Calculating Waves
Most arithmetic involved in waves concerns itself with 3 main properties.
Wavelength, represented by λ (lambda)
Speed, represented by c
Frequency, represented by f
Any property can be worked out by knowing the values for the two other properties via the formula:
λ = c / f
This means that you can calculate the speed of the wave by multiplying the wavelength with the frequency, however this leads to the incorrect assumption that altering the frequency affects the speed of the wave.
Assume a 1KHz sound wave is travelling through air at 20*C at sea level, which carries sound at 343m/s. The wavelength of a 1KHZ sound wave is 34.2cm, or 0.343m.
Speed = Wavelength * Frequency
Speed = 0.343 * 1000
Speed = 343m/s
Assume the above conditions, but for a 10Hz wave. The wavelength of a 10Hz sound wave is 34.3m
Speed = 34.3 * 10
Speed = 343m/s
As you can not change the frequency without also changing the wavelength, you can not alter the speed of a wave by changing its frequency.
The speed of sound itself is dependent on the material through which it is travelling. Here is a table of figures, taken from http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html
| Medium | Velocity | |
| (m/s) | (ft/s) | |
| Aluminum | 6420 | 21063 |
| Brass | 3475 | 11400 |
| Brick | 4176 | 13700 |
| Concrete | 3200 - 3600 | 10500 - 11800 |
| Copper | 3901 | 12800 |
| Cork | 366 - 518 | 1200 - 1700 |
| Diamond | 12000 | 39400 |
| Glass | 3962 | 13000 |
| Glass, Pyrex | 5640 | 18500 |
| Gold | 3240 | 10630 |
| Hardwood | 3962 | 13000 |
| Iron | 5130 | 16830 |
| Lead | 1158 | 3800 |
| Lucite | 2680 | 8790 |
| Rubber | 40 - 150 | 130 - 492 |
| Steel | 6100 | 20000 |
| Water | 1433 | 4700 |
| Wood (hard) | 3960 | 13000 |
| Wood | 3300 - 3600 | 10820 - 11810 |
Mathematically, what effect does this have on the wavelength on a given frequency? Let's try the 1KHz sound wave again, this time through a typical hard wood.
Wavelength = 3960 / 1000
Wavelength = 3.96m
For the same frequency then, the wavelength is longer if the medium through which the sound is travelling propagates sound more quickly. The inverse is also true, if a wave is travelling through a medium that propagates a wave more slowly the wavelength for the same frequency is shorter.
The following is a link to an experiment which supports this notion: http://www.ap.smu.ca/demonstrations/index.php?option=com_content&view=article&id=147&Itemid=85
Interference
When two waves propagating through the same medium pass through each other, interference will occur. The effect of the interference depends on which parts of the phase of each wave are colliding in a given area. All following cases assume the waves are of the same frequency and amplitude.
 |
| The blue wave is two sound waves of the precise same amplitude, wavelength and frequency occupying the same space and in the same phase. The green wave is the resultant wave from the interference. |
If one wave is meeting another halfway through its phase, then the result is the waves effectively cancel each other out.
This is because the resultant wave is the sum of the amplitudes of the waves that are superimposing on each other during interference. The interference is said to be constructive if the amplitude of the resultant wave is higher than that of the interference waves. The interference is said to be destructive if the resultant wave's amplitude is lower.
As the type of interference is dependant on the particular phases the waves happen to be meeting on, take a scenario where you have two speakers emitting the same sound as each other. If you walk away from the speakers you will notice the sound get louder as you are in an area where the peaks of the waves meet and get quieter as you move into an area where peaks are meeting valleys. If you continue to move, you'll enter another area where the phases of the waves line up and again the sound will be louder. I confirmed the effect for myself by moving around the lab while a 1KHz tone was being played through two speakers.
One bizarre effect of this phenomenon is that if you are in an area where phases are out of line with each other and thusly can't hear anything, covering a speaker will remove the interference and actually cause the sound to get louder.
Amplitude and Loudness
Amplitude, typically the vertical scale when illustrating a longitudinal wave, is the magnitude of change in a particle's position, at least when talking about sound. It can be measured as either peak amplitude (0 to the wave's peak) or peak-to-peak (crest to valley).
A higher amplitude is associated with a louder wave, but this isn't entirely straightforward as this section will explain. Amplitude is the measure of the amount of force applied over an area, it is related to intensity and both are related to the power of a sound.
