Earthquakes and decibels
Tonight's idea
Most scales add. Two metres more is two metres more, wherever you start. Earthquake magnitudes and decibels are different. Each step up means multiply.
A magnitude is read off a seismograph, which records how far the ground moves. Going up one unit of magnitude multiplies that movement, the amplitude, by 10. Two units is ×100, three is ×1000: a magnitude 7 moves the ground a thousand times further than a magnitude 4. The energy released grows faster still, by ×1000 for every two units. One unit is therefore the number that multiplies by itself to make 1000: \(\sqrt{1000} \approx 31.6\).
Sound works the same way. A decibel level compares a sound's intensity (the energy per second reaching your ear) with the quietest sound a person can hear, which is set at 0 dB. Every 10 dB multiplies the intensity by 10. So 20 dB is ×100, 60 dB is a million, and 120 dB is a trillion.
On a multiplying scale, a difference in readings is a ratio in the real thing. Subtract the readings, then count powers of ten.
Watch one: Haiti 2010 and Japan 2011
The 2010 Haiti earthquake measured 7.0. The 2011 earthquake off Japan measured 9.0. The difference is 2 units. Shaking: \(10^2 = 100\) times the ground movement. Energy: two units is ×1000, so the Japanese quake released about a thousand times the energy of the Haitian one.
Decibels run the same way. Heavy traffic at 80 dB and a whisper at 30 dB differ by 50 dB. That is five steps of ×10, so the traffic carries \(10^5 = 100{,}000\) times the intensity. Neither answer needed anything beyond a subtraction and a power of ten.
Your turn
- A magnitude 5 earthquake happens somewhere most days; a magnitude 8 is a catastrophe. How many times bigger is the ground movement in the magnitude 8, and how many times more energy does it release?
- Normal conversation is about 60 dB and a rock concert about 110 dB. (a) How many times more intense is the concert? (b) A friend says "110 is less than double 60, so it is not even twice as loud." What have they assumed, and why is it wrong?
- Magnitudes come with decimals. Compare a magnitude 7.5 with a 7.0. (a) By what factor is the ground movement multiplied? (b) By what factor is the energy? Hint: two half-steps make one whole step, so a half-step is the number that squares to give the whole step.
- (a) An earthquake releases 100 times the energy of a magnitude 6.0. Two units of magnitude multiply the energy by \(1000 = 10^3\), so each unit adds 1.5 to the power of ten. You need the number of units \(d\) with \(1.5d = 2\). Find the magnitude. (b) A headline says "magnitude 8 is twice as powerful as magnitude 4". Give the true factor for shaking and for energy.
Stretch: why doubling is 3 dB
Ten doublings is \(2^{10} = 1024\), almost exactly 1000, which is three steps of ×10, or 30 dB. So one doubling of intensity adds about 3 dB. Use this: (a) a second identical speaker next to the first adds how many dB? (b) How many dB is ×500? (c) How many 90 dB lawnmowers together make 100 dB?
Answers
- Three units. Ground movement \(10^3 = 1000\) times. Energy \(31.6^3\), which is \(1000 \times 31.6 \approx 31{,}600\) times, about thirty thousand magnitude-5 earthquakes at once.
- (a) 50 dB is five steps of ×10: \(10^5 = 100{,}000\) times the intensity. (b) They treated decibels as an adding scale and divided the readings. On a multiplying scale you subtract the readings, and each 10 of the difference is a factor of 10.
- (a) A whole step is ×10, so a half-step is \(\sqrt{10} \approx 3.16\): about three times the shaking. (b) A whole step in energy is ×31.6, so a half-step is \(\sqrt{31.6} \approx 5.6\).
- (a) \(d = 2/1.5 = 4/3\), so the magnitude is \(6 + 1.33 \approx 7.3\). Check: \(1.5 \times \tfrac43 = 2\), and \(10^2 = 100\). (b) Four units: shaking \(10^4 = 10{,}000\) times, energy \(1000 \times 1000 = 10^6\), a million times. Not twice.
Stretch. (a) Two speakers give twice the intensity: about 3 dB more, not twice the decibels. (b) \(500 = 1000 \div 2\): 30 dB minus 3 dB, about 27 dB. (c) 100 dB is 10 dB more than 90 dB, which is ten times the intensity: ten lawnmowers.