Sound Velocity Dependent on Temperature

In summary, The velocity of sound depends on the temperature of the medium in which it travels. This can be modeled using a simple linear lattice model with nearest neighbor interaction, where the dispersion relation is given by w^2=4c/M * sin^2 (0.5ka) and v=w/k. The force constant (c) is the only variable that depends on temperature in this model. For a crude approximation of sound velocity at low temperatures, c_T = A*c_0*T^{1/2}. For the temperature dependence of electrical conductivity, the Drude model can be used, where sigma = ne^2t/m and t is the time between collisions. At room temperature, resistivity is proportional to temperature and t
  • #1
JohanL
158
0
How does the velocity of sound depend on the temperarure.
If you set up a simple model of a linear lattice with nearest neighbour
interaction only you get the dispersion relation:

w^2=4c/M * sin^2 (0.5ka)

and v=w/k.
where a is the spacing between atoms, c is the force constant, k the wavevector and M the mass of an atom.
Of these only c dependes on temperature?
How do you model, explain, show how c dependes on temperature?

Maybe you know a better way to show how sound velocity dependes on
temperature.

Thank you.
 
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  • #2
Okay, in a dispersive medium, you have something like

[tex]c = \sqrt{\frac{E}{\rho}} [/tex]

What follows is my simplistic guess for a crude model.

At low enough temperatures that you do not undergo a phase transformation, E (Young's modulus) decreases almost like the inverse square of the atomic spacing. Also, the density [itex]\rho [/itex] decreases as the inverse cube of the atomic spacing. The atomic spacing is a nearly linear function of temperature, determined by the thermal expansion coefficient, [itex]\alpha[/itex].

So, my guess is that [itex]c_T = A~`c_0~T^{1/2} [/itex] to a first order approximation, at low temperatures.
 
  • #3
thx for your answer.

I have another question.
How do you with a simple model explain the temperature dependence of the
electrical conductivity.
If you use the Drude model you get for the electrical conductivity

sigma = ne^2t / m

where n is the density of mobile electrons and t is the relaxation time.
t is the time between collisions and must be the only variabel here that depends on temperature. How can you estimate t(T).

Maybe there is a better model that describes the temperature dependence of the electrical conductivity.
 
  • #4
Practically, one way to deal with temperature and conductivity is through resistivity which is 1/sigma. Resistivity is fairly easy to measure. At room temperature, resistivity is proportional to temperature (rho =constant + A*T) The coefficient, A, is often tabulated and its temperature range of validity is given.
Since rho = m/(n*e^2*t), t is inversely proportional to temperature.

One way to look at this is that at room temperature, the ions vibrate and electrons "collide" with these ions. As the temperature increases, the vibrations increase and more collisions occur and the average time (t) between collisions decreases. So t is inversely proportional to temperature. Likewise, the conductivity declines with temperature for Drude metals.

At extremely low temperatures, the vibrations are so small that the mean free path is dominated by impurities or defects in the material and becomes almost constant with temperature.

Note that all of this applies only to substances that follow the Drude model. For semiconductors, n does depend on temperature (~exp(-1/T)). Thus, for semiconductors conductivity is a rapidly increasing function of temperature. Although collisions still occur, n tends to dominate in semiconductors.
 

1. What is sound velocity?

Sound velocity refers to the speed at which sound travels through a medium, such as air, water, or solids.

2. How does temperature affect sound velocity?

As the temperature of a medium increases, the molecules within the medium have more energy and vibrate faster, causing sound waves to travel faster. Conversely, as temperature decreases, molecules have less energy and vibrate slower, resulting in a decrease in sound velocity.

3. What is the relationship between sound velocity and temperature?

The relationship between sound velocity and temperature is described by the Newton-Laplace equation, which states that sound velocity is directly proportional to the square root of temperature.

4. Why is it important to consider sound velocity when studying the ocean or other bodies of water?

Sound velocity plays a crucial role in underwater acoustics, as it affects the propagation of sound through the ocean. Understanding how sound velocity changes with temperature is essential for accurately measuring ocean properties, such as depth and temperature.

5. How is sound velocity dependent on temperature measured?

Sound velocity dependent on temperature can be measured using various techniques, such as using acoustic instruments, laboratory experiments, or theoretical models. These methods involve measuring sound velocity at different temperatures and then using the data to determine the relationship between sound velocity and temperature.

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