Amplitude can be measured in Newtons per square meter (N/m2)
Intensity can be measured in Watts per square meter (W/m2)
Power can be measured in Watts
Intensity is directly related to the distance from the source via inverse square law. As a wave spreads from its source its energy is spread out over a wider area, thus its watts per square meter will be lower. A more powerful initial sound will be able to travel further than a less powerful one before becoming inaudible.
Loudness is a matter of perception. Firstly, doubling the power of a wave isn't nearly enough to double the volume given the sheer range of powers that the human ear can detect. The threshold of pain requires a sound that is around 1 billion times more powerful than a whisper. A whisper itself is 1000 times more powerful than the threshold of human hearing.
The below table was extracted from http://www.sengpielaudio.com/TableOfSoundPressureLevels.htm, in order to support the above paragraphs.
| Table of sound levels L (loudness) and corresponding sound pressure and sound intensity |
||||
| Sound Sources (Noise) Examples with distance |
Sound Pressure Level Lp dB SPL |
Sound Pressure p N/m2 = Pa sound field quantity |
Sound Intensity I W/m2 sound energy quantity |
|
| Jet aircraft, 50 m away | 140 | 200 | 100 | |
| Threshold of pain | 130 | 63.2 | 10 | |
| Threshold of discomfort | 120 | 20 | 1 | |
| Chainsaw, 1 m distance | 110 | 6.3 | 0.1 | |
| Disco, 1 m from speaker | 100 | 2 | 0.01 | |
| Diesel truck, 10 m away | 90 | 0.63 | 0.001 | |
| Kerbside of busy road, 5 m | 80 | 0.2 | 0.000 1 | |
| Vacuum cleaner, distance 1 m | 70 | 0.063 | 0.000 01 | |
| Conversational speech, 1 m | 60 | 0.02 | 0.000 001 | |
| Average home | 50 | 0.006 3 | 0.000 000 1 | |
| Quiet library | 40 | 0.002 | 0.000 000 01 | |
| Quiet bedroom at night | 30 | 0.000 63 | 0.000 000 001 | |
| Background in TV studio | 20 | 0.000 2 | 0.000 000 000 1 | |
| Rustling leaves in the distance | 10 | 0.000 063 | 0.000 000 000 01 | |
| Threshold of hearing | 0 | 0.000 02 | 0.000 000 000 001 | |
Dealing with such a wide range of digits isn't easy for the human mind to cope with, so instead of expressing sound levels in sound intensity it's expressed in Decibels (sound pressure level) which is a logarithmic scale. A logarithm is just a way of expressing large numbers by writing how many powers of a given base a number is. For example, the logarithm of 1000 is 3. A power level of 1,000,000,000 can be expressed as 9.
Decibels are a measurement of the ratio of two sounds, it can be worked out with the following formula:
dB = 10 * log10(Power1 / Power2)
A sound of power 1000W / 1W produces:
dB = 10 * log10(1000 / 1)
dB = 10 * log10(1000)
dB = 10 * 3 = 30
If we double the power..
dB = 10 * log10(2000 / 1)
dB = 33.01 rounded
http://www.indiana.edu/~emusic/acoustics/amplitude.htm
So doubling the power raises the sound by 3dB. Likewise a 6dB raise corresponds to 4 times the power. As it happens, 6fB appears to be the amount you need to increase a sound by in order to double its percieved volume (http://www.practicalpc.co.uk/computing/sound/dBeasy.htm).
There are two other factors that affect percieved loudness, however. Time and frequency.
The human ear levels a sound over a 600-1000ms window. Anything after that will not appear to get louder, but sounds shorter than 600ms will appear to be quieter, even though their dB are the same.
The other factor is Frequency. Human ears can, on average, detect between 20-20,000Hz sounds. It does not however hear all these sounds at equal percieved loudness. Rather than sounds suddenly cutting to silence if their frequency was to wander off either end of the scale they appear to fade out.
This can be demonstrated on an equal loudness graph, where the sound pressure level (dB) required to achieve an equal loudness across a range of frequencies is plotted out. The lower the dB, the more sensitive to that frequency the human ear is.
The chart indicates that human ears are most sensitive to frequencies in the 2-4KHz range.
Decibel calculator for power levels: http://www.radio-electronics.com/info/formulae/decibels/dB-decibel-calculator.php
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