Speed, Pressure and Intensity of Sound

Transcript

0:00Introduction

Hello everyone! Let's start with a famous mistake. Newton calculated the speed of sound in air and got about two hundred and eighty metres per second. But the measured speed was about three hundred and thirty two metres per second at zero degrees Celsius. So what did Newton miss? Keep that puzzle in mind. The speed, pressure and intensity of sound come up often in JEE, so let's begin.

0:27Speed of sound: Newton and Laplace

So, here is the speed of sound in a fluid: v equals the square root of capital B over rho. Capital B is the bulk modulus, how hard the medium is to squeeze, and rho, the Greek letter like a p with a tail, is the density. In a thin rod, capital Y, Young's modulus, replaces the bulk modulus. Okay? Now, Newton assumed that the temperature stays the same as sound squeezes the air, so the compressions are isothermal. Then capital B equals p, the gas pressure, giving two hundred and eighty metres per second; the small minus one on the s means per second. Right? But here is the mistake from our puzzle. Sound squeezes and stretches the air very fast. So tell me, does the heat get time to flow out? It does not, that is correct! That was Laplace's correction: the compressions are adiabatic. Then capital B equals gamma times p, where gamma, the Greek letter like a small y, is the ratio of the specific heats, one point four for air. So two hundred and eighty times root one point four gives about three hundred and thirty one metres per second; the wavy equals sign means approximately. That is very close to the measured value, so our puzzle is solved. Clear? So here is Laplace's formula: v equals the root of gamma p over rho, and by the gas law, the root of gamma R capital T over capital M, where R is the gas constant, capital T the temperature in kelvin, and capital M the molar mass. Got it?

2:02What the speed depends on

Now, a classic trap. We double the pressure of the air, but keep the temperature, capital T, fixed. Does the speed change? It does not change at all! At fixed temperature, doubling the pressure doubles the density too, so p over rho stays the same. Many students slip here. With me? But temperature does matter: v is proportional to root capital T; this sign, like a stretched alpha, means proportional to. And moist air has a smaller molar mass, so sound travels faster in it. Okay? And in general, sound is fastest in solids, slower in liquids, and slowest in gases. Right? Now sound passes from air into water, and its speed changes. So what stays the same, the frequency or the wavelength? The frequency f stays the same, yes, exactly! The source decides it. So the wavelength, lambda, the Greek letter like an upside down y, is proportional to v. Clear?

3:06Displacement wave and pressure wave

Now, a sound wave can be described in two ways. First, the displacement: s equals s sub zero, times sine of, omega t minus k x. Here s is how far a layer of air moves, and s sub zero, the s with a small zero, is the displacement amplitude. k is the wave number, and omega, the Greek letter like a rounded w, is the angular frequency. Okay? Second, the pressure. The little triangle is capital delta, so delta p is the change in pressure. Delta p equals p sub zero, times cos of, omega t minus k x, where p sub zero is the pressure amplitude. Right? The two amplitudes are linked: p sub zero equals capital B times k times s sub zero, which is also rho times v times omega times s sub zero. Clear? Now let me ask you something. Where the displacement is zero, is the change in pressure also zero? No, it is just the opposite! Where the sine is zero, the cos is plus or minus one, its largest size. So pressure is a quarter-wave out of step with displacement, and displacement nodes, where the air does not move, are pressure antinodes, where the pressure changes the most. This comes back in organ pipes. Got it?

4:29Intensity and the decibel scale

Next, the intensity, capital I, the energy the sound carries each second through one square metre. Capital I equals p sub zero squared, over two times rho times v. Okay? For loudness we use the level, beta, the Greek letter like a tall b. Beta equals ten times log to the base ten of capital I over capital I sub zero, in decibels, the d B on screen. Capital I sub zero is ten to the power minus twelve watts per square metre; the small minus two means per square metre. Right? Now a quick one. If the intensity is made one hundred times, by how many decibels does the level rise? It rises by twenty decibels, that is correct! Intensity times one hundred adds ten times log of one hundred, which is ten times two, so twenty decibels. The level does not become one hundred times bigger, so do not fall for that. With me? And for a point source, capital I is proportional to one over r squared, where r is the distance from the source. Clear?

5:36Example 1: speed of sound in air at 27 degrees C

Now our first example. Find the speed of sound in air at twenty seven degrees Celsius; the small circle means degrees. Gamma is one point four, and capital M is twenty nine grams per mole. Pause the video and try it first. Done? Okay, let's check. Can we put twenty seven and twenty nine straight into Laplace's formula? No, no, wait, that is the trap! The temperature must be in kelvin, the capital K on screen: twenty seven plus two hundred and seventy three gives three hundred kelvin. And the molar mass must be in kilograms per mole, so twenty nine grams per mole becomes zero point zero two nine. Okay? Now we substitute: v equals the root of, one point four times eight point three one four times three hundred, over zero point zero two nine. Here eight point three one four is R, the gas constant, in joules per mole per kelvin. Right? So v is about three hundred and forty seven metres per second. Remember, kelvin for temperature and kilograms per mole for molar mass, every time. Got it?

6:46Example 2: pressure amplitude of a 1000 Hz sound

Now a harder one. Sound of frequency one thousand hertz travels in air of density one point two nine kilograms per cubic metre, at three hundred and forty metres per second; the small minus three means per cubic metre. Its displacement amplitude is ten to the power minus six metres. Find its pressure amplitude. Pause the video and try it first. Done? Okay, let's check. Which form of p sub zero fits this data? The form with rho, v and omega, that is correct! But careful, omega is not one thousand. Omega is two pi f, so it is two thousand pi radians per second. With me? Now we substitute: p sub zero equals one point two nine, times three hundred and forty, times two thousand pi, times ten to the power minus six. Okay? That gives about two point seven six pascals. So a displacement of one millionth of a metre makes a pressure swing of less than three pascals. Clear?

7:53Quick revision: Laplace formula, pressure wave, decibels

So, quick revision. First, v equals the root of gamma R capital T over capital M; v grows as root capital T, and at fixed temperature it does not depend on pressure. Second, p sub zero equals rho times v times omega times s sub zero, and pressure is a quarter-wave out of step with displacement. Third, capital I equals p sub zero squared, over two times rho times v, beta is ten log of capital I over capital I sub zero, and for a point source, capital I goes as one over r squared. Now try the practice problems on this page. All the best!