a sound was not. In order to find intensity level beta (b) from the equation one just takes the logarithm of and multiplies it by ten. The finding of the intensity level of a quiet radio by knowing its intensity will be the example to test the method.
After carrying out this process and the other to find all the missing values, a relationship between the increase of the values can be discerned. By carefully analyzing the table above, it can is concluded that while Intensity Level increase ten units, Intensity increase 10 times. By using a graphing utility to graph this can be easily noticed. By looking at the change in the graph between points (1,120) and (10,130) it is seen how the X-values (Intensity) increase ten times and how Y-values (Intensity Level) increase 10 units.
Given the formula it is very simple to show that the relationship holds true. In order to show that it is true, one must create two functions, and . is nothing but the original formula, and has the intensity I multiplied by 10. This multiplication by ten is meant to help prove that when I increases by ten times, B...
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