Elliot Kim
Sound is a mechanical wave that requires a medium such as air, water, or solid to propagate. The speed of sound is influenced by the characteristics of the medium through which it travels, including the medium's state, density, temperature, and humidity. At high altitudes, the standard equations for the speed of sound become less applicable due to the shorter wavelength of sound waves relative to the mean free path of molecules in a gas. The speed of sound in an ideal gas is proportional to the square root of its absolute temperature, and it increases with the stiffness of the material while decreasing with density. The speed of sound formula calculates the distance travelled per unit time by a sound wave as it moves through an elastic medium, with the SI unit being meters per second. While there is no simple formula for the speed of sound in water, the speed of sound in an ideal gas can be derived using classical mechanics.
| Characteristics | Values |
|---|---|
| Speed of sound equation | speed of sound = the square root of (the coefficient ratio of specific heats x the pressure of the gas / the density of the medium) |
| Variables | L = length, T = time required for a sound wave to travel |
| Medium | Air, water, solids |
| Dependent on | Temperature of the gas, molecular weight, humidity, wavelength of the sound wave, mean free path of molecules in a gas |
| Speed | 1482 m/s (for 20 °C) |
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What You'll Learn
The speed of sound in a medium
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. In simpler terms, the speed of sound is the speed of vibrations. Sound waves in solids comprise compression waves and another type of sound wave called a shear wave, which occurs only in solids.
The speed of sound is also directly proportional to the temperature of the medium. As the temperature rises, the rate of collision among the particles of the medium increases, thus making the transfer of sound energy easier. In general, the more rigid or less compressible the medium, the faster the speed of sound. The speed of sound in air is low because air is easily compressible. Liquids and solids are relatively rigid and very difficult to compress, so the speed of sound in these media is generally greater than in gases.
The speed of sound can change when sound travels from one medium to another, but the frequency usually remains the same. This is similar to the frequency of a wave on a string being equal to the frequency of the force oscillating the string. If the speed of sound changes and the frequency remains the same, then the wavelength must change. That is, because v = f*lambda, the higher the speed of sound, the greater its wavelength for a given frequency.
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The effect of temperature and humidity
The speed of sound is dependent on the medium through which it travels. For example, the speed of sound in solids is greater than that in liquids or gases because molecules in solids are tightly packed and more rigid. The speed of sound in gases, such as air, is influenced by factors such as temperature and humidity.
The speed of sound increases with higher temperatures. In the case of gases, an increase in temperature causes the molecules to move faster, leading to an increase in the speed of sound. This relationship is described by the equation:
> v = 331 m/s * sqrt(1 + T_C / 273 °C) = 331 m/s * sqrt(T_K / 273 K)
Where v is the speed of sound, T_C is the temperature in degrees Celsius, and T_K is the temperature in Kelvin. This equation is valid for air at sea level.
The speed of sound is also affected by humidity, with higher humidity levels leading to slower sound propagation due to the presence of water vapour in the air. The difference in the speed of sound between 0% and 100% humidity at standard pressure and temperature is approximately 1.5 m/s. However, the impact of humidity becomes more significant with increasing temperature. Humidity causes an increase in the speed of sound of about 0.1%-0.6%.
It is worth noting that the speed of sound in Earth's atmosphere varies with altitude, with lower temperatures at higher altitudes leading to a decrease in the speed of sound. This variation in the speed of sound with altitude creates an acoustic shadow, where sound is refracted upward, away from listeners on the ground. However, in the stratosphere above 20 km, the speed of sound increases with height due to an increase in temperature from ozone heating.
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The role of density and pressure
The speed of sound in a fluid medium is used as a relative measure for the speed of an object moving through that fluid. The speed of sound depends on the density of the material through which it is travelling, and this density depends on the temperature. Thus, there is a relationship between temperature and the speed of sound.
In an ideal gas, the speed of sound depends on temperature and composition only, not on pressure or density. This is because these two factors change in lockstep at a given temperature and cancel each other out. However, in non-ideal gas behaviour, there is a slight dependence of sound velocity on gas pressure.
In fluids, the density of the liquid and the compressibility of the gas affect the speed of sound. In gases, adiabatic compressibility is directly related to pressure through the heat capacity ratio (adiabatic index). Pressure and density are inversely related to temperature and molecular weight. In solids, compression waves depend on compressibility and density, but in gases, density contributes to compressibility in a way that makes the speed of sound dependent on temperature, molecular weight, and heat capacity ratio.
The speed of sound is inversely proportional to the density of the medium. This means that the greater the density of a medium, the slower the speed of sound. The speed of sound increases with increasing pressure.
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Newton's early calculations
In the latter half of the 17th century, Sir Isaac Newton published his famous work, "Principia Mathematica" (1687), which included a computation of the speed of sound in air. Newton believed that he had correctly predicted the speed of sound through any medium—solid, liquid, or gas. His formula, now known as the Newton-Laplace equation, was based on the assumption that sound waves are isothermal.
Newton's formula considered the density of the medium and the pressure acting on the sound wave. He believed that by calculating the square root of the pressure divided by the medium's density, he could determine the speed of sound. This formula can be expressed as:
> c = speed of sound
> K = elastic bulk modulus
> p = density of the medium
Newton's work in this field predated most of the development of thermodynamics, and his assumption of isothermal sound waves was later proven incorrect by Laplace in 1816. Laplace demonstrated that sound waves are adiabatic and modified Newton's formula by multiplying the gamma (heat component) by the pressure. This correction brought Newton's calculations closer to the accepted value for the speed of sound in air.
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Ideal gas laws
The speed of sound in an ideal gas depends on its temperature and composition. It has a weak dependence on frequency and pressure in dry air, deviating slightly from ideal behaviour. The speed of sound is calculated as the distance travelled per unit of time by a sound wave as it propagates through an elastic medium.
The speed of sound can be calculated using the ideal gas law, which states that pressure for an ideal gas can be expressed as a simple function of density and a function of molecular structure or the ratio of specific heats. The ideal gas law is expressed as:
> pV = nRT = (m/M)RT
> p = (m/V)(RT/M) = ρ(RT/M)
The speed of sound is then equal to the square root of the derivative of pressure with respect to density:
> v = sqrt(dp/dρ)
The speed of sound in gases is related to the average speed of particles in the gas. It is also influenced by the rigidity or compressibility of the medium, with sound generally travelling faster in less compressible media. The speed of sound in air is relatively low because air is easily compressible.
The speed of sound in an ideal gas can also be determined using the laws of thermodynamics, Newton's laws of motion, and the continuity equation of fluid dynamics.
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Frequently asked questions
The equation for the speed of sound in a medium depends on the square root of the restoring force or the elastic property divided by the inertial property.
The speed of sound is influenced by the medium it travels through, the temperature, and the humidity of the surroundings. The speed of sound is faster in solids compared to liquids and gases due to the tight molecular packing in solids.
The speed of sound progressively loses its range of applicability at high altitudes. The standard equations for the speed of sound are reasonably accurate when the wavelength of the sound wave is significantly longer than the mean free path of molecules in a gas.
There is no simple formula for calculating the speed of sound in water. The equations derived from experimental data are complex and involve higher-order polynomials and numerous coefficients.
