After the necessary alterations were made, apparatus appeared as it does below.
Figure 3 ↑
Figure 4 ↑
Safety
The results are to be taken in a laboratory environment; the apparatus of the experiment pose no serious danger. When I was assembling the equipment I found the single retort stand to be a little unstable. I clamped masses to one side of the cross member in an effort to counterbalance the weight of the SPL meter, this worked, the equipment was stable.
Working with SPLs above 70dB for any significant length of time can damage hearing so I will wear protective earplugs for the duration of the experiment.
Theories
Due to the wide range, (e.g. from about N/m at 1000 Hz (threshold of audibility) totimes greater) SPL measurements are made on a logarithmic (decibel) scale.
SPL = RMS x Amplitude
Intensity Level (L) = Sound Pressure Level (SPL)
Sound Pressure Level (SPL) N/m Where (threshold of audibility)
and p = root mean squared (RMS)
(VRMS)2 is proportional to 10^Intensity Level (SPL)
SPL is proportional to log10 (VRMS)
Results
Initial lab time was used to take results of amplitudes between 10mV - 200mV.
Figure 5 ↑
These results can be more easily interpreted with the use of a graph.
Figure 6 ↑
The graph shows the relationship between SPL and amplitude to be liner. It was felt that some of these results were inaccurate; this could be due to the SPL meter. The readings are inaccurate when nearing the end of each setting range.
At this stage the graph is inconclusive and cannot be relied upon to suggest any particular theories. There are insufficient results to be able to judge any sort of relationship. It was decided at this point that to obtain a more accurate understanding of the relationship, a greater range of amplitudes would need to be plotted.
Figure 7 ↑
These results were then plotted.
Figure 8 ↑
This graph shows a greater range of signal amplitudes, a logarithmic curve has been placed over the results and it fits very well, some of the points are not actually on the trend line but shape is very similar.
Intensity level is given by:
L = log10(N/k)
=> L = log10(P/A/k)
=> L = log10(P/(Ak))
since P=IV
=> L = log10[(IV)/(Ak)]
and V=IR
=> L = log10[(V2/R)/(Ak)]
=> L = log10[V2/(RAk)]
raise 10 to each side
=> 10L = 10log10[V^2/(RAk)]
since alogab = b
=> 10L = V2/(RAk)
Since R, A and k are constant, 10L is proportional to V2.
VRMS is given by:
VRMS = V x 0.50.5 [0.50.5 is root of a half]
=> 10L = (VRMS/0.50.5)/(RAk)
=> 10L = [(VRMS)2/0.5]/[RAk]
=> 10L = 2(VRMS)2/(RAk)
Where:
-
N = Intensity (in Wm-2)
- L = Intensity Level (in dB)
-
k = Intensity at threshold of hearing, 1 pWm-2 (1x10^-12 Wm2)
- P = Power (in W)
-
V = Maximum amplitude, aka Sound Pressure Variation (in V)
-
VRMS = Root mean squared amplitude, aka Effective Pressure Variation (in V)
Therefore, 10L should also be proportional to (VRMS)2.
To discern whether the results show this, a graph must be plotted; 10L against root mean squared amplitude2.
Figure 9 ↑
The graph does not show a straight line as I would expect it to if they are proportional. It appears as if there are only one or two points which do not fit the liner pattern however this is not the case. Each time the apparently ‘problem’ result is taken away, the result previous to it takes its place and the graph looks almost exactly the same.
More readings needed to be taken in order to draw conclusions from the results so I decided to vary the distance between the speaker and SPL meter.
Taking the results as I did for the graph (figure 8) took a great deal of time; because the amplitude had to be varied precisely the CRT had to be reconfigured after most readings.
Due to the time constraints imposed by the lab I decided to take fewer readings, I wanted to maintain the range of amplitudes so readings were taken less frequently than before.
The results for the different distances can be seen on the next page followed by the graph of these results (Figures 10 & 11 respectively).
Figure 10 ↑
Figure 11 ↑
This graph shows that the SPL decreases when the distance between the speaker and the sensor increases. More importantly however it shows that the relationship between SPL and amplitude which was obvious that a distance of 3.5cm continues for all other results (although the curve is not always apparent).
The graph also shows that the amount by which the SPL decreases (as the distance increases) is roughly a regular amount. If more results were taken a rule could be established with relation to the source – sensor distance and the SPL at certain amplitudes.
It has been established that the using simplified sound system (1 speaker) there is a relationship between SPL at a given distance and amplitude of the signal.
Is there still the same relationship if there are two sources?
Figure 13 ↑
This graph shows that with two sources, the relationship is not so apparent. In theory there should be a certain about of wave superposition but this cannot be proven with these results.
Due to time constraints it was not possible to complete anymore than two sets of results for a two source system. These results are inconclusive because they are so few and also because in order to have the drivers equidistant from the sensor, the distance between the two had to be greater than previously.
Evaluation
Some of the anomalous results can be explained by considering the operation of the SPL meter, SPL is calculated by setting the dial on the meter to the desired level e.g. ‘50’ then adding to this the meter reading e.g. 50 this method is laborious and time consuming.
As with any meter there are problems with the accuracy when the extremes of the range are reached. When measuring an SPL of 70dB, on the 60 setting it reads +8 and on the 70 setting it reads +2. In the example outlined above I decided to take an average and recorded an SPL of 70dB however estimates such as these are inaccurate and often lead to errors. Inaccuracies in readings at the end of each scale are a possible reason for jumps and steps which seem to be appearing in my results.
The small display on the SPL meter was difficult to read so I connected the output terminals to a voltmeter with a more accurate Vu dial. I now think that this may not have been a good idea. The scale on the meter’s dial was logarithmic however I connected it to a dial with a liner scale, this could account for my unexpected results.
I did not have access to the instruction manual for the SPL meter so I have no way of knowing if the signal from the output terminals was only true if it was read from a logarithmic scale. It would also have been useful for me to know the error rating of the meter, (information I did not have access to) in order to calculate how incorrect some of the readings were.
In hindsight I should have realised the limitations of the SPL meter I was using much earlier in the investigation. If I had, I could have changed the investigation so the SPL could be measured by the computer using one of the ‘LIVE’ boxes. Using the computer could have brought about more accurate results because the scale available is much greater. I decided not to use the computer to measure SPL because I could not make the meter sensitive enough to register the lower amplitudes.
I thought that I would have a great deal of problems with ‘background noise’, (any sound level which could be detected by the SPL meter which was not part of the investigation) however it was not as much of a problem as I had imagined. The general background level in the lab was approximately 55dB for the duration of the investigation. Very loud anomalous readings e.g. doors slamming could easily be ignored and the results recorded after this event.
Conclusion
I have succeeded in what I set out to do; I have taken results and proven that the relationship between SPL and amplitude is non-liner. This said, I have not proven what I had expected to, the formulae I have included previously indicate what the results should be, and they are not. The failure to prove the formulae correct are due to a number of reasons, as outlined above.