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 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. dB. 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.
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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. Enter ENTER 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 30 m 140 Threshold of pain 120 1 Loud indoor rock concert 120 Siren at 30 m 100 Busy street traffic 70 Quiet radio 40 Whisper 20 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.
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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 increases by ten units. By the application of the quotient rule for logarithms, one separates into two logarithms. Once this is done, one must apply the quotient rule once more to the function. This time it is only applied to one part of the function. The can be separated into two other logarithms.
At this point, the 10 log 10 is simplified to -0 via the use of a Ti-82 graphing calculator. ENTER Once the 10 is found, the quotient rule for logarithms is applied once more to make one logarithm out of the two remaining logarithms. Once this is done it is evident that for any value of I that equals b, if I is multiplied by ten, then is 10 plus b. The graph of b as a function of I can be sketched via the use of Graphing Calculator 1. 3 for the Macintosh.
The following graph is the graph for b (I) = where is the constant. The graph has its X-min at 0 and max at 290, while it has its Y-min at 0 and its max at 190. When the intensity of a sound wave is to be found, the same process as used before is carried out. In order to find the intensity of normal conversation on must set beta to its given values, in this case there are three intensities that must be found so three different betas; 6 dB, 12 dB and 18 dB. As always, the constant is set to be. 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 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 b- becomes an exponent of ten. After doing this, one knows that. This process is worked out in its entirety on the next page in order to find the individual values for I.
If one wanted to find how much more intense would the 18 dB sound be compared to the 6 dB sound one would have to divide the value of I for 18 dB by the value of I for 6 dB. times. A dog’s threshold of hearing, or, is. This is different to the threshold of human hearing.
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A dog’s threshold of hearing is 10 times smaller than that of humans. The graph for the function changes when the value for is changed. The graphs of (human hearing’s reception level function) and dog hearing’s reception level function) are different in that the dog’s function is translated 10 units up on the Y-axis from the human’s function. By looking at the following graphs, one can easily notice that the second curve is 10 units up than the first curve.
Human Hearing Dog Hearing The relationship in between the intensity and intensity level described before is very important to know and understand when working with the formula. But since a new threshold was added to the investigation one might ask, does the relationship change? Well, the relationship does not change because the relationship is found with only using constants. Now, if the constants change, then the values of the relationship change. For example with human hearing, the intensity level of a wave with an intensity I of 1 is 130 dB, while with dog hearing it is 140 dB. Now, for a human, when the intensity level of a wave with an intensity I of 10 is 140 dB, and for a dog it is 150 dB.
It has just been proven that even though the has changed, if the intensity level increases by ten units, then the intensity increases ten times. When me and my dog cross a busy street the noise does not seem equally loud to both of us. This because the dog has a higher threshold of hearing than me. This is explained best by the fact that the graph for the intensity level in terms of intensity for dogs is translated ten units up from the one for humans. The vertical translation signifies that for any given intensity, the intensity level is ten dB more for dogs than for humans. Knowing the relationship between intensity I and the intensity level b is very important to understanding how all beings hear.
By understand the relationship, we know that the threshold of something’s hearing is what affects loudness with which it hears the sound. Furthermore, by understanding the relationship we know that the curve for the relationship will always have the same shape, but the only difference will be that for different thresholds, it will be translated along the Y-axis. Throughout this project, it was researched how to use properties of logarithms in order to assist one in finding intensity by knowing intensity level,.
